Penholodigital Square Equation

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Theorem

The following equations, which include each digit from $1$ to $9$ inclusive, are the only ones of their kind:

\(\ds 567^2\) \(=\) \(\ds 321 \, 489\)
\(\ds 854^2\) \(=\) \(\ds 729 \, 316\)


Proof

The square of a $2$-digit integer cannot have more than $4$ digits:

$99^2 = 9801$

The square of a $4$-digit integer has at least $7$ digits:

$1000^2 = 1 \, 000 \, 000$

Hence we only need to inspect $3$-digit integers, with a corresponding $6$-digit square.


A lower bound is given by $\ceiling {\sqrt {123 \, 456}} = 352$, where $123 \, 456$ is the smallest $6$-digit integer without repeating digits or $0$.

An upper bound is given by $\floor {\sqrt {876 \, 543}} = 936$, where $876 \, 543$ is the largest $6$-digit integer without repeating digits or $9$.

It is seen that it is not necessary to investigate $6$-digit squares beginning with $9$, because its square root would also begin with $9$.


It is noted that integers ending in $1$, $5$ or $6$ have squares ending in those same digits.

Such numbers can be eliminated from our search, as they will duplicate the appearance of those digits.

We also cannot have $0$ as a digit.


Moreover, observe that an integer $\bmod 9$ and its square $\bmod 9$ have the following pattern:

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}

\hline n \bmod 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline n^2 \bmod 9 & 0 & 1 & 4 & 0 & 7 & 7 & 0 & 4 & 1 \\ \hline \end{array}$

By Congruence of Sum of Digits to Base Less 1, the sum of digits of $n$ and $n^2 \pmod 9$ are congruent to $n$ and $n^2 \pmod 9$ respectively.

We must require their sum to be congruent to $0 \bmod 9$ since the sum of all their digits is:

$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \equiv 0 \pmod 9$

Hence $n \equiv 0$ or $8 \pmod 9$.


There are $52$ numbers left to check:

For $n \equiv 0 \pmod 9$:

\(\ds 369^2\) \(=\) \(\ds 136 \, 161\)
\(\ds 378^2\) \(=\) \(\ds 142 \, 884\)
\(\ds 387^2\) \(=\) \(\ds 149 \, 769\)
\(\ds 423^2\) \(=\) \(\ds 178 \, 929\)
\(\ds 432^2\) \(=\) \(\ds 186 \, 624\)
\(\ds 459^2\) \(=\) \(\ds 210 \, 681\)
\(\ds 468^2\) \(=\) \(\ds 219 \, 024\)
\(\ds 513^2\) \(=\) \(\ds 263 \, 169\)
\(\ds 549^2\) \(=\) \(\ds 301 \, 401\)
\(\ds 567^2\) \(=\) \(\ds 321 \, 489\) is penholodigital
\(\ds 594^2\) \(=\) \(\ds 352 \, 836\)
\(\ds 612^2\) \(=\) \(\ds 374 \, 544\)
\(\ds 639^2\) \(=\) \(\ds 408 \, 321\)
\(\ds 648^2\) \(=\) \(\ds 419 \, 904\)
\(\ds 657^2\) \(=\) \(\ds 431 \, 649\)
\(\ds 684^2\) \(=\) \(\ds 467 \, 856\)
\(\ds 693^2\) \(=\) \(\ds 480 \, 249\)
\(\ds 729^2\) \(=\) \(\ds 531 \, 441\)
\(\ds 738^2\) \(=\) \(\ds 544 \, 644\)
\(\ds 783^2\) \(=\) \(\ds 613 \, 089\)
\(\ds 792^2\) \(=\) \(\ds 627 \, 264\)
\(\ds 819^2\) \(=\) \(\ds 670 \, 761\)
\(\ds 837^2\) \(=\) \(\ds 700 \, 569\)
\(\ds 864^2\) \(=\) \(\ds 746 \, 496\)
\(\ds 873^2\) \(=\) \(\ds 762 \, 129\)
\(\ds 918^2\) \(=\) \(\ds 842 \, 724\)
\(\ds 927^2\) \(=\) \(\ds 859 \, 329\)

For $n \equiv 8 \pmod 9$:

\(\ds 359^2\) \(=\) \(\ds 128 \, 881\)
\(\ds 368^2\) \(=\) \(\ds 135 \, 424\)
\(\ds 413^2\) \(=\) \(\ds 170 \, 569\)
\(\ds 458^2\) \(=\) \(\ds 209 \, 764\)
\(\ds 467^2\) \(=\) \(\ds 218 \, 089\)
\(\ds 512^2\) \(=\) \(\ds 262 \, 144\)
\(\ds 539^2\) \(=\) \(\ds 290 \, 521\)
\(\ds 548^2\) \(=\) \(\ds 300 \, 304\)
\(\ds 584^2\) \(=\) \(\ds 341 \, 056\)
\(\ds 593^2\) \(=\) \(\ds 351 \, 649\)
\(\ds 629^2\) \(=\) \(\ds 395 \, 641\)
\(\ds 638^2\) \(=\) \(\ds 407 \, 044\)
\(\ds 647^2\) \(=\) \(\ds 418 \, 609\)
\(\ds 674^2\) \(=\) \(\ds 454 \, 276\)
\(\ds 683^2\) \(=\) \(\ds 466 \, 489\)
\(\ds 692^2\) \(=\) \(\ds 478 \, 864\)
\(\ds 719^2\) \(=\) \(\ds 516 \, 961\)
\(\ds 728^2\) \(=\) \(\ds 529 \, 984\)
\(\ds 764^2\) \(=\) \(\ds 583 \, 696\)
\(\ds 782^2\) \(=\) \(\ds 611 \, 524\)
\(\ds 827^2\) \(=\) \(\ds 683 \, 929\)
\(\ds 854^2\) \(=\) \(\ds 729 \, 316\) is penholodigital
\(\ds 863^2\) \(=\) \(\ds 744 \, 769\)
\(\ds 872^2\) \(=\) \(\ds 760 \, 384\)
\(\ds 917^2\) \(=\) \(\ds 840 \, 889\)

Hence the result by inspection.

$\blacksquare$


Sources

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