A number which is simultaneously a pentagonal number and triangular number . Such numbers exist when
(1)
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Completing the square gives
(2)
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Substituting and gives the Pell-like quadratic Diophantine equation
(3)
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which has solutions , (19, 11), (71, 41), (265, 153), .... In terms of , these give (1, 1), (10/3,5), (12, 20), (133/3, 76), (165, 285), ..., of which the whole number solutions are , (12, 20), (165, 285), (2296, 3976), ... (OEIS A046174 and A046175), corresponding to the pentagonal triangular numbers 1, 210, 40755, 7906276, 1533776805, ... (OEIS A014979).