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Prime power pairs (Posted on 2023-07-16)

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

No Solution Yet Submitted by Danish Ahmed Khan

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| Comment 2 of 7 |

LHS is odd so p is odd.

Say q is odd as well. Then 2q^p-1 + 19=1 mod4 but 3p^q=3 mod4. So any solution requires q=2 and the equation becomes 3p^2-2^p=19.

However 3p^2 < 2^p when p>8 so we only have to check primes 3,5,7.

When p=3, 3*9-8 = 19.

When p=5, 3*25-32=43.

When p=7, 3*49-128=19.

Solutions are (3,2) and (7,2).

Posted by xdog on 2023-07-17 10:02:33

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