Decomposing 40 billion integers by four tetrahedral numbers

Decomposing 40 billion integers by four tetrahedral numbers

Based upon a computer search performed on a massively parallel supercomputer, we found that any integer n less than 40 billion (40B) but greater than 343, 867 can be written as a sum of four or fewer tetrahedral numbers. This result has established a new upper bound for a conjecture compared to an older one, 1B, obtained a year earlier. It also gives more accurate asymptotic forms for partition...

Approximation of Complex Numbers by Cyclotomic Integers

We present a new method of approximating complex numbers by cyclotomic integers in Ze 2i=2 n ] whose coeecients with respect to the basis given by powers of e 2i=2 n are bounded in absolute value by a given integer M. It has been suggested by Cozzens and Finkelstein 5] that such approximations reduce the dynamic range requirements of the discrete Fourier transform. For xed n our algorithm gives...

The First Four Billion Years

Molecular biology by numbers … four

The tetrahedron has four triangular faces and, because the threefold axes of symmetry also pass through the four vertices, it is the dual of itself. We know it best as a description of how the four bonds of the carbon atom are disposed in space. I was once told by a theoretical physicist that the values of the fundamental physical constants need not have been the same during the evolution of th...

Algebraic Numbers and Algebraic Integers

c © W W L Chen, 1984, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...