Problem 4

The positive integers

a

and

b

are such that

15a+16b

and

16a-15b

are both squares of positive integers. What is the least possible value of the smaller of these two squares?

Step 2 of 4: Use residues modulo

13

Previous step Next step

In plain words

Use residues modulo $13$

m\equiv n\equiv0\pmod{13}

Detailed analysis

Modulo $13$ , the first identity is $2m^2+3n^2\equiv0$ , so if $n\not\equiv0$ then $m^2/n^2\equiv5\pmod{13}$ . Since $5$ is a quadratic nonresidue modulo $13$ , we must have $n\equiv0$ , and then $m\equiv0$ .