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    First five positive deficient square pentagonal numbers

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    The m-th number is the number Pm = 1/2 m (3m - 1 ).
    A pentagonal number is a deficient square if Pm = n^2 - 1 for some integer n.
    Find the first five positive deficient square pentagonal numbers.

    The answer should demonstrate in a table with 3 columns how to get the corresponding m, n and Pm for each of the 5 numbers.

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    Solution Preview

    ** Please see the attachment for the complete solution **

    We wish to find the first five positive deficient square pentagonal numbers. This amounts to finding the five smallest positive integer solutions to the Diophantine equation
    (please see the attached file)

    which we may write as:
    (please see the attached file)

    Multiplying both sides by 24, we obtain:
    (please see the attached file)

    Adding 1 to both sides yields:
    (please see the attached file)

    (please see the attached file)

    This is a Pell-like equation, which we know how to solve. Let (please see the attached file) and (please see the attached file). Then we have:
    (please see the attached file)

    By inspection, we see that (please see the attached file) is a solution ...

    Solution Summary

    In this solution we use a modified version of Pell's equation to find the first five deficient square pentagonal numbers.