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On generalised Carmichael numbers.

Abstract : For arbitrary integers $k\in\mathbb Z$, we investigate the set $C_k$ of the generalised Carmichael number, i.e. the natural numbers $n< \max\{1, 1-k\}$ such that the equation $a^{n+k}\equiv a \mod n$ holds for all $a\in\mathbb N$. We give a characterization of these generalised Carmichael numbers and discuss several special cases. In particular, we prove that $C_1$ is infinite and that $C_k$ is infinite, whenever $1-k>1$ 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 $a^n\equiv a \mod n$ 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.
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L Halbeisen, N Hungerbühler. On generalised Carmichael numbers.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1999, Volume 22 - 1999 (2), pp.8-22. ⟨10.46298/hrj.1999.138⟩. ⟨hal-01109575⟩

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