On Functions Taking Only Prime Values

For n = 1, 2, 3, . . . define S(n) as the smallest integer m > 1 such that those 2k(k − 1) mod m for k = 1, . . . , n are pairwise distinct; we show that S(n) is the least prime greater than 2n− 2 and hence the value set of the function S(n) is exactly the set of all prime numbers. For every n = 4, 5, . . . , we prove that the least prime p > 3n with p ≡ 1 (mod 3) is just the least positive integer m such that 18k(3k− 1) (k = 1, . . . , n) are pairwise distinct modulo m. For d ∈ {4, 6, 12} and n = 3, 4, . . . , we show that the least prime p > 2n − 1 with p ≡ −1 (mod d) is the smallest integer m such that those (2k − 1)d for k = 1, . . . , n are pairwise distinct modulo m. We also pose several challenging conjectures on primes. For example, we find a surprising recurrence for primes, namely, for every n = 10, 11, . . . the (n + 1)-th prime pn+1 is just the least positive integer m such that 2sk (k = 1, . . . , n) are pairwise distinct modulo m where sk = ∑k j=1(−1)pj . We also conjecture that for any positive integer m there are consecutive primes pk, . . . , pn (k < n) not exceeding 2m + 2.2 √ m such that m = pn − pn−1 + · · ·+ (−1)pk.