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Abstract. We prove a conjecture of Kenyon and Smillie concerning the nonexistence of acute rational-angled triangles with the lattice property. Mathematics Subject Classification (2000). 58F99, 11N25.

Let h À p be the relative class number of the p-th cyclotomic field. We

We show that there is no set A of integers, such that (P − 1) ֤ A + A ֤ P ∪ (P − 1), where P denotes the set of primes.Let N be a set of integers. Following Wirsing[2], N is called additively decomposable, if there are sets A , B, such that A + B = {a + b|a ∈ A , b ∈ B} = N and both A and B have at least two elements. He showed that if N is probabilistically constructed with P(n ∈ N ) = 1 2 , we have with probability 1 that N is indecomposable, where N is any set which equals N up to finitely many elements. Let P be the set of primes. It is still unknown, whether P is decomposable. For any set A , we will use A (x) to denote the number of elements of A which are Ϲ x. With this notation A. Hofmann and D. Wolke [1] showed that if A is a set such that A + (A + 1) = P , then x log xx 1/2 . In this note we will show that no such A exists. Theorem 1. There is no setEspecially, if we had A + (A + 1) = P , the set A would contradict our theorem.P r o o f. Define A to be the set of residue classes (mod 30), such that A contains infinitely many elements from this class, B be the corresponding set for A + A and P for P. Then by the prime number theorem for arithmetic progressions and our assumption we get

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