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Nonagonal Pentagonal Number


A number which is simultaneously a nonagonal number N_m and pentagonal number P_n and therefore satisfies the Diophantine equation

 1/2m(7m-5)=1/2n(3n-1).
(1)

Completing the square and rearranging gives

 3(14n-5)^2-7(6m-1)^2=68.
(2)

Defining x=14n-5 and y=6m+1 gives the Pell-like equation

 3x^2-7y^2=68.
(3)

This has solutions in (x,y) corresponding to solutions which are integral in m and n of (m,n)=(1,1), (14, 21), (7189, 10981), (165026, 252081), (86968201, 132846121), ... (OEIS A048913 and A048914), giving the nonagonal pentagonal numbers 1, 651, 180868051, 95317119801, 26472137730696901, ... (OEIS A048915).


See also

Nonagonal Number, Pentagonal Number

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References

Sloane, N. J. A. Sequences A048913, A048914, and A048915 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Nonagonal Pentagonal Number

Cite this as:

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

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