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    Cayley numbers with arbitrarily many distinct prime factors

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    A positive integer nn is a Cayley number if every vertex-transitive graph of order nn is a Cayley graph. In 1983, Dragan Maru\v{s}i\v{c} posed the problem of determining the Cayley numbers. In this paper we give an infinite set SS of primes such that every finite product of distinct elements from SS is a Cayley number. This answers a 1996 outstanding question of Brendan McKay and Cheryl Praeger, which they "believe to be the key unresolved question" on Cayley numbers. We also show that, for every finite product nn of distinct elements from SS, every transitive group of degree nn contains a semiregular element
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