PROOF OF A CONJECTURE BY ERDOS AND GRAHAM CONCERNING THE PROBLEM OF FROBENIUS

被引：17

作者：

DIXMIER, J ^{[1
]}

机构：

[1] INST HAUTES ETUD SCI,F-91440 BURES SUR YVETTE,FRANCE

来源：

JOURNAL OF NUMBER THEORY | 1990年 / 34卷 / 02期

关键词：

D O I：

10.1016/0022-314X(90)90150-P

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Let x1 < x2 ... < xb be integers ≥ 1 such that gcd(x1, ..., xb) = 1. Let S be the additive subsemigroup of N generated by x1, ..., xb. Let G(x1, ..., xb) be the greatest element of the (finite) set N|S. Let g(b, a) = sup G(x1, ..., xb), where the upper bound is taken over all systems x1, ..., xb such that 1 ≤ x1 < ... < xb ≤ a, gcd(x1, ..., xb) = 1. According to a conjecture of Erdös and Graham, it is proved that g(b, a) is roughly equal to a2 (b - 1), and the exact value of g(b, a) is computed when b - 1 divides a or a - 1 or a - 2. It is proved that S {frown} ](k - 1)a, ka] contains at least inf(a, kb - k + 1) elements for k = 1, 2, .... © 1990.

引用

页码：198 / 209

页数：12