Heptagonal Square Number

A number which is simultaneously a heptagonal number 👁 H_n
and square number 👁 S_m
. Such numbers exist when

Completing the square and rearranging gives

Substituting 👁 x=10n-3
and 👁 y=2m
gives the Pell-like quadratic Diophantine equation

which has basic solutions 👁 (x,y)=(7,2)
, (13, 4), and (57, 18). Additional solutions can be obtained from the unit Pell equation, and correspond to integer solutions when 👁 (n,m)=(1,1)
, (6, 9), (49, 77), (961, 1519), ... (OEIS A046195 and A046196), corresponding to the heptagonal square numbers 1, 81, 5929, 2307361, 168662169, 12328771225, ... (OEIS A036354).

See also

Heptagonal Number, Square Number

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References

Sloane, N. J. A. Sequences A036354, A046195, and A046196 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Heptagonal Square Number

Cite this as:

Weisstein, Eric W. "Heptagonal Square Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HeptagonalSquareNumber.html

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