Abstract
34, while the right side is not. Hence P1 > 3. Sincep2+p+1O0 mod 3 wheneverp--1 mod 3, andp2+p+1I1 mod 3 whenever p =-1 mod 3, we deduce that exactly one of P1,. , Pk is congruent to 1 modulo 3; denote this prime by P. Since p2 + p + 1 +4 3P, we conclude that p2 + P + 1 has a prime factor q congruent to -1 modulo 3. Since q divides p2 + P + 1, also q divides P3 1. On the other hand, q and P are relatively prime, so Fermat's Theorem yields pq-1 = 1 mod q. Since 3 does not divide (q 1), the fact that both P3 and pq-1 are congruent to 1 modulo q requires P -1 mod q. Hence p2 + p + 1 3 mod q, which contradicts q being a prime factor of p2 + P + 1. Hence there is no such n.
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