If the equation of the title has an integer solution with k ≥ 2, then m > 109.3·10 6 . This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark m > 1010 7 . Here we achieve m > 1010 9 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N , wit...

P. del Amo Sanchez, J. P. Lees, V. Poireau, E. Prencipe, V. Tisserand, J. Garra Tico, E. Grauges, M. Martinelli, A. Palano, M. Pappagallo, G. Eigen, B. Stugu, L. Sun, M. Battaglia, D.N. Brown, B. Hooberman, L. T. Kerth, Yu. G. Kolomensky, G. Lynch, I. L. Osipenkov, T. Tanabe, C.M. Hawkes, A. T. Watson, H. Koch, T. Schroeder, D. J. Asgeirsson, C. Hearty, T. S. Mattison, J. A. McKenna, A. Khan, A...

A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are satisfied [9]. Here we investigate a wider class of groups of fractions, called thin, which are those associated with monoids where minimal common multiples exist, b...

Estimation of admixture fractions has become one of the most commonly used computational tools in population genomics. However, there is remarkably little population genetic theory on their statistical properties. We develop theoretical results that can accurately predict means and variances of admixture proportions within a population using models with recombination and genetic drift. Based on...

This article presents some results about Sturmian words and continued fractions — none of them new — which are needed in my not yet written paper about regular expressions. I have tried to find as direct a route as possible to the important results starting from first principles. The style is somewhat terse, and the reader (if reader there be) will find it useful to work out examples of the def...

14 Cancers arise as a result of genetic and epigenetic alterations. These accumulate in cells during the 15 processes of tissue development, homeostasis and repair. Many tumor types are hierarchically orga16 nized and driven by a sub-population of cells often called cancer stem cells. Cancer stem cells are 17 uniquely capable of recapitulating the tumor and can be highly resistant to radioand c...

Given a continued fraction, we construct a certain function that is discontinuous at every rational number p/q. We call this discontinuity the “gap”. We then try to characterize the gap sizes, and find, to the first order, the size is 1/q2, and that, for higher orders, the gap appears to be perfectly ’randomly’ distributed, in that it is Cauchy-dense on the unit square, and thus, this function ...

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic fl...

For matrix power series with coefficients over a field, the notion of a matrix power series remainder sequence and its corresponding cofactor sequence are introduced and developed. An algor!thm for constructing these sequences is presented. It is shown that the cofactor sequence yields directly a sequence of Padd fractions for a matrix power series represented as a quotient B(z)-lA(z). When B(z...