منابع مشابه
On Odd Covering Systems with Distinct Moduli
A famous unsolved conjecture of P. Erdős and J. L. Selfridge states that there does not exist a covering system {as(mod ns)}s=1 with the moduli n1, . . . , nk odd, distinct and greater than one. In this paper we show that if such a covering system {as(mod ns)}s=1 exists with n1, . . . , nk all square-free, then the least common multiple of n1, . . . , nk has at least 22 prime divisors.
متن کاملOn Odd Covering Systems with Distinct
Abstract. A famous unsolved conjecture of P. Erdős and J. L. Selfridge states that there does not exist a covering system {as(mod ns)}ks=1 with the moduli n1, . . . , nk odd, distinct and greater than one. In this paper we show that if such a covering system {as(mod ns)}ks=1 exists with n1, . . . , nk all square-free, then the least common multiple of n1, . . . , nk has at least 22 prime divisors.
متن کاملCovers of the Integers with Odd Moduli
In this paper we construct a cover {as(mod ns)}s=1 of Z with odd moduli such that there are distinct primes p1, . . . , pk dividing 2 n1 − 1, . . . , 2k − 1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2 ± p w...
متن کاملCovers of the Integers with Odd Moduli and Their Applications
In this paper we construct a cover {as(mod ns)}s=1 of Z with odd moduli such that there are distinct primes p1, . . . , pk dividing 2 n1 − 1, . . . , 2k − 1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2 ± p w...
متن کاملNew Linear Codes with Covering Radius 2 and Odd Basis
On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R = 2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r = 2k + 1 (the case q = 3, r = 4k + 1 was considered ...
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 2005
ISSN: 0196-8858
DOI: 10.1016/j.aam.2005.01.004