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March 20th, 2016, 07:02 AM   #1
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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.
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March 20th, 2016, 11:40 AM   #2
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$\displaystyle f(x,y)\neq \frac{f(x+z,y+z)+f(x-z,y-z)}{2} $
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March 20th, 2016, 12:30 PM   #3
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Originally Posted by idontknow View Post
$\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.
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March 20th, 2016, 12:46 PM   #4
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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.
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October 28th, 2018, 02:38 AM   #5
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Quote:
Originally Posted by magicterry View Post
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)
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October 28th, 2018, 03:12 AM   #6
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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!
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October 28th, 2018, 03:19 AM   #7
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Originally Posted by magicterry View Post
. . . simply x^2 + y^2
Did you mean x² - y²?
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October 28th, 2018, 08:40 AM   #8
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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
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