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)
[This "proof" is not correct, see above responses]
If q = 2, then RHS = 14 and there is no solution.
Therefore q is odd which means RHS is even.
Which means p is even.
(in fact RHS is even whether q is even or odd)
Which means p = 2
The equation becomes:
2(16 + 4 + 10*q) = q(q^2+3)
40 + 20*q = q^3+3q
q^3 - 17q - 40 = 0
(q - 5)(q^2 + 5q + 8) = 0
q = 5
(p,q) = (2,5) is the only solution
[below here is correct]
To confirm, testing primes up to 100,000 finds only one solution.
When (p,q) = (2,5) then both expressions = 140
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primes = [p for p in range(100000) if isprime(p)]
for p in primes:
for q in primes:
if p*(p**4 + p**2 + 10*q) == q*(q**2 + 3):
print(p,q)
Edited on February 18, 2024, 9:25 am
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Posted by Larry
on 2024-02-17 18:34:10 |
Analytic solution with computer verification
