The conjecture that Euler's totient has no nontrivial prime congruences

From papers

Let p>2p>2 be prime, and let AA' and BB' be coprime integers. A Ramanujan-type congruence for Euler's totient function cvarphicvarphi would have the form

φ(An+B)0(modp)\varphi(A'n+B')\equiv0\,\,({\rm mod}\,\,{p})

for every n0n\geq0. The totient congruence conjecture. There are no coprime integers A,BA',B' for which this congruence holds for all n0n\geq0. The claim expresses that Euler's totient function has no nontrivial Ramanujan-type congruences of this prime-modulus form; the source gives no resolution.

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Sources & referencesView supporting material

Primary source

William Craig and Mircea Merca, “On Ramanujan-type Congruences for Multiplicative Functions”, arXiv:2112.05649 (2022).

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