Each of p and 8p
2 + 1 is a prime number.
Is 8p2 - 1 a prime number?
- If not, provide an example.
- If so, prove it.
Computer program and spreadsheet aided methodologies are welcome, but an analytical solution is preferred.
*** Adapted from a problem appearing in a Russian Mathematical Olympiad.
True,
8p2 - 1 is a prime number.
Proof
Consider mod 3.
If p is not divisible by 3, then it is 1 or 2 mod 3. In either case, 8p^2 + 1 is 0 mod 3, and therefore not prime.
So the only case where 8p2 + 1 and p are both prime is where p = 3, making 8p2 + 1 = 73. Then 8p2 - 1 = 71, which is also prime.
q.e.d.
Edited on December 10, 2022, 10:33 am
Analytical solution (spoiler)
