A number which is simultaneously a heptagonal number and square number . Such numbers exist when
(1)
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Completing the square and rearranging gives
(2)
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Substituting and gives the Pell-like quadratic Diophantine equation
(3)
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which has basic solutions , (13, 4), and (57, 18). Additional solutions can be obtained from the unit Pell equation, and correspond to integer solutions when , (6, 9), (49, 77), (961, 1519), ... (OEIS A046195 and A046196), corresponding to the heptagonal square numbers 1, 81, 5929, 2307361, 168662169, 12328771225, ... (OEIS A036354).