I tried $p$ for $2, 3$ and $5$ and they are not primes for both cases. How can I find all these prime numbers that satisfy those conditions?
[Math] Find all prime numbers p such that both numbers $4p^2+1$ and $6p^2+1$ are prime numbers
number theoryprime numbers
Best Answer
Only $p = 5$ satisfies both $4p^2 + 1$ and $6p^2 + 1$ being prime, giving the primes $101$ and $151$.
For other odd primes either $4p^2 + 1$ or $6p^2 + 1$ can be prime, but not both. Given an odd prime $p \neq 5$, we have $p^2 \equiv 1$ or $9 \bmod 10$. The former gives $4p^2 + 1 \equiv 5 \bmod 10$ which is clearly composite on account of being divisible by $5$, while the latter gives $6p^2 + 1 \equiv 5 \bmod 10$, also a multiple of $5$.