
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 8th, 2017, 04:42 PM  #1 
Newbie Joined: Jan 2017 From: Mangalore, Karnataka, India Posts: 11 Thanks: 0  Integer solutions to (x^2+x) + (y^2+y) = (z^2+z)
Is there any integer solution to the equation, (x^2+x) + (y^2+y) = (z^2+z) ?

January 8th, 2017, 07:12 PM  #2 
Senior Member Joined: Feb 2010 Posts: 624 Thanks: 98 
$\displaystyle (3^2+3)+(5^2+5)=(6^2+6)$

January 8th, 2017, 11:26 PM  #3 
Newbie Joined: Jul 2015 From: tbilisi Posts: 26 Thanks: 7  
January 9th, 2017, 10:24 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 17,734 Thanks: 1360 
(x, y, z) = (0, t, t), where t is an integer is the easiest solution. A better question would be to find all the solutions in positive integers. Let t be a natural number, then some (not necessarily distinct) solutions in positive integers are given below. (x, y, z) = (3t, 4t + 1, 5t + 1) (x, y, z) = (3t + 2, 4t + 2, 5t + 3) (x, y, z) = (t + 2, (t + 1)(t + 4)/2, (t + 2)(t + 3)/2) (x, y, z) = (3t + 5, (3t + 2)(t + 3)/2, (3tÂ² + 11t + 12)/2) One might speculate that every solution with x < y can be given in a similar way. 
January 9th, 2017, 11:03 AM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,146 Thanks: 685 
z = {1 + SQRT[1 + 4(x^2 + x + y^2 + y)]} / 2 Kinda interesting to "look at"!! 

Tags 
integer, solutions 
Search tags for this page 
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Integer Solutions to Z²sZ+p=0  AlephUser  Elementary Math  0  January 3rd, 2014 05:42 AM 
Integer solutions  proglote  Number Theory  4  June 8th, 2011 01:41 PM 
Some equations having integer solutions  proglote  Number Theory  4  April 19th, 2011 09:06 AM 
all integer solutions of 3^m2^n=+1  calligraphy  Number Theory  7  April 13th, 2011 12:56 PM 
x^5+31=y^2 has no integer solutions  elim  Number Theory  4  April 5th, 2010 01:05 PM 