L Halbeisen ; N Hungerbühler - On generalised Carmichael numbers.

hrj:138 - Hardy-Ramanujan Journal, January 1, 1999, Volume 22 - 1999 - https://doi.org/10.46298/hrj.1999.138
On generalised Carmichael numbers.Article

Authors: L Halbeisen 1; N Hungerbühler 2

For arbitrary integers kZ, we investigate the set Ck of the generalised Carmichael number, i.e. the natural numbers n<max{1,1k} such that the equation an+kamodn holds for all aN. We give a characterization of these generalised Carmichael numbers and discuss several special cases. In particular, we prove that C1 is infinite and that Ck is infinite, whenever 1k>1 is square-free. We also discuss generalised Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers n which satisfy the equation anamodn only for a=2, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares.


Volume: Volume 22 - 1999
Published on: January 1, 1999
Imported on: March 3, 2015
Keywords: square-free numbers,Fermat congruence,Korselt's criterion,generalised Carmichael numbers,[MATH]Mathematics [math]

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