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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r=⌊n/3⌋,n=3r2+h,0≤h<6r+3r=\left\lfloor\sqrt{n/3}\right\rfloor,\quad n=3r^2+h,\quad0\le h<6r+3
Detailed analysis

Let r=⌊n/3⌋r=\lfloor\sqrt{n/3}\rfloor and write n=3r2+hn=3r^2+h with 0≤h<6r+30\le h<6r+3. The case n=2n=2 is vacuous; otherwise r≥1r\ge1.