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 3 of 4: Check both factors

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In plain words

Factoring alone is not enough: compositeness requires two nontrivial factors, which the sum-of-squares forms guarantee.

n^2+2rn+2r^2=(n+r)^2+r^2>1,\qquad n^2-2rn+2r^2=(n-r)^2+r^2>1

Detailed analysis

Both factors are sums of squares. Since $r\ge2$ , each contains the positive term $r^2\ge4$ , so both factors are integers strictly greater than $1$ for every natural number $n$ .