Problem 1

Prove that there are infinitely many natural numbers

a

such that

n^4+a

is not prime for every natural number

n

Step 4 of 4: Conclude infinitude

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In plain words

The single family $a=4r^4$ already proves the required infinitude.

n^4+a=(n^2+2rn+2r^2)(n^2-2rn+2r^2),\qquad a=4r^4

Detailed analysis

Thus $n^4+a$ is a product of two integers greater than $1$ for every natural $n$ , so it is never prime. Since $r=2,3,4,\ldots$ supplies infinitely many distinct $a$ , the proof is complete.