Covers of the integers with odd moduli and their applications to the forms ๐ฅ^{๐}-2โฟ and ๐ฅยฒ-๐น_{3๐}/2
WU Ke-jian, ZhiโWei Sun ยท Mathematics of Computation ยท 2009
In this paper we construct a cover { a s ( mod โก n s ) } s = 1 k \{a_{s}(\operatorname {mod} \ n_{s})\}_{s=1}^{k} of Z \mathbb {Z} with odd moduli such that there are distinct primes p 1 , โฆ , p k p_{1},\ldots ,p_{k} dividing 2 n 1 โ 1 , โฆ , 2 n k โ 1 2^{n_{1}}-1,\ldots ,2^{n_{k}}-1 respectively. Using this cover we show that for any positive integer m m divisible by none of 3 , 5 , 7 , 11 , 13 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the m m th powers of whose terms are never of the form 2 n ยฑ p a 2^{n}\pm p^{a}