Find all pairs of prime numbers p and q that satisfy the equation 3p^q-2q^p-1=19.

Checking integers up to 4000, I found two solutions for (p,q):
(3, 2)
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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

solutions = []
primes = [i for i in range(4000) if isprime(i)]
from itertools import combinations
for co in combinations(primes,2):
    p=co[0]
    q=co[1]
    if 3*p**q - 2*q**(p-1) == 19:
        if [p,q] not in solutions:
            print([p,q])
            solutions.append([p,q])
    if 3*q**p - 2*p**(q-1) == 19:
        if [q,p] not in solutions:
            print([q,p])
            solutions.append([q,p])

Posted by Larry on 2023-07-16 11:53:03