Notes on degenerate numbers

Notes on Algebraic Numbers

This is a summary of my 1994–1995 course on Algebraic Numbers. (Revised and improved on 1993– 1994!) The background assumed is standard elementary number theory—as found in my Level III course—and a little (Abelian) group theory. Corrections and suggestions for improvement are welcome, and will be credited in future editions! I first learned algebraic number theory from Stewart & Tall’s book ([...

Notes on Generalized Fermat Numbers

There are two different definitions of generalized Fermat numbers (GFN), one of which is more general than the other. In [5], Ribenboim defines a generalized Fermat number as a number of the form Fa,n = a n + 1 with a > 2, while Riesel ([6]) further generalizes, defining it to be a number of the form a n + b n . Both definitions generalize the usual Fermat numbers Fn = 2 n + 1. The only known F...

Notes on p-adic numbers

as one can check using induction on l. The usual absolute value function |x| satisfies these conditions with the ordinary triangle inequality (4). If N(x) = 0 when x = 0 and N(x) = 1 when x 6= 0, then N(x) satisfies these conditions with the ultrametric version of the triangle inequality. For each prime number p, the p-adic absolute value of a rational number x is denoted |x|p and defined by |x...

Ramsey numbers of degenerate graphs

A graph is d-degenerate if all its subgraphs have a vertex of degree at most d. We prove that there exists a constant c such that for all natural numbers d and r, every d-degenerate graph G of chromatic number r has Ramsey number at most 2 cr |V (G)|. This solves a conjecture of Burr and Erdős from 1973.

Notes on Degenerate Twisted (h, q)-Tangent Numbers and Polynomials Associated with the p-Adic Integral on Zp