Parity check:
If p and q are both odd primes, then LHS is even and RHS is odd, so there are no solutions when both p and q are odd.
If q=2, 28p + p = 8 + 43p^3 + 1
43p^3 - 29p + 9 = 0 which has no integer solutions.
If p=2, 14q^2 + 2 = q^3 + 43*8 + 1
q^3 + 43*8 + 1 - 14q^2 - 2 = 0
q^3 - 14q^2 + 343 = 0
343 = 7^3
q=7 is a solution
q^3 - 14q^2 + 343 = (q-7)(q^2 - 7q - 49)
the roots are q = {7, 7Φ, 7/Φ}
(p,q) = (2,7) is the only solution
No programming required
(but of course I did anyway, and only the one solution was found)
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Posted by Larry
on 2024-08-08 09:41:50 |
Analytic solution