The positive integers a, b are such that 15a + 16b and 16a - 15b areboth squares of positive integers. What is

Question

the least possible value thatcan be taken on by the smaller of these two squares?

Answer 1

If you accept 0 as the least possible value that can be taken on by the smaller of the two squares, the answer you got from gianlino is correct.

However, there exist many sets of values for "a" and "b", for which both 15a + 16b and 16a - 15b have equal non-zero perfect square values. Here are they;

b = 481, a = 31*481
b = 4*481, a = 31*4*481
b = 9*481, a = 31*9*481
b = 16*481, a = 31*16*481 an so on.

In general b = (n^2)*481, a = 31*b, with n = 1, 2, 3, 4, 5....

The smallest possible square in this case is 231361 (=481^2)

Answer 2

I have a problem with both the above answers.
The first, 12 and 9, doesn't satisfy 16a -15b = a square.
16*12-15*9 = 57.
The second one gives a square for both quantities,
but we don't know if it yields the smallest value
that the smaller of the 2 squares can take on.
This looks like a "search and destroy" problem.
I'll try to find some conditions to speed up the
search and some back here.

Answer 3

0 take a = 15* 481 and b= 16* 481.