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Pentagonal Triangular Number


A number which is simultaneously a pentagonal number P_n and triangular number T_m. Such numbers exist when

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

Completing the square gives

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

Substituting x=6n-1 and y=2m+1 gives the Pell-like quadratic Diophantine equation

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

which has solutions (x,y)=(5,3), (19, 11), (71, 41), (265, 153), .... In terms of (n,m), these give (1, 1), (10/3,5), (12, 20), (133/3, 76), (165, 285), ..., of which the whole number solutions are (n,m)=(1,1), (12, 20), (165, 285), (2296, 3976), ... (OEIS A046174 and A046175), corresponding to the pentagonal triangular numbers 1, 210, 40755, 7906276, 1533776805, ... (OEIS A014979).


See also

Pentagonal Number, Pentagonal Square Triangular Number, Triangular Number

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References

Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996.Sloane, N. J. A. Sequences A014979, A046174, and A046175 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Pentagonal Triangular Number

Cite this as:

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

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