October 10th, 2012, 05:15 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, 06:46 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 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, 07:06 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,595 Thanks: 1493 
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, 09:24 AM  #4  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,672 Thanks: 741  Re: Diophantine Fun Quote:
 

Tags 
diophantine, fun 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Another diophantine equation  harrypham  Number Theory  0  May 31st, 2012 06:49 PM 
diophantine x³ + 8x²  6x + 8 = y³  Pell's fish  Number Theory  9  March 23rd, 2012 12:07 AM 
Fun Diophantine Equations  icemanfan  Number Theory  1  March 14th, 2012 03:52 PM 
diophantine equation  islam  Number Theory  4  January 11th, 2012 04:28 AM 
Diophantine Fun  mathbalarka  Algebra  1  December 31st, 1969 04:00 PM 