A number which is simultaneously a pentagonal number and a square number . Such numbers exist when
(1)
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Completing the square gives
(2)
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(3)
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(4)
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Substituting and gives the Pell equation
(5)
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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).