MathLabs

Problem 4

The positive integers a a and b b are such that 15a+16b15a+16b and 16a−15b16a-15b are both squares of positive integers. What is the least possible value of the smaller of these two squares?
Step 1 of 4: Name the square roots
Previous stepNext step
In plain words

Name the square roots

15a+16b=m2,16a−15b=n215a+16b=m^2,\quad16a-15b=n^2
Detailed analysis

Let m,n m,n be positive integers with 15a+16b=m215a+16b=m^2 and 16a−15b=n216a-15b=n^2. Eliminating b b and then a a gives 15m2+16n2=481a15m^2+16n^2=481a and 16m2−15n2=481b16m^2-15n^2=481b , where 481=13⋅37481=13\cdot37.