On a Lehmer problem concerning Euler's totient function

Grytczuk, Aleksander; Wójtowicz, Marek

doi:10.3792/pjaa.79.136

Cited by 7 publications

(7 citation statements)

References 5 publications

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“…Up till now no solution has been found and the lower bound has been pushed much higher. (A good discussion of this problem may be found in [4].) For this reason it is most likely that φ −1 (m) contains no n ≡ 1(mod m), when m + 1 is composite.…”

Section: Factorialsmentioning

confidence: 99%

Some remarks on Euler's totient function

Coleman

2012

Preprint

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The image of Euler's totient function is composed of the number 1 and even numbers. However, many even numbers are not in the image. We consider the problem of finding those even numbers which are in the image and those which are not.If an even number is in the image, then its preimage can have at most half its elements odd. However, it may contain only even numbers. We consider the structure of the preimage of certain numbers in the image of the totient function.Classification: 11A25, 11A41.Euler's totient function φ is the function defined on the positive natural numbers N * in the following way, and so on. The aim of this article is to study certain aspects of the image of the function φ.

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Section: Factorialsmentioning

confidence: 99%

Some remarks on Euler's totient function

Coleman

2012

Preprint

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“…This question remains still open (see, e.g., [3,Problem B37]); and (under the hypothesis L = ∅) the structure of L was studied in a number of papers (see [2,4,12] and the references given therein).…”

Section: Introductionmentioning

confidence: 99%

Appendix to the Note “The structure of the set of numbers with the Lehmer property”

Wójtowicz

Skonieczna

2009

Comptes Rendus. Mathématique

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“…The problem has been studied by many mathematicians (we refer to [1][2][3][4][5] for some results on this topic), and up to now it is known that if any counterexample exists, then it must be bigger than 10 30 , and it must be Carmichael (and hence odd and square-free). We recall that a composite number n is called Carmichael if b n ≡ b (mod n)…”

Section: Introductionmentioning

confidence: 99%

A group-theoretical approach to Lehmer's totient problem

Norman

2021

Preprint

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Lehmer's totient problem asks whether there exists any composite number n such that ϕ(n) | (n − 1), where ϕ is Euler totient function. It is known that if any such n exists, it must be Carmichael and n > 10 30 . In this paper, we develop a new approach to the problem via some recent results in group theory related to a function ψ (the sum of order of elements of a group) and show that if kϕ(n) = n − 1 for some integer k, then k must be ≥ 3, and actually, if 5, 7, 11, 13 |n, k ≥ 4. This implies that any counterexample must be such that n > 10 8171 and ω(n) ≥ 1991.

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On a Lehmer problem concerning Euler's totient function

Cited by 7 publications

References 5 publications

Some remarks on Euler's totient function

Some remarks on Euler's totient function

Appendix to the Note “The structure of the set of numbers with the Lehmer property”

A group-theoretical approach to Lehmer's totient problem

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