Abstract

Let S and T be the sets of Pisot and Salem numbers, respectively. We prove that the set mT ∩ T is empty for every positive integer m ⩾ 2 , i.e., that no sum of several Salem numbers is a Salem number. We also obtain a result which implies that the sets mT ∩ S and mS ∩ T are nonempty for every m ⩾ 2 , i.e., that certain Salem numbers can sum to a Pisot number and that certain Pisot numbers can sum to a Salem number. As an explicit example, the Salem number ( 22 + 10 5 + 5 10 5 + 14 + 50 5 + 70 ) / 8 = 10.99925 … is expressed by a sum of two Pisot numbers.

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