October 10th, 2012, 04:15 AM  #1 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Diophantine Fun This one's pretty challenging but not as hard as we think : Prove that has no solution in positive integers Hint : Consider the equation modulo 4 Balarka . 
October 10th, 2012, 05:46 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: Diophantine Fun
Pretty easy once you divide out common factors.

October 10th, 2012, 06:06 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,528 Thanks: 1750 
If 3 divides a² + b², both a and b must be divisible by 3, so 9 divides a² + b², which implies c is divisible by 3. Hence one could divide all three variables by 3 to get another solution in integers. After sufficient repetition, one must get a = b = c = 0, so this is true to start with, showing that there is no solution in positive integers. 
October 11th, 2012, 08:24 AM  #4  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,126 Thanks: 914  Re: Diophantine Fun Quote:
 

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