Carmichael numbers | ScienceGate

A base a pseudoprime is an integer n such that1A Carmichael number is a composite integer n such that (1) is true for all a such that (a, n ) = l. It was shown by Carmichael [1] that, if n is a Carmichael number, then n is the product of k(>2) distinct primes P1,P2,P3, … Pk, and Pi-1|n-1(i=1, 2, 3, …, k).

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Carmichael Numbers with a Square Totient

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-001-7 ◽

2009 ◽

Vol 52 (1) ◽

pp. 3-8 ◽

Cited By ~ 3

Author(s):

W. D. Banks

Keyword(s):

Euler Function ◽

Carmichael Numbers

AbstractLet φ denote the Euler function. In this paper, we show that for all large x there are more than x0.33 Carmichael numbers n ⩽ x with the property that φ(n) is a perfect square. We also obtain similar results for higher powers.

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Carmichael numbers and Lucas tests

Finite Fields: Theory, Applications and Algorithms - Contemporary Mathematics ◽

10.1090/conm/225/03221 ◽

1999 ◽

pp. 193-202 ◽

Cited By ~ 3

Author(s):

Siguna M. S. Müller

Keyword(s):

Carmichael Numbers

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The extension of euler-fermat’s theorem and the rationale of carmichael numbers

Austrian Journal of Technical and Natural Sciences ◽

10.20534/ajt-15-9.10-18-20 ◽

2015 ◽

pp. 18-20

Author(s):

V. V. Drushinin ◽

A. A. Lazarev

Keyword(s):

Carmichael Numbers

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INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788712000080 ◽

2012 ◽

Vol 92 (1) ◽

pp. 45-60 ◽

Cited By ~ 7

Author(s):

AARON EKSTROM ◽

CARL POMERANCE ◽

DINESH S. THAKUR

Keyword(s):

Elliptic Curve ◽

Elliptic Curves ◽

Residue Class ◽

Complex Multiplication ◽

Weak Form ◽

Arithmetic Progressions ◽

Carmichael Numbers ◽

The Third ◽

Fermat’S Little Theorem

AbstractIn 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 (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

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