Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)
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Abstract We show that for some even $k\leqslant 3570$ and all $k$ with $442720643463713815200|k$, the equation $\phi (n)=\phi (n+k)$ has infinitely many solutions $n$, where $\phi $ is Euler’s totient function. We also show that for a positive proportion of all $k$, the equation $\sigma (n)=\sigma (n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao, and PolyMath.
1970 ◽
Vol 28
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pp. 478-479
1993 ◽
Vol 51
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pp. 966-967
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1921 ◽
Vol 3
(2supp)
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pp. 182-182
1894 ◽
Vol 37
(949supp)
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pp. 15170-15171
1904 ◽
Vol 58
(1490supp)
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pp. 23878-23880
1915 ◽
Vol 80
(2063supp)
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pp. 44-46
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