congruences - Congruences in Number Theory Jonathan L.F....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Congruences in Number Theory Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA squash@math.ufl.edu Webpage http://www.math.ufl.edu/ squash/ 30 April, 2006 (at 22:02 ) Entrance. Let Primes( N ) mean the set of primes that divide N . An arithmetic progression means a set T + G Z of integers, where the gap G is a posint and translation T an integer. Use comb , also, for “arithmetic progression”. A comb C := T + G Z is co-prime if T G . Divisibility Conundra Here is a soln to LeVeque’s # 7 P 63 . Short soln. Fix a co-prime comb T + G Z and posint N . Let F be the maximum factor of N st. F G . Letting Q := N F , then, Primes( Q ) Primes( G ) . 1 : Since F G , the CRT applies to produce an integer x with x G T and x F 1 . 2 : So in order to show that x N , we need show that x Q . FTSOC, suppose p is a prime with p •| x and p •| Q . This latter forces p •| G , by (1). Now LhS(2) forces T |• p . This contradicts that T
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

Ask a homework question - tutors are online