MathLabs

Problem 1

Prove that there are infinitely many natural numbers aa such that n4+an^4+a is not prime for every natural number nn.
Step 1 of 4: Find a factorization
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In plain words

The added term is chosen so the quartic becomes a difference of two squares.

n4+4r4=(n2+2r2)2−(2rn)2=(n2+2rn+2r2)(n2−2rn+2r2)n^4+4r^4=(n^2+2r^2)^2-(2rn)^2=(n^2+2rn+2r^2)(n^2-2rn+2r^2)
Detailed analysis

The difference-of-squares identity gives Sophie Germain's factorization n4+4r4=(n2+2rn+2r2)(n2−2rn+2r2)n^4+4r^4=(n^2+2rn+2r^2)(n^2-2rn+2r^2). It works for every pair of integers n,rn,r.