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Pentagonal Square Number

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A pentagonal square number is a number which is simultaneously a pentagonal number P_n and a square number S_m. For positive integers n and m, such numbers exist when

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

Completing the square gives

1/2n(3n-1)=3/2(n^2-1/3n)=3/2(n-1/6)^2-3/(72)=m^2
(2)
3/2(6n-1)^2-3/2=36m^2
(3)
(6n-1)^2-24m^2=1.
(4)

Substituting x=6n-1 and y=2m gives the Pell equation

x^2-6y^2=1,
(5)

which has solutions (x,y)=(5,2), (49, 20), (485, 198), .... In terms of (n,m), these give (1,1), (25/3, 10), (81, 99), (2401/3, 980), (7921, 9701), ..., of which the positive integer solutions are (n,m)=(1,1), (81, 99), (7921, 9701), (776161, 950599), ... (OEIS A046172 and A046173), corresponding to the pentagonal square numbers 1, 9801, 94109401, 903638458801, 8676736387298001, ... (OEIS A036353).

See also

Pentagonal Number, Pentagonal Square Triangular Number, Square 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 A036353, A046172, and A046173 in "The On-Line Encyclopedia of Integer Sequences."

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Pentagonal Square Number

Cite this as:

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

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