Numbers whose Square is Palindromic with Even Number of Digits

Theorem

The sequence of positive integers whose square is a palindromic number with an even number of digits begins:

$836, 798 \, 644, 64 \, 030 \, 648, 83 \, 163 \, 115 \, 486, 6 \, 360 \, 832 \, 925 \, 898, \ldots$

This sequence is A016113 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

\(\ds 836^2\)

\(=\)

\(\ds 698 \, 896\)

$6$ digits

\(\ds 798 \, 644^2\)

\(=\)

\(\ds 637 \, 832 \, 238 \, 736\)

$12$ digits

\(\ds 64 \, 030 \, 648^2\)

\(=\)

\(\ds 4 \, 099 \, 923 \, 883 \, 299 \, 904\)

$16$ digits

\(\ds 83 \, 163 \, 115 \, 486^2\)

\(=\)

\(\ds 6 \, 916 \, 103 \, 777 \, 337 \, 773 \, 016 \, 196\)

$22$ digits

\(\ds 6 \, 360 \, 832 \, 925 \, 898^2\)

\(=\)

\(\ds 40 \, 460 \, 195 \, 511 \, 188 \, 111 \, 559 \, 106 \, 404\)

$26$ digits

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $836$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $698,896$
• 1988: R. Ondrejka: A Palindrome (151) of Palindromic Squares (J. Recr. Math. Vol. 20, no. 1: pp. 68 – 71)
• 1990: C. Ashbacher: More on palindromic squares (J. Recr. Math. Vol. 22, no. 2: pp. 133 – 135)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $836$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $698,896$