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congruences

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

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Congruences in Number Theory Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 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 G .
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