perplexus.info :: Numbers : Smallest Prime Settlement

Smallest Prime Settlement (Posted on 2022-08-14)

Determine the minimum value of a prime number p such that there are positive integers x and y satisfying this equation:

         x² + p³ = y⁴

*** As an extra challenge only, solve this puzzle without using a computer program/spreadsheet aided method.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Possible Solution

| Comment 2 of 3 |

p^3 = (y^2-x)(y^2+x)

p's only factors are 1 and p, so y^2-x = 1, or x = (y^2-1)

Then p^3 = (y^2-(y^2-1))(y^2+(y^2-1)) = 2y^2-1, a familiar form.

{1,1}, {23,78} are the smallest integer solutions.

Checking: 23^3=2*78^2-1

Posted by broll on 2022-08-14 12:07:50

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