| 董平川,董 浙,姜海益.k-素数和唯一分解*[J].数学年刊A辑,2023,44(2):211~224 |
| k-素数和唯一分解* |
| k-Primes and the Unique Factorization |
| Received:June 13, 2021 Revised:December 08, 2022 |
| DOI：10.16205/j.cnki.cama.2023.0015 |
| 中文关键词: 素数, 唯一k-素因数分解, $k${-}组合条件, 费马定理, 素数定理 |
| 英文关键词:k-Prime, Unique factorization, k-Combination condition, Fermat’s theorem, Prime number theorem |
| 基金项目:本文受到国家自然科学基金 (No.11871423) |
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| 中文摘要: |
| 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) Sk中唯一k-素因数分解成立, (3)Sk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S3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S3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π3(x) , 它表示小于等于x的3-素数个数.由素数定理, 作者得到π3(x)的一个具体公式以及一些近似公式. |
| 英文摘要: |
| 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) Sk has the unique factorization into k-primes; (3) Fermat’s theorem in Sk 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 S3 have the unique factorization into 3-primes? In Section 3, the authors obtain a sufficient and necessary condition for which integers in S3 have the unique factorization into 3-primes. In the end of Section 3, the authors introduce π3(x) which represents the number of 3-primes less than or equal to x. By the prime number theorem, the authors obtain a concrete formula and some approximate formulae for π3(x). |
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