将一个偶数表为两个素数的二进伪和

Expressing an Even Number as the Dyadic Pseudo-sum of Two Primes

进一步研究了作者最近提出的一个猜想:每个正偶数,均可表示成两个奇素数的二进伪和,且其表法有无穷多种。利用个人计算机,将猜想的验证范围由107扩大至109,并对表法数目的渐近公式进行了探讨。在某些方面,新猜想与著名的孪生素数猜想有一定的相似之处:如,当偶数为2的幂时,其两种表法在给定的素数范围内的数目几乎相同,且都比其他偶数的要小;对同一个偶数,两种表法数目随素数范围增大的速率基本一致。但在另一些方面,新猜想又表现出与孪生素数猜想迥异的特点,如一个偶数表成两个素数二进伪和的表法数目,并不取决于它的奇素因子,而似乎与其最大奇因子或二进制表示有关。

The following conjecture proposed recently by the author is further studied: every positive even number can be expressed as the dyadic pseudosum of two odd primes, and the number of ways of such expression is infinite. Using personal computer, the conjecture has been verified from 107 up to 109. In addition, asymptotic formula for the number of the expressions is also explored. At several aspects, the new conjecture is similar to the wellknown twin prime conjecture. For given range of primes, the number of ways of expressing an even number which is a power of 2 as the dyadic pseudosum of two primes is almost equal to the one of expressing the same even number as the difference between two primes, and less than that of other even numbers. As the primes get larger the number of representations in both conjectures grows at an approximately equal rate. At some other aspects, however, the new conjecture possesses very different properties relative to the twin prime conjecture. For instance, the number of ways of expressing an even number as the dyadic pseudosum of two primes is not dependent with its odd prime divisors, but likely with its maximum odd divisor or its binary representation.