k-素数和唯一分解  

k-Primes and the Unique Factorization

作　　者：董平川 董浙[2] 姜海益[2] DONG Pingchuan;DONG Zhe;JIANG Haiyi(Department of Mathematics,New York University,NY 10012-1110,USA.;Corresponding author.School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China)

机构地区：[1]纽约大学数学系,美国纽约10012-1110 [2]浙江大学数学科学学院,杭州310058

出　　处：《数学年刊（A辑）》2023年第2期211-224,共14页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.11871423)的资助.

摘　　要：在本文中,作者揭示了唯一k-素因数分解的更深层原因.在第二节中,首先引入Sk中的k-组合条件和费马定理;并证明了下面4论断是等价的:(1)k-组合条件成立,(2)中唯一k-素因数分解成立,(3)S_(k)中费马定理成立,(4)k=1或2.为了更好地理解k-素数,在第三节中作者考察了一类特殊的k-素数,即3-素数.众所周知唯一3-素因数分解一般是不成立的,那么S_(3)中的哪些正整数具有唯一3-素因数分解性质呢?在第三节中,作者得到一个S_(3)中的整数具有唯一3-素因数分解的充要条件.在第三节最后,作者引入π_(3)(x),它表示小于等于x的3-素数个数.由素数定理,作者得到π_(3)(x)的一个具体公式以及一些近似公式.In this paper,the authors try to find out the deeper reasons for the unique factorization into k-primes.In Section 2,the authors introduce the k-combination condition and Fermat's theorem in Sk.One major result of this paper is that the following 4 assertions are equivalent(1)the k-combination condition holds;(2)Sh has the unique factorization into k-primes;(3)Fermat's theorem in Sh is true;(4)k=1 or 2.In order to understand k-primes more precisely,in Section 3 the authors investigate a special case of k-primes,i.e.3-primes.It is well-known that the unique factorization into 3-primes fails in general.However which integers in S_(3)have the unique factorization into 3-primes?In Section 3,the authors obtain a sufficient and necessary condition for which integers in S_(3)have the unique factorization into 3-primes.In the end of Section 3,the authors introduce π_(3)(x) which represents the number of 3-primes less than or equal to c.By the prime number theorem,the authors obtain a concrete formula and some approximate formulae for π_(3)(c).

关 键 词：k-素数 唯一k-素因数分解 k-组合条件 费马定理 素数定理 

分 类 号：O156.1[理学—数学]