# One question about the sequence $a+0d$, $a+d, a+2d, a+3d,\dots$ where $a,d$ are natural numbers

This is an exercise from a textbook in Portuguese.

Let $a,d$ be natural numbers and consider the sequence $a+0d$, $a+d, a+2d, a+3d,\dots$. Show that there is no square or there exist infinitely many squares.

I am not allowed to use $\gcd$, congruences or the Fundamental theorem of arithmetic, just Euclidean Division.

If there exists one square, then it is either of the form $3k$ or $3k+1$. But it doesn't help me.

Hint: If $n^2 = a+dk$, show that $(n+d)^2 = a+dj$ for some $j$. –  Thomas Andrews Aug 17 '12 at 13:56
Hint: If $n^2 = a+dk$, show that $(n+d)^2 = a+dj$ for some $j$.