March 20th, 2016, 06:02 AM  #1 
Member Joined: Aug 2015 From: Chiddingfold, Surrey Posts: 57 Thanks: 3 Math Focus: Number theory, Applied maths  A surprising? discovery
A SURPRISING DISCOVERY: I have only recently discovered the following equations of a certain function of a pair of two integers, which surprised me: f(x, y) = {f(x+z, y+z) + f(xz, yz)}/2 1 f(x, y) = {f(x+1, y+1) + f(x1, y1)}/22 .2 Where z = any value, positive or negative. Of course the first equation is true for f(x,y) = x+/y, and the second is true for f(x,y) = x+/y2, but there is another surprising function (non linear) which I will leave the reader to discover for now. 
March 20th, 2016, 10:40 AM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 541 Thanks: 82 
$\displaystyle f(x,y)\neq \frac{f(x+z,y+z)+f(xz,yz)}{2} $

March 20th, 2016, 11:30 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  magicterry did not say that was true for all functions! He said that there exist functions that do. He pointed out f(x,y)= x+ y and f(x,y)= x y satisfy it and challenged people to find a third, nonlinear, equation that satisfies it.

March 20th, 2016, 11:46 AM  #4 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16 
any f that is linear on all lines x=y+constant will do. I think that if f is differentiable one can show that its restriction to the line x=y+c has to be linear.

October 28th, 2018, 01:38 AM  #5  
Member Joined: Aug 2015 From: Chiddingfold, Surrey Posts: 57 Thanks: 3 Math Focus: Number theory, Applied maths  Quote:
 
October 28th, 2018, 02:12 AM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,826 Thanks: 646 Math Focus: Yet to find out.  
October 28th, 2018, 02:19 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,757 Thanks: 2138  
October 28th, 2018, 07:40 AM  #8 
Member Joined: Aug 2015 From: Chiddingfold, Surrey Posts: 57 Thanks: 3 Math Focus: Number theory, Applied maths  

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