منابع مشابه
On a Conjecture of Carmichael
V. L. KLEE, JR. 1 Carmichael [ l ] 2 conjectured that for no integer n can the equation (x)=n ( being Euler's totient) have exactly one solution. To support the conjecture, he showed that each n for which there is a unique solution must satisfy a restriction which implies w>10. In this note we prove the validity of restrictions considerably stronger than those of Carmichael, and raise the...
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Here, we show that if E is a CM elliptic curve with CM field different from Q( √ −1), then the set of n for which the nth Fibonacci number Fn is elliptic Carmichael for E is of asymptotic density zero.
متن کاملOn the Distribution of Carmichael Numbers
Pomerance conjectured that there are x 1− {1+o(1)} log log log x log log x Carmichael numbers up to x. At the time, his data tables up to 25 · 109 appeared to support his conjecture. However, Pinch extended this data and showed that up to 1021, Pomerance's conjecture did not appear well-supported. Thus, we build upon the work of Carl Pomerance and others to formulate an alternative conjecture r...
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Assuming a weak version of a conjecture of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m > 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.
متن کاملCarmichael numbers and pseudoprimes
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: Journal of Number Theory
سال: 1998
ISSN: 0022-314X
DOI: 10.1006/jnth.1998.2227