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 Number Theory Number Theory Math Forum

 January 31st, 2016, 01:21 AM #1 Senior Member   Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT Sum of Composites Here's are a fairly simple problem. I know the solution and I thought others might enjoying solving it: Prove that every integer n > 11 is the sum of two composite numbers. January 31st, 2016, 03:48 AM #2 Senior Member   Joined: Jul 2014 From: भारत Posts: 1,178 Thanks: 230 Proof (by contradiction): Assume there exists a number n such that n > 11 and n is not the sum of 2 composite numbers. We have n = (n − 4) + 4. Notice that 4 = 2 · 2 is a composite number. Since, by assumption n is not the sum of two composite numbers, we conclude that (n − 4) is not a composite number. Similarly, we have n = (n − 6) + 6. Notice that 6 = 2.3 is a composite number. Since, by assumption n is not the sum of two composite numbers, we conclude that (n − 6) is not a composite number. Once more, we have n = (n − 8) + 8. Notice that 8 = 4.2 is a composite number. Since, by assumption n is not the sum of two composite numbers, we conclude that (n − 8) is not a composite number. We have proved that none of (n − 4),(n − 6) or (n − 8) is composite. By the division theorem, there exist integers q and r such that n = 3.q + r and 0 ≤ r < 3. We consider 3 cases: 1. If r = 0, then n − 6 = 3q − 6 = 3 · (q − 2) 2. If r = 1, then n − 4=3q + 1 − 4 = 3 · (q − 1) 3. If r = 2, then n − 8 = 3q + 2 − 8 = 3 · (q − 2) In all three cases, we have shown that one of (n − 4),(n − 6) or (n − 8) is a multiple of 3. Since n > 11, we also know that (n−4),(n−6) and (n−8) are all bigger than 3. It follows that one of (n−4),(n−6),(n−8) composite. But this a contradiction, because we had proved that none of (n−4),(n−6),(n−8) is a composite. So, our initial assumption that there exists an integer n > 11 which is not the sum of two composite numbers must be false. This proves that every integer bigger than 11 is the sum of two composite numbers. Thanks from Petek Tags composites, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post interestedinmaths Linear Algebra 2 January 26th, 2013 11:43 PM johnr Number Theory 2 May 21st, 2012 06:38 AM johnr Number Theory 20 May 11th, 2012 04:16 PM funsize999 Calculus 5 March 17th, 2009 04:23 AM kaushiks.nitt Number Theory 16 February 10th, 2009 07:30 PM

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