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Hexagonal Pentagonal Number


A number which is simultaneously pentagonal and hexagonal. Let P_n denote the nth pentagonal number and H_m the mth hexagonal number, then a number which is both pentagonal and hexagonal satisfies the equation P_n=H_m, or

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

Completing the square and rearranging gives

 (6n-1)^2-3(4m-1)^2=-2.
(2)

Therefore, defining

x=6n-1
(3)
y=4m-1
(4)

gives the Pell-like equation

 x^2-3y^2=-2
(5)

The first few solutions are (x,y)=(1,1), (5, 3), (19, 11), (71, 41), (265, 153), (989, 571), .... These give the solutions (n,m)=(1/3,1/2), (1, 1), (10/3, 3), (12, 21/2), (133/3, 77/2), (165, 143), ..., of which the integer solutions are (1, 1), (165, 143), (31977, 27693), (6203341, 5372251), ... (OEIS A046178 and A046179), corresponding to the pentagonal hexagonal numbers 1, 40755, 1533776805, 57722156241751, ... (OEIS A046180).


See also

Hexagonal Number, Pentagonal Number

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References

Sloane, N. J. A. Sequences A046178, A046179, and A046180 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Hexagonal Pentagonal Number

Cite this as:

Weisstein, Eric W. "Hexagonal Pentagonal Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HexagonalPentagonalNumber.html

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