On a Problem of Additive Number Theory
On a Problem of Additive Number Theory
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DOI：
10.1112/jlms/s1-31.1.67
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发表时间：
1956
影响因子：
1.2
通讯作者：
P. Erdös;W. Fuchs
中科院分区：
文献类型：
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作者：
P. Erdös;W. Fuchs
其中每个A的所有素因子都具有给定的形式。对文献的搜索似乎表明，各种定理已经被证明，但没有一个得到实际证明。欧拉在没有证明的情况下指出，形式为47+2的每一个整数都是形式为47 + 1的两个素数的和。即使是较弱的陈述，即每一个整数的形式4/+2是一个总和的两个整数，其所有的素因子的形式4/+1尚未被证明。鉴于文献中没有任何明确的结果，似乎值得指出的是，一些非常有趣的定理可以用初等方法得到。这是在本文的第一部分中完成的，结果总结在下面的定理1，2和3中。在第二部分中，我们使用Viggo BrunJ的方法来证明一个一般定理，并由此推导出下面的定理4和定理5。
where all the prime factors of each A,are of a given form. A search of the literature seemed to indicate that various theorems had been conjectured but none actually proved.f For example, L. Euler stated without proof that every integer of the form 47+2 is a sum of two primes each of the form 47 + 1. Even the weaker statement that every integer of the form 4/+2 is a sum of two integers which have all their prime factors of the form 4/+1 has not yet been proved. In view of the absence of any definite results in the literature it seems worthwhile to point out that some very interesting theorems can be obtained in an elementary way. This is done in Part I of this paper and the results are summarized in Theorems 1, 2, and 3 below. In Part II we use the method of Viggo BrunJ to prove a general theorem and from this we deduce Theorems 4 and 5 below.