# First five positive deficient square pentagonal numbers

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

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which we may write as:

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Multiplying both sides by 24, we obtain:

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Adding 1 to both sides yields:

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whence:

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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:

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(1)

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.