Mathematics – Number Theory
Scientific paper
2012-02-29
Mathematics
Number Theory
21 pages. Add Remark 1.1(b) and Remark 1.2
Scientific paper
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 d=4,6,12 and n=3,4,...., we prove that the least prime p>2n-2 with p=-1 (mod d) is the smallest integer m such that those (2k-1)^d with k=1,...,n are pairwise distinct modulo m. The paper also contains several challenging conjectures on primes. For example, we find a surprising recurrence for primes, namely, for any positive integer n different from 1,2,4,9, the (n+1)-th prime p_{n+1} is just the least positive integer m such that 2s_k^2 (k=1,...,n) are pairwise distinct modulo m where s_k=sum_{j=1}^k(-1)^{k-j}p_j. We also conjecture that for any positive integer m there are consecutive primes p_k,...,p_n not exceeding 2m+2.2*sqrt(m) such that m = p_n-p_{n-1}+...+(-1)^{n-k}p_k.
No associations
LandOfFree
On functions taking only prime values does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On functions taking only prime values, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On functions taking only prime values will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-524990