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Let $a,d \in \mathbb{N}$. When will the arithmetic sequence $(a+dn)_n$ contain a square number?
Obviously, we need to have that $a$ is a square $\mod d$. But is this a sufficient condition?

My try:

For example when $a =3$ and $d=11$, for finding a square in the sequence we get the diophantine equation

$$y^2-11x-3=0$$

which has solutions $x=11t^2+10t+2$ or $x=11t^2+12t+3$, $t\in\mathbb{Z}$. But I don't know how to solve the equation for general $a$ and $d$.

ploosu2
  • 8,707

1 Answers1

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It is equivalent. If $m^2\equiv a\pmod d$, then $\exists n$ such that $m^2=a+dn$.

If you want $n>0$, replace $m$ by for example $m+ad$.

Bart Michels
  • 26,355