TOPICS
Search

Pentagonal Square Number


A number which is simultaneously a pentagonal number P_n and a square number S_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 whole number 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

Explore with Wolfram|Alpha

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."

Referenced on Wolfram|Alpha

Pentagonal Square Number

Cite this as:

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

Subject classifications