Problem 6

Let n be an integer at least 2. Prove that if

k^2+k+n

is prime for every integer k with

0\le k\le\sqrt{n/3}

, then it is prime for every integer k with

0\le k\le n-2

Step 4 of 5: Step 4

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N_s-(i+2s+1)(2r-i)=n+(r-i-s-1)(r-i-s)

Detailed analysis

Every integer from $r+s+1$ to $2r+2s$ can be written as $2r-i$ . The identity transports a common factor to $N_{s'=r-i-s-1}$ , an earlier value. Induction and the initial prime values rule this out; the exceptional equality would imply $s'^2=s-(n-r-1)<0$ .