MathLabs

Problem 6

Let n be an integer at least 2. Prove that if k2+k+nk^2+k+n is prime for every integer k with 0≤k≤n/30\le k\le\sqrt{n/3}, then it is prime for every integer k with 0≤k≤n−20\le k\le n-2.
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Ns=n+(r+s)(r+s+1)N_s=n+(r+s)(r+s+1)
Detailed analysis

For s=1,…,n−r−2s=1,\ldots,n-r-2, define Ns=n+(r+s)(r+s+1)N_s=n+(r+s)(r+s+1). We prove these values prime inductively; expanding with n=3r2+hn=3r^2+h gives (2r+2s+1)2>Ns(2r+2s+1)^2>N_s.