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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.

I would appreciate your help!

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    $\begingroup$ Hint: If $n^2 = a+dk$, show that $(n+d)^2 = a+dj$ for some $j$. $\endgroup$ – Thomas Andrews Aug 17 '12 at 13:56
  • $\begingroup$ @ThomasAndrews: It worked out! Please, if you do not mind, add your comment as an answer. $\endgroup$ – spohreis Aug 17 '12 at 14:02
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Hint: If $n^2 = a+dk$, show that $(n+d)^2 = a+dj$ for some $j$.

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