Question : Can we find every set of the five consective terms of an arithmetic sequence such that the four terms of them are squares?

Motivation : It has been known that four consecutive terms in an arithmetic sequence cannot all be squares (see here).

This got me interested in the above question. By using computer, I got the following examples where $CD$ represents the common difference :

$$7^2,{13}^2,{17}^2,409,{23}^2\ (CD=120)$$ $${647}^2,4688329,{2993}^2,{3637}^2,{4183}^2\ (CD=8539440)$$

However, I don't have any good idea. Can anyone help?

  • $\begingroup$ The link you posted analyzes this problem quite well. $\endgroup$ – vadim123 Nov 18 '13 at 15:11
  • $\begingroup$ Oh, I have read only the top part of the link. Thank you very much. I'll study it by myself. $\endgroup$ – mathlove Nov 18 '13 at 15:18

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