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On the Number of Partitions of an Integer in the m -bonacci Base

Marcia Edson, Luca Q. Zamboni (2006)

Annales de l’institut Fourier

For each m ≥ 2 , we consider the m -bonacci numbers defined by F k = 2 k for 0 ≤ k ≤ m - 1 and F k = F k - 1 + F k - 2 + ⋯ + F k - m for k ≥ m . When m = 2 , these are the usual Fibonacci numbers. Every positive integer n may be expressed as a sum of distinct m -bonacci numbers in one or more different ways. Let R m ( n ) be the number of partitions of n as a sum of distinct m -bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for R m ( n ) involving sums of binomial coefficients modulo 2 . In addition we show that this formula may be used to determine the number of partitions...

On the number of places of convergence for Newton’s method over number fields

Xander Faber, José Felipe Voloch (2011)

Journal de Théorie des Nombres de Bordeaux

Let f be a polynomial of degree at least 2 with coefficients in a number field K , let x 0 be a sufficiently general element of K , and let α be a root of f . We give precise conditions under which Newton iteration, started at the point x 0 , converges v -adically to the root α for infinitely many places v of K . As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v -adically to any given root of f for infinitely many places v . We also conjecture that...

On the Olson and the Strong Davenport constants

Oscar Ordaz, Andreas Philipp, Irene Santos, Wolfgang A. Schmid (2011)

Journal de Théorie des Nombres de Bordeaux

A subset S of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of S is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, p -groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general...

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