Find all primes $\displaystyle p$ such that $\displaystyle 4p^2 + 1$ and $\displaystyle 6p^2 + 1$ are both primes.
Hint: Work mod 10. In other words, think about the final digit of p. Every prime number apart from 2 and 5 ends with 1, 3, 7 or 9. What does that tell you about $\displaystyle 4p^2 + 1$ and $\displaystyle 6p^2 + 1$?
Hint: Work mod 10. In other words, think about the final digit of p. Every prime number apart from 2 and 5 ends with 1, 3, 7 or 9. What does that tell you about $\displaystyle 4p^2 + 1$ and $\displaystyle 6p^2 + 1$?
It tells me that for every prime $\displaystyle p$ apart from 2 and 5, either $\displaystyle 4p^2 + 1$ or $\displaystyle 6p^2 + 1$ is divisible by 5.