We characterize all nonvanishing multiplicative functions f for which f (p) + f (q) = f (p + q) for all odd primes p, q. As a corollary, a multiplicative function f is the identity function if and only if f (3) = 3 and f (p) + f (q) = f (p + q) for all odd primes p, q. Two questions posed by Claudia A. Spiro in 1992 are answered negatively. Two new conjectures are posed.

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Let G be a finite abelian group of order n and let A ⊆ Z be non-empty.

In 1934, Romanoff proved that there are a positive proportion natural numbers which can be expressed as the sum of a prime and a power of 2. In this paper, a quantitative version of this theorem is given. We show that the proportion is larger than 0.0868 and for a positive proportion of odd integers the number of such representations is between 1 and 16. r

For a set A of nonnegative integers the representation functionsR 2 (A, n), R 3 (A, n) are defined as the number of solutions of the equation n = a + a , a, a ∈ A with a < a , a a , respectively. Let D(0) = 0 and let D(a) denote the number of ones in the binary representation of a. Let A 0 be the set of all nonnegative integers a with even D(a) and A 1 be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R 2 (A, n) = R 2 (N \ A, n) for all n 2N − 1, then R 2 (A, n) = R 2 (N \ A, n) 1 for all n 12N 2 − 10N − 2 except for A = A 0 or A = A 1 ; (b) if R 3 (A, n) = R 3 (N \ A, n) for all n 2N − 1, then R 3 (A, n) = R 3 (N \ A, n) 1 for all n 12N 2 + 2N. Several problems are posed in this paper.

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In this paper we prove that the set of positive odd integers k such that k&2 n has at least three distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to k2 n +1 is also true. Academic Press

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