Pentagonal Square Number

A number which is simultaneously a pentagonal number  and a square number . Such numbers exist when

(1)

Completing the square gives

(2)

(3)

(4)

Substituting  and  gives the Pell equation

(5)

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

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."