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 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(x-z, y-z)}/2 ----------------1 f(x, y) = {f(x+1, y+1) + f(x-1, y-1)}/2-2 .-----------------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+/-y-2, 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: 636 Thanks: 91 $\displaystyle f(x,y)\neq \frac{f(x+z,y+z)+f(x-z,y-z)}{2}$ March 20th, 2016, 11:30 AM   #3
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 Originally Posted by idontknow $\displaystyle f(x,y)\neq \frac{f(x+z,y+z)+f(x-z,y-z)}{2}$
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, non-linear, 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
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 Originally Posted by magicterry 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(x-z, y-z)}/2 ----------------1 f(x, y) = {f(x+1, y+1) + f(x-1, y-1)}/2-2 .-----------------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+/-y-2, but there is another surprising function (non linear) which I will leave the reader to discover for now.
I've almost forgotten the other function which is simply x^2 + y^2 ( or X^2 - Y^2 + 2) October 28th, 2018, 02:12 AM   #6
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 Originally Posted by magicterry I've almost forgotten the other function which is simply x^2 + y^2 ( or X^2 - Y^2 + 2)
We've been waiting this whole time!  October 28th, 2018, 02:19 AM   #7
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 Originally Posted by magicterry . . . simply x^2 + y^2
Did you mean x² - y²? October 28th, 2018, 07:40 AM   #8
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 Originally Posted by Joppy We've been waiting this whole time! Correction :X^2 + Y^2 applies to equation 2 and X^2 - Y^2 applies to equation 1 Tags discovery, surprising Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Carl James Mesaros Algebra 1 December 15th, 2014 05:52 AM momo Number Theory 5 April 7th, 2009 05:57 AM ipzig__1963 Applied Math 0 December 31st, 1969 04:00 PM

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