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
This simple approach works.
2c is a difference of squares so 2c=(x+y)(xy).
With prime c, the larger factor on RHS can only equal 2c or c.
The first case implies 2x=2c+1, an impossibility.
In the second case, 2x=c+2 which is impossible for odd c, so c=2 and x=2. Then a^b +2=4 and a=2, b=1, and y=0.

Posted by xdog
on 20180203 07:20:31 