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Heptagonal Square Number


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

 1/2n(5n-3)=m^2.
(1)

Completing the square and rearranging gives

 (10n-3)^2-40m^2=9.
(2)

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

 x^2-10y^2=9,
(3)

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 Web Resource. https://mathworld.wolfram.com/HeptagonalSquareNumber.html

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