ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Nomura, Yasutoshi, Okayama University of Science, Japan Arithmetic functions associated with arithmetic-geometric mean quotients For natural numbers x_{1}, ..., x_{n} except x_{1} = ... = x_{n}, we denote the arithmetic-geometric mean quotient of the n-th powers of them by Q(x_{1}, .., x_{n}). Given positive integers n and x greater than 2, we define m(n) to be the least integralQ(x_{1}, ..., x_{n}) (if such x_{1}, ..., x_{n} do not exist, we assign an infinity to it) and e(x) the least n for which there exits a factorization x = x_{1}...x_{n} with integral Q(x_{1}, ..., x_{n}). Several families with integral Q(x_{1}, ..., x_{n}) and upper bounds of quotients are given. References: Y. Nomura, Mathematics Magazine 68 No. 5 (1995), 399 J. S. Sumner, and K. L. Dove, Mathematics Magazine 69 No.5 (1996), 386-388