Find the value of $b$ in the smallest triplet of perfect squares $(a,b,c)$ such that $a+5k=b$ and $b+5k=c$ for positive integers $a,b,c,k.$

I started by defining $a=a_1^2, b=b_1^2, c=c_1^2$ for convenience as they are perfect squares. Then, we know that $$a_1^2=a_1^2, \;(a_1^2+5k)=b_1^2, \;(a_1^2+10k)=c_1^2.$$ I'm not completely sure on how to continue from here, but I assume we have to solve the equaions in the problem and then try to minimize $b$ from there. May I have some help? Thanks in advance.

When I say smallest triplet, I mean a triplet that has its value of $b$ as minimal as possible.

## Best Answer

It depends on what you mean by minimal. The triple $25, 625,1225$ has a smaller $a$, but $49,169,289$ beats it for $b$ and $c$.