By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2 p + 1 divides the Mersenne number M _( p ) . The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents M _( p ) .
展开▼
机译：通过将两种算术运算扩展为有限的自然数集，从整个自然数集中依次删除一些以质数为模的残差类，我们发明了一种对自然数及其集进行递归筛分的方法或算法。该算法以机械方式产生一系列集合，该集合收敛于所有素数 p的集合，以使2 p +1除以梅森数M _（p）。与集合序列相对应的基本序列严格增加。因此，在没有任何估计的情况下，我们已经捕获了足够多的可用结构，这些结构的现有理论使我们能够证明确切的结果：存在无限多个质数指数M _（p）的梅森复合数。
展开▼
