Pandigital Pairs whose Squares are Pandigital

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Theorem

The elements of the following pandigital pairs of integers each have squares which are themselves pandigital:

$\left({35 \, 172, 60 \, 984}\right), \left({57 \, 321, 60 \, 984}\right), \left({58 \, 413, 96 \, 702}\right), \left({59 \, 403, 76 \, 182}\right)$

The sequence of the first elements is A085545 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\ds 35 \, 172^2\) \(=\) \(\ds 1 \, 237 \, 069 \, 584\)
\(\ds 60 \, 984^2\) \(=\) \(\ds 3 \, 719 \, 048 \, 256\)


\(\ds 57 \, 321^2\) \(=\) \(\ds 3 \, 285 \, 697 \, 041\)
\(\ds 60 \, 984^2\) \(=\) \(\ds 3 \, 719 \, 048 \, 256\)


\(\ds 58 \, 413^2\) \(=\) \(\ds 3 \, 412 \, 078 \, 569\)
\(\ds 96 \, 702^2\) \(=\) \(\ds 9 \, 351 \, 276 \, 804\)


\(\ds 59 \, 403^2\) \(=\) \(\ds 3 \, 528 \, 716 \, 409\)
\(\ds 76 \, 182^2\) \(=\) \(\ds 5 \, 803 \, 697 \, 124\)


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In particular: It remains to be shown there are no further such pairs.
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Sources

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