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Prime to perfect square (Posted on 2024-03-04)

Find all prime numbers p for which 71p + 1 is a perfect square.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

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Let x^2 be the square. Then 71*p + 1 = x^2.

Rearrange and factor into 71*p = (x-1)*(x+1).

71 and p are prime factors, so then x+1 and x-1 are 71 and p in some order.

x+1=71 and x-1=p implies p=69, but 69 is composite.

x-1=71 and x+1=p implies p=73, and 73 is prime.

p=73 is the only solution.

Posted by Brian Smith on 2025-03-02 19:24:52

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