Pandigital Pairs whose Squares are Pandigital

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

1976: Martin Gardner: The Incredible Dr. Matrix
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $57,321$