On a Lehmer problem concerning Euler's totient function
نویسندگان
چکیده
منابع مشابه
On a Generalisation of a Lehmer Problem
that is, #Uq = φ(q), the Euler function. For n ∈ Uq we use n to denote the modular inverse of n, that is, nn ≡ 1 (mod q), n ∈ Uq. The classical question of D. H. Lehmer (see [9, Problem F12]) about the joint distribution of the parity of n and n has been solved by W. Zhang [19, 20]. Recently this question has been generalised by E. Alkan, F. Stan and A. Zaharescu [1] as follows. Given vector a ...
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Giuga has conjectured that if the sum of the (n− 1)-st powers of the residues modulo n is −1 (mod n), then n is 1 or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if φ(n) divides n−1, with φ being Euler’s function, must it be true that n is 1 or prime. No examples are known, but a composite number with this property must be a Carmichael number. We show that...
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Euler's totient function φ is the function defined on the positive natural numbers N * in the following way: if n ∈ N * , then φ(n) is the cardinal of the set {x ∈ N * : 1 ≤ x ≤ n, (x, n) = 1}, where (x, n) is the pgcd of x and n. Thus φ(1) = 1, φ(2) = 1, φ(3) = 2, φ(4) = 2, and so on. The principle aim of this article is to study certain aspects of the image of the function φ. 1 Elementary pro...
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We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution of the residues of any fixed number of negative and positive powers of integers in prescribed arithmetic progressions. In particular, we improve and generali...
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We study subsets of [1, x] on which the Euler φ-function is monotone (nondecreasing or nonincreasing). For example, we show that for any > 0, every such subset has size < x, once x > x0( ). This confirms a conjecture of the second author.
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ژورنال
عنوان ژورنال: Proceedings of the Japan Academy, Series A, Mathematical Sciences
سال: 2003
ISSN: 0386-2194
DOI: 10.3792/pjaa.79.136