Each of A and B is a positive integer.

Find the relationship between A and B such that each of the expressions:
x² + A*x + B and x² + A*x + B + 2 is resolvable into factors of the form (x+p)(x+q), for positive integers p and q.

Each quadratic must have a discriminant that is a perfect square:
a^2-4b and a^2-4b-8
The only perfect squares that differ by 8 are 1 and 9.
So a^2 - 4b=9
Since 4b is even a^2 and thus a must be odd.
So for any odd a
b=(a^2 - 9)/4
[odd squares are congruent to 1 mod 8 so the division will yield an integer.  also a must be at least 5 for b to be positive]

There's probably more to the story but that will suffice for now.  I may play with this more tomorrow.

Posted by Jer on 2015-10-21 20:22:38