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!

  • 4
    $\begingroup$ 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

Hint: If $n^2 = a+dk$, show that $(n+d)^2 = a+dj$ for some $j$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.