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

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

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

for every n≥0n\geq0. The totient congruence conjecture. There are no coprime integers A′,B′A',B' for which this congruence holds for all n≥0n\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.

References

Primary source

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

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