perplexus.info :: Numbers : Prime Pairs satisfy Quintic = Cubic

Prime Pairs satisfy Quintic = Cubic (Posted on 2024-02-17)

Given that each of p and q is a prime number:

Determine all possible pairs (p,q) that satisfy this equation:

        p(p⁴+p²+10q) = q(q²+3)

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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p=2

| Comment 4 of 7 |

q=2 yields no solution.

Odd q makes RHS divisible by 4.

For odd p, p^2+1 and 10q can only be divisible by 2 so p^2(p^2+1) + 10q can only be divisible by 2.

Therefore, only p=2 is a possible (and actual, as shown) solution.

****Sorry. The above is pretty much nonsense.****

Edited on February 18, 2024, 4:27 pm

Posted by xdog on 2024-02-17 21:50:59

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