Fermat Set is Diophantine Quadruple

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Theorem

The Fermat set $F = \left\{{1, 3, 8, 120}\right\}$ is a Diophantine quadruple:

$\forall a, b \in F: a \ne b: a b + 1 = n^2$

for some $n \in \Z$.


Proof

\(\ds 1 \times 3 + 1\) \(=\) \(\ds 4\)
\(\ds \) \(=\) \(\ds 2^2\)


\(\ds 1 \times 8 + 1\) \(=\) \(\ds 9\)
\(\ds \) \(=\) \(\ds 3^2\)


\(\ds 1 \times 120 + 1\) \(=\) \(\ds 121\)
\(\ds \) \(=\) \(\ds 11^2\)


\(\ds 3 \times 8 + 1\) \(=\) \(\ds 25\)
\(\ds \) \(=\) \(\ds 5^2\)


\(\ds 3 \times 120 + 1\) \(=\) \(\ds 361\)
\(\ds \) \(=\) \(\ds 19^2\)


\(\ds 8 \times 120 + 1\) \(=\) \(\ds 961\)
\(\ds \) \(=\) \(\ds 31^2\)

$\blacksquare$


Sources