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 April 20th, 2013, 08:48 PM #1 Newbie   Joined: Apr 2013 Posts: 1 Thanks: 0 Number Theory Is there a solution to the Diophantine equation (a+b)(c+d)=ab+cd ?
 April 20th, 2013, 10:37 PM #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: Number Theory Sure, a = b = c = d = 0. Not sure if there are any nontrivial solutions.
April 20th, 2013, 10:54 PM   #3
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Re: Number Theory

Quote:
 Originally Posted by Guatama Is there a solution to the Diophantine equation (a+b)(c+d)=ab+cd ?
Lot's of, probably infinitely many. Take for example, (1, 1, 3, 5) and (1, 2, 4, 10).

April 21st, 2013, 05:50 AM   #4
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Re: Number Theory

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by Guatama Is there a solution to the Diophantine equation (a+b)(c+d)=ab+cd ?
Lot's of, probably infinitely many. Take for example, (1, 1, 3, 5) and (1, 2, 4, 10).
Nice. How did you come up with these?

 April 21st, 2013, 07:22 AM #5 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: Number Theory One infinite family: a = b = c = 0.
 April 21st, 2013, 11:25 AM #6 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 Re: Number Theory Oh, I suddenly realize that this question may be highly relevant to something I was trying to do many years ago. It's gone take some work finding that stuff again in the clutter of my mind. Kinda like trying to remember whether some document you need is buried somewhere in the cluttered basement, cluttered attic or cluttered garage. Depressing, but it may pop to mind. Anyway, in anticipation of remembering what I was trying to do, I would be eager to see ways of dredging up NON-trivial solutions!
April 21st, 2013, 12:27 PM   #7
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Re: Number Theory

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 Originally Posted by johnr I would be eager to see ways of dredging up NON-trivial solutions!
I'll wait for someone to define nontrivial in this context.

April 21st, 2013, 01:15 PM   #8
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Re: Number Theory

Quote:
 Originally Posted by Guatama Is there a solution to the Diophantine equation (a+b)(c+d)=ab+cd ?
Try b = c, then ad + bc = 0 will work.

April 21st, 2013, 01:27 PM   #9
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Re: Number Theory

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Originally Posted by CRGreathouse
Quote:
 Originally Posted by johnr I would be eager to see ways of dredging up NON-trivial solutions!
I'll wait for someone to define nontrivial in this context.
Ok, how about solutions that don't involve zeroing everything out!

April 21st, 2013, 01:39 PM   #10
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Re: Number Theory

Quote:
 Originally Posted by johnr Ok, how about solutions that don't involve zeroing everything out!
I gave an infinite family of solutions (or four families, by symmetry) that did not zero everything out. But I suspect you'd still call them trivial.

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