ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Garcia, Mariano, Touro College, N. Y. City, USA New type (i, 1) amicable pairs with greatest common factor a power of 2 or a prime times a power of 2 Two natural numbers M and N are amicable if S(M) = S(N) = M + N, where S(M) is the sum of the divisors of M. The amicable pairs are of type (i, 1) if M = E ∗ (p1 ∗ p2 ∗ ⋯ pi) and N = E ∗ q, where E is the greatest common factor (GCF) of M and N, and q and each pj are different primes not dividing E for j = 1, 2, ⋯ i. We use a result of Borho and Hoffmann to obtain several new pairs of type (2, 1) with GCF = a prime times a power of 2. We also list several new pairs of type (3, 1) with GCF = a power of 2. Two new theorems are given concerning numbers of the latter type. Reference: W. Borho and H. Hoffmann, Breeding natural numbers in Abundance, Math. Comp. 46 (1986), 281-293.