Show that there can never be an infinite arithmetic progression whose terms are made of distinct integers all of which are squares. Also, if you can, find the maximum size (number of terms) of such a progression.

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TopNewestIf \(a,b∈N\) then:

\(f(m)=a+mb\)

is an increasing function of m. Moreover, \(f(m+1)−f(m)=b\).

Suppose that \(f(m)=n^2_m\) for a sequence of terms \(n_m∈N\). Necessarily nm is an increasing sequence of integers. We know that the gap between two consecutive squares is given by:

\((n+1)^2−n^2=2n+1.\) For sufficiently large \(N, 2n+1>b \quad \forall \quad n>N\). Now let \(m\) be large enough so that \(n_m>N\). Thus we have

\(f(m+1)−f(m)=(n_{m+1})^2−(n_m)^2≥(n_m+1)^2−(n_m)^2=2n_m+1>b\)

Fermat has proven that there cannot exist such a progression beyond 3 terms and I didn't make head or tails of his proof.Just in case anyone was interested. – Arian Tashakkor · 2 years ago

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– Vishnu C · 2 years ago

Did Fermat prove that it cannot exist with 3 terms as well? Or is there a progression with 3 terms?Log in to reply

– Arian Tashakkor · 2 years ago

Do you want an example of a progress with 3 terms?\(1^2,5^2,7^2\) Here you go.But Fermat proved that such progression cannot extend over 3 terms.Log in to reply

Say \(l^2\) is the middle term and \((l-k)^2\) is the term below it. So now the common diference is \(2kl-k^2\). Say now that \((l+m)^2\) is the next term. So the common difference can now be expressed as \(2lm+m^2\).

Equating the common differences, we get that \(l=\frac{k^2+m^2}{2(k-m)}\). And then I kinda got stuck – Vishnu C · 2 years ago

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– Arian Tashakkor · 2 years ago

@Vishnu C sorry there was a typo.I edited it nowLog in to reply

– Vishnu C · 2 years ago

Nice solution! Exploiting the lack of consistency in the difference between two squares. I thought that there couldn't be more than 2 terms in such a sequence. So I tried to disprove it for 3 terms. Wrong approach.Log in to reply

– Arian Tashakkor · 2 years ago

Thanks!Post some more proof problems while I have to time to solve will ya? xDLog in to reply