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Lehmer's Totient Problem

Lehmer's totient problem asks if there exist any composite numbers such that , where is the totient function? No such numbers are known. However, any such an would need to be a Carmichael number, since for every element in the integers (mod ), , so and is a Carmichael number.In 1932, Lehmer showed that such an must be odd and squarefree, and that the number of distinct prime factors must satisfy . This was subsequently extended to . The best current result is and , improving the lower bound of Cohen and Hagis (1980) since there are no Carmichael numbers less than having distinct prime factors; Pinch). However, even better results are known in the special cases , in which case (Wall 1980), and , in which case and (Lieuwens 1970).

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