Density of Carmichael numbers with three prime factors
نویسندگان
چکیده
منابع مشابه
Carmichael Numbers With Three Prime Factors
A Carmichael number (or absolute pseudo-prime) is a composite positive integer n such that n|an − a for every integer a. It is not difficult to prove that such an integer must be square-free, with at least 3 prime factors. Moreover if the numbers p = 6m + 1, q = 12m + 1 and r = 18m + 1 are all prime, then n = pqr will be a Carmichael number. However it is not currently known whether there are i...
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We get an upper bound of O(x 5=14+o(1)) on the numberof Carmichael numbers x with exactly three prime factors.
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We get an upper bound ofO(x5/14+o(1)) on the number of Carmichael numbers ≤ x with exactly three prime factors.
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Mohammed Ben Alhocain, in an Arab manuscript of the 10th century, stated that the principal object of the theory of rational right triangles is to find a square that when increased or diminished by a certain number, m becomes a square [Dickson LE (1971) History of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In modern language, this object is to find a rational point of infinite ...
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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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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1997
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-97-00857-0