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Prime Crossed Factor Puzzle (Posted on 2023-03-18) Difficulty: 3 of 5
Find all possible pairs (x, y) of prime numbers, with x less than or equal to y, and each of x and y being less than 2023, such that:
• y divides x2 + 8 and,
• x divides y2 + 8
Provide valid reasoning as to why there are no further solutions.

*** Extra credit for an analytical solution.

Note: This puzzle is adapted from a problem appearing in the Zhautykov Olympiad.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts Computer solution without extra credit | Comment 1 of 3
Three solutions found given the constraints:
(2,2)
(3,17)
(89,881)

def isprime(n):
    '''check if integer n is a prime'''
    n = abs(int(n))
    if n < 2:
        return False
    if n == 2:
        return True    
    if not n & 1:
        return False
    for x in range(3, int(n**0.5)+1, 2):
        if n % x == 0:
            return False
    return True

primes = [i for i in range(2024) if isprime(i)]
for x in primes:
    for y in primes:
        if y < x:
            continue
        if (x**2+8)%y == 0 and (y**2+8)%x == 0:
            print('({},{})'.format(x,y))

  Posted by Larry on 2023-03-18 08:23:01
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