A number which is simultaneously pentagonal and hexagonal. Let denote the th pentagonal number and the th hexagonal number, then a number which is both pentagonal and hexagonal satisfies the equation , or
(1)
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Completing the square and rearranging gives
(2)
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Therefore, defining
(3)
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(4)
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gives the Pell-like equation
(5)
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The first few solutions are , (5, 3), (19, 11), (71, 41), (265, 153), (989, 571), .... These give the solutions , (1, 1), (10/3, 3), (12, 21/2), (133/3, 77/2), (165, 143), ..., of which the integer solutions are (1, 1), (165, 143), (31977, 27693), (6203341, 5372251), ... (OEIS A046178 and A046179), corresponding to the pentagonal hexagonal numbers 1, 40755, 1533776805, 57722156241751, ... (OEIS A046180).