论文标题
几乎素数，奇偶校验问题和连续整数II的ERD猜想之间的差距很小
Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers II
论文作者
论文摘要
本文旨在作为论文arxiv的续集：0803.2636这里的四个合着者写了。在本文中，他们证明了Erdős-Mirksy的一种更强的猜想，该猜想指出，有无限的正整数$ x $，因此$ d（x）= d（x+1）$其中$ d（x）$表示$ x $的分区数量。这种猜想首先由希思布朗（Heath-Brown）于1984年证明，但该方法并未揭示此类$ x $的值$ d（x）$的性质。特别是，无法得出结论，$ d（x）= d（x+1）= a $ a $ cobly频繁的$ a $ a $ a $ a $ a $ a。在上一篇论文ARXIV：0803.2636中，作者表明，有很多正整数$ x $，因此$ x $和$ x+1 $都有指数pattert $ \ {2,1,1,1,1 \} $，所以$ d（x）= d（x）= d（x+1）= 24 $。类似的结果以某些班次$ n $，即$ x $和$ x+n $具有相同的固定指数模式而闻名。这是针对$ n $的换班，即使是一对双素素的产物都可以分开或不排除。本文的目的是简单地证明了指数模式的结果，以进行任意转移$ n $。
This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erdős-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper arXiv:0803.2636, the authors showed that there are infinitely many positive integers $x$ such that both $x$ and $x+1$ have exponent pattern $\{2,1,1,1\}$, so $d(x)=d(x+1)=24$. Similar results were known for certain shifts $n$, i.e., $x$ and $x+n$ have the same fixed exponent pattern infinitely often. This was done for shifts $n$ which are either even or not divisible by the product of a pair of twin primes. The goal of this paper is to give simple proofs of results on exponent patterns for an arbitrary shift $n$.
