BdMO National Junior 2020 P11

[Mursalin]

\(n\)-এর সকল সম্ভাব্য মানের যোগফল বের করো যাতে \(n\), \(n^2+10\), \(n^2-2\), \(n^2-8\), \(n^3+6\)-এর সবগুলো মৌলিক সংখ্যা হয়। (হিন্ট: এরকম অন্তত একটি \(n\) রয়েছে)।

Find the sum of all possible \(n\) such that \(n\), \(n^2+10\), \(n^2-2\), \(n^2-8,\) \(n^3+6\) are all prime numbers. (Hint: there is at least one such \(n\).)

[Enthurelxyz]

Let, $n\neq 7$

So, $n \equiv 1,2,3,4,5,6 (mod$ $7)$ $ $n^2 \equiv 1,2,4 (mod$ $7)$.

If $n^2 \equiv 1 (mod$ $7)$ then $n ^2-8\equiv 0 (mod$ $7)$. So, $n^2=115$ but $15$ is not a perfect square.

If $n^2\equiv 2 (mod$ $7)$ then $n^2-2=7 n=3$ $but $3^2-8=1$ which is not a prime.

If $n^2 \equiv 4 (mod$ $7)$ then $n^2+10\equiv 0(mod$ $7)$ $ n^2+10=7$ but it has no real solution.

If $n=7$ then $n^2+10,n^2-2,n^2-8,n^3+6$ are all prime.

So, $n=7$ and answer is $7$.