论文标题
$ ϕ（n）的解决方案= ϕ（n+k）$和$σ（n）=σ（n+k）$
Solutions of $ϕ(n)=ϕ(n+k)$ and $σ(n)=σ(n+k)$
论文作者
论文摘要
我们表明，对于一些$ k \ le 3570 $和全部$ k $，带有$ 442720643463713815200 | k $，方程$ ϕ（n）= ϕ（n+k）$具有无限的许多solutions $ n $，$ n $，$ ϕ $是Euler的正常函数。我们还表明，对于所有$ k $的正比例，方程$σ（n）=σ（n+k）$具有无限的许多解决方案$ n $。证据取决于张，梅纳德，道和polymath的Prime $ k $ tupers的最新进展。
We show that for some $k\le 3570$ and all $k$ with $442720643463713815200|k$, the equation $ϕ(n)=ϕ(n+k)$ has infinitely many solutions $n$, where $ϕ$ is Euler's totient function. We also show that for a positive proportion of all $k$, the equation $σ(n)=σ(n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao and PolyMath.
