On functions taking only prime values

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• 作者：Zhi-Wei Sun ^{zwsun@nju.edu.cn}
• 关键词：primary ; 11A41 ; 11Y11 ; secondary ; 05A10 ; 11A07 ; 11B75 ; 11N13
• 刊名：Journal of Number Theory
• 出版年：2013
• 出版时间：August, 2013
• 年：2013
• 卷：133
• 期：8
• 页码：2794-2812
• 全文大小：272 K

文摘

For define as the smallest integer such that those mod m for are pairwise distinct; we show that is the least prime greater than and hence the value set of the function is exactly the set of all prime numbers. For every ?, we prove that the least prime with is just the least positive integer m such that () are pairwise distinct modulo m. For and ?, we show that the least prime with is the smallest integer m such that those for 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 the -th prime is just the least positive integer m such that () are pairwise distinct modulo m where . We also conjecture that for any positive integer m there are consecutive primes () not exceeding such that .