Infinitude of k-Lehmer numbers which are not Carmichael
نویسندگان
چکیده
منابع مشابه
Infinitude of Elliptic Carmichael Numbers
In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in...
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A Carmichael number is a positive integer N which is composite and the congruence aN ≡ a (mod N) holds for all integers a. The smallest Carmichael number is N = 561 and was found by Carmichael in 1910 in [6]. It is well– known that there are infinitely many Carmichael numbers (see [1]). Here, we let k be any odd positive integer and study the presence of Carmichael numbers in the sequence of ge...
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We now establish a pleasantly simple description of Carmichael numbers, due to Korselt. First, we need the following notion. Let a and p be coprime (usually, p will be prime, but this is not essential). The order of a modulo p, denoted by ordp(a), is the smallest positive integer m such that a ≡ 1 mod p. Recall [NT4.5]: If ordp(a) = m and r is any integer such that a ≡ 1 mod p, then r is a mult...
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We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indica...
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2016
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042116501153