Find all integers a, b, c where c is a prime number such that
a^b + c and
a^b  c are square numbers.
Source: Pedro Henrique O. Pantoja, Brazil
The difference between the two squares would be 2*c, and therefore even.
But the difference between two squares equals the sum of their square roots times the difference of their square roots. Since that difference, and therefore the product, is even, and since the sum and difference of the square roots have the same parity, both factors of the product must be even, so the difference of two squares is always a multiple of 4. When you divide it by two to get c, c must then also be even. The only even prime is 2, but the difference between two squares is never 2*2=4; therefore no values meet the criterion.

Posted by Charlie
on 20180202 13:18:35 