p-Integral harmonic sums
p-Integral harmonic sums
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DOI：
10.1016/0012-365x(90)90234-9
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发表时间：
1991-09
期刊：
影响因子：
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通讯作者：
A. Eswarathasan;E. Levine
中科院分区：
文献类型：
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作者：
A. Eswarathasan;E. Levine
对于n> 0，设Σ n k= 1 1 k= a (n) b (n)其中a (n)和b (n)是相对素数正整数。对于素数p，让J (p)={n小于0| p除以a (n)}，其中a(0)根据定义= 0。我们推测J (p)对所有p都是有限的。这里，对于p≤7证明了这个猜想，但对于p= 11仍然没有证明。属于J(7)的最大整数n= 102728。证明了对于p >2, J (p)、{0,p−1,p (p−1),p 2−1}。当J (p)={0, p−1,p (p−1)，p2−1}时，p撇p称为谐波。许多小素数是谐波的。我们推测调和素数的集合是无限的。
For n> 0, let Σ n k= 1 1 k= a (n) b (n) where a (n) and b (n) are relatively prime positive integers. For a prime p, let J (p)={n⩾ 0| p divides a (n)} where a (0)= 0 by definition. It is conjectured that J (p) is finite for all p. Here, the conjecture is proved for p⩽ 7, but remains unproved for p= 11. The largest integer belonging to J (7) is n= 102728. It is shown that for p> 2, J (p)⊃{0, p− 1, p (p− 1), p 2− 1}. When J (p)={0, p− 1, p (p− 1), p 2− 1}, the p rime p is called harmonic. Many small primes are harmonic. It is conjectured that the set of harmonic primes is infinite.