Euler 070 partial replacement of numpy loops.

[quant12345]

Describe your change:

Removed loop for filling 'totients' values. Instead, use: np.arange. The cycle for division without a remainder has also been removed: totients[j] // i. Instead, at each iteration, an array of indexes is created to select from the 'totients' array.
Speeds up calculations by 30%.

tests

"""
Project Euler Problem 70: https://projecteuler.net/problem=70
Euler's Totient function, φ(n) [sometimes called the phi function], is used to
determine the number of positive numbers less than or equal to n which are
relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than
nine and relatively prime to nine, φ(9)=6.
The number 1 is considered to be relatively prime to every positive number, so
φ(1)=1.
Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation
of 79180.
Find the value of n, 1 < n < 10^7, for which φ(n) is a permutation of n and
the ratio n/φ(n) produces a minimum.
-----
This is essentially brute force. Calculate all totients up to 10^7 and
find the minimum ratio of n/φ(n) that way. To minimize the ratio, we want
to minimize n and maximize φ(n) as much as possible, so we can store the
minimum fraction's numerator and denominator and calculate new fractions
with each totient to compare against. To avoid dividing by zero, I opt to
use cross multiplication.
References:
Finding totients
https://en.wikipedia.org/wiki/Euler's_totient_function#Euler's_product_formula
"""
from __future__ import annotations
import numpy as np
import datetime
def get_totients_new(max_one: int) -> list[int]:
"""
Calculates a list of totients from 0 to max_one exclusive, using the
definition of Euler's product formula.
>>> get_totients(5)
[0, 1, 1, 2, 2]
>>> get_totients(10)
[0, 1, 1, 2, 2, 4, 2, 6, 4, 6]
"""
totients = np.arange(max_one)
for i in range(2, max_one):
if totients[i] == i:
x = np.arange(i, max_one, i)#array of indexes to select
totients[x] -= totients[x] // i
return totients.tolist()
def get_totients(max_one: int) -> list[int]:
"""
Calculates a list of totients from 0 to max_one exclusive, using the
definition of Euler's product formula.
>>> get_totients(5)
[0, 1, 1, 2, 2]
>>> get_totients(10)
[0, 1, 1, 2, 2, 4, 2, 6, 4, 6]
"""
totients = [0] * max_one
for i in range(max_one):
totients[i] = i
for i in range(2, max_one):
if totients[i] == i:
for j in range(i, max_one, i):
totients[j] -= totients[j] // i
return totients
def has_same_digits(num1: int, num2: int) -> bool:
"""
Return True if num1 and num2 have the same frequency of every digit, False
otherwise.
>>> has_same_digits(123456789, 987654321)
True
>>> has_same_digits(123, 23)
False
>>> has_same_digits(1234566, 123456)
False
"""
return sorted(str(num1)) == sorted(str(num2))
def solution_new(max_n: int = 10000000) -> int:
"""
Finds the value of n from 1 to max such that n/φ(n) produces a minimum.
>>> solution(100)
21
>>> solution(10000)
4435
"""
min_numerator = 1 # i
min_denominator = 0 # φ(i)
totients = get_totients_new(max_n + 1)
for i in range(2, max_n + 1):
t = totients[i]
if i * min_denominator < min_numerator * t and has_same_digits(i, t):
min_numerator = i
min_denominator = t
return min_numerator
def solution(max_n: int = 10000000) -> int:
"""
Finds the value of n from 1 to max such that n/φ(n) produces a minimum.
>>> solution(100)
21
>>> solution(10000)
4435
"""
min_numerator = 1 # i
min_denominator = 0 # φ(i)
totients = get_totients(max_n + 1)
for i in range(2, max_n + 1):
t = totients[i]
if i * min_denominator < min_numerator * t and has_same_digits(i, t):
min_numerator = i
min_denominator = t
return min_numerator
if __name__ == "__main__":
import doctest
doctest.testmod()
now = datetime.datetime.now()
print(f"{solution() = }")
print('old_time', datetime.datetime.now() - now)
now = datetime.datetime.now()
print(f"{solution_new() = }")
print('new_time', datetime.datetime.now() - now)

Output:

solution() = 8319823
old_time 0:00:16.189952
solution_new() = 8319823
new_time 0:00:10.938119

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
• If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes #ISSUE-NUMBER".

[quant12345]

@cclauss two errors occur:

Error: project_euler/problem_070/sol1.py:31:1: I001 Import block is un-sorted or un-formatted
ERROR project_euler/problem_070/sol1.py - ModuleNotFoundError: No module named 'numpy'

numpy import connected why error? And it is not clear what the first error means?

[cclauss]

@dhruvmanila is the use of numpy no allowed in the solution of Euler projects?

[dhruvmanila]

I think it's fine to use external libraries but usually it's better to implement the solution on our own.

That said, the CI is failing because it doesn't install any of the dependencies from requirements.txt. The reason is to speed up the job and that no external libraries have been used in any of the solutions.

If we were to go the route of installing external dependencies, I would take it case by case and declare a new requirements.txt. This will also involve adding cache to the Project Euler and Validate solutions job because it currently doesn't have it.

[quant12345]

@cclauss@dhruvmanila
I looked through all sol and none of them contain numpy. I think that pure Python is used together with numba and it speeds up the code a lot. But the first start is warming up and it is slow. I made comparisons: the first launch on numba lasted 2 seconds (even with a small array). And if the algorithm is accessed once, then it makes sense in numpy. Also, not all numpy functions work with numba, just like the built-in functions that come out of the Python box.

[cclauss]

Yeah! Let's land this one!!

[quant12345]

Do I understand correctly that my changes have been accepted? If so, then this is my first pull request)

[cclauss]

This is merged into master so this code has been assimilated into the codebase.