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.
Step 5 of 5: Step 5
Previous stepNext step
Ns prime for s=1,…,n−r−2N_s\text{ prime for }s=1,\ldots,n-r-2
Detailed analysis

The lemma proves every NsN_s prime. Together with the assumed prime values for k=0,…,rk=0,\ldots,r, these cover exactly k=r+1,…,n−2k=r+1,\ldots,n-2, completing the proof.