My Math Forum Proof of a subset of integer numbers and linear combinations

 Number Theory Number Theory Math Forum

 February 4th, 2010, 04:03 AM #1 Newbie   Joined: Sep 2008 Posts: 5 Thanks: 0 Proof of a subset of integer numbers and linear combinations Assume that the set S is a subset of integer numbers. Assume it has a property that if numbers x and y belong to S then their integer linear combination nx + my also belongs to S. Prove that either S consists from one number zero or there is a positive number d such that any other number in S is proportional to d. I already proved that it consists only of zero when x=y=0 and that otherwise there is a positive number that belongs to S, but I can't figure out how to prove the last part. I'm pretty sure that d must be the gcd, but I'm not sure if I have to prove that too.
 February 4th, 2010, 06:27 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Proof of a subset of integer numbers and linear combinations I'm not sure what you say you've proved. It consists only of zero, except when it doesn't? How about this: for every x, y in S, x - y is in S (because their linear combination with (n, m) = (1, -1) is in S). Following the Euclidean algorithm, you can see that gcd(x, y) is in S.

 Tags combinations, integer, linear, numbers, proof, subset

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post munjo5746 Real Analysis 2 June 7th, 2013 06:20 PM santosingh Advanced Statistics 1 May 16th, 2012 04:58 PM jstarks4444 Number Theory 1 October 31st, 2011 02:30 PM kiwifruit Linear Algebra 1 January 10th, 2010 01:16 PM boxerdog246 Real Analysis 3 October 6th, 2008 10:35 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top