Unify O(sqrt(N))is_prime functions under project_euler

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]:

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]:

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]:

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]:

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]:

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]:

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]:

project_euler/problem_003/sol1.py

@@ -13,9 +13,11 @@
 import math
 
 
-def is_prime(num: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,23 +28,21 @@ def is_prime(num: int) -> bool:
  >>> is_prime(2999)
  True
  >>> is_prime(0)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  >>> is_prime(1)
- Traceback (most recent call last):
- ...
- ValueError: Parameter num must be greater than or equal to two.
+ False
  """
 
- if num <= 1:
- raise ValueError("Parameter num must be greater than or equal to two.")
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- for i in range(3, int(math.sqrt(num)) + 1, 2):
- if num % i == 0:
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol1.py

@@ -15,29 +15,37 @@
 from math import sqrt
 
 
-def is_prime(num: int) -> bool:
- """
- Determines whether the given number is prime or not
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
  >>> is_prime(0)
  False
+ >>> is_prime(1)
+ False
  """
 
- if num == 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif num % 2 == 0:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- else:
- sq = int(sqrt(num)) + 1
-for i in range(3, sq, 2):
- if num % i == 0:
-return False
+
+# All primes number are in format of 6k +/- 1
+ for i in range(5, int(sqrt(number) + 1), 6):
+if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_007/sol2.py

@@ -11,22 +11,39 @@
 References:
  - https://en.wikipedia.org/wiki/Prime_number
 """
+import math
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether the given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- for i in range(2, int(number**0.5) + 1):
- if number % i == 0:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
  return False
  return True
 

project_euler/problem_007/sol3.py

@@ -16,20 +16,37 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Determines whether a given number is prime or not
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
- >>> is_prime(15)
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(29)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator():

project_euler/problem_010/sol1.py

@@ -11,12 +11,14 @@
  - https://en.wikipedia.org/wiki/Prime_number
 """
 
-from math import sqrt
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -26,13 +28,24 @@ def is_prime(n: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if 1 < n < 4:
+ if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- elif n < 2 or not n % 2:
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return not any(not n % i for i in range(3, int(sqrt(n) + 1), 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def solution(n: int = 2000000) -> int:

project_euler/problem_010/sol2.py

@@ -16,8 +16,10 @@
 
 
 def is_prime(number: int) -> bool:
- """
- Returns boolean representing primality of given number num.
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
 
  >>> is_prime(2)
  True
@@ -27,11 +29,24 @@ def is_prime(number: int) -> bool:
  False
  >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
  """
 
- if number % 2 == 0 and number > 2:
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  return False
- return all(number % i for i in range(3, int(math.sqrt(number)) + 1, 2))
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def prime_generator() -> Iterator[int]:

project_euler/problem_027/sol1.py

@@ -23,22 +23,39 @@
 import math
 
 
-def is_prime(k: int) -> bool:
- """
- Determine if a number is prime
- >>> is_prime(10)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+ A number is prime if it has exactly two factors: 1 and itself.
+ Returns boolean representing primality of given number num (i.e., if the
+ result is true, then the number is indeed prime else it is not).
+
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
  False
- >>> is_prime(11)
+ >>> is_prime(2999)
  True
+ >>> is_prime(0)
+ False
+ >>> is_prime(1)
+ False
+ >>> is_prime(-10)
+ False
  """
- if k < 2 or k % 2 == 0:
- return False
-elif k == 2:
+
+if 1 < number < 4:
+ # 2 and 3 are primes
  return True
- else:
- for x in range(3, int(math.sqrt(k) + 1), 2):
- if k % x == 0:
- return False
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
  return True
 
 

project_euler/problem_037/sol1.py

@@ -1,4 +1,7 @@
 """
+Truncatable primes
+Problem 37: https://projecteuler.net/problem=37
+
 The number 3797 has an interesting property. Being prime itself, it is possible
 to continuously remove digits from left to right, and remain prime at each stage:
 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
@@ -11,28 +14,46 @@
 
 from __future__ import annotations
 
-seive = [True] * 1000001
-seive[1] = False
-i = 2
-while i * i <= 1000000:
- if seive[i]:
- for j in range(i * i, 1000001, i):
- seive[j] = False
- i += 1
+import math
 
 
-def is_prime(n: int) -> bool:
- """
- Returns True if n is prime,
- False otherwise, for 1 <= n <= 1000000
- >>> is_prime(87)
+def is_prime(number: int) -> bool:
+ """Checks to see if a number is a prime in O(sqrt(n)).
+
+ A number is prime if it has exactly two factors: 1 and itself.
+
+ >>> is_prime(0)
  False
  >>> is_prime(1)
  False
- >>> is_prime(25363)
+ >>> is_prime(2)
+ True
+ >>> is_prime(3)
+ True
+ >>> is_prime(27)
+ False
+ >>> is_prime(87)
+ False
+ >>> is_prime(563)
+ True
+ >>> is_prime(2999)
+ True
+ >>> is_prime(67483)
  False
  """
- return seive[n]
+
+ if 1 < number < 4:
+ # 2 and 3 are primes
+ return True
+ elif number < 2 or number % 2 == 0 or number % 3 == 0:
+ # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
+ return False
+
+ # All primes number are in format of 6k +/- 1
+ for i in range(5, int(math.sqrt(number) + 1), 6):
+ if number % i == 0 or number % (i + 2) == 0:
+ return False
+ return True
 
 
 def list_truncated_nums(n: int) -> list[int]: