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A prime and a cube (Posted on 2020-07-13)

What prime numbers p exist such that 16p+1 is a perfect cube?

See The Solution Submitted by Brian Smith

Rating: 4.0000 (3 votes)

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solution

| Comment 5 of 11 |

16p + 1 = x^3

x is odd, and since p=2 isn't a solution p is odd as well.

Factoring, 16p = (x-1)(x^2 + x + 1)

(x-1) is even and won't divide odd p so it divides 16.

Thus, (x-1) can only be 2,4,8,or 16.

The only solution is when x=17 giving p=307 which is prime.

Posted by xdog on 2020-07-14 07:33:30

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