Question 1209952
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The function f(n) is defined for all integers n, such that

f(x) + f(y) = f(x + y) - 4xy - 1 + f(x^2) + f(y^2)
for all integers x and y, and f(1) = 1.  Find f(n).
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        The solution in the post by @CPhill,  giving the answer  f(n) = {{{(3/2)*n*(n-1))}}},  is  INCORRECT.

        I will show it below in this my post.



<pre>
(a)  Substitute x=0, y=0 into the basic formula.  You will get

     f(0) + f(0) = f(0) - 0 - 1 + f(0) + f(0).


Cancel f(0) in both sides everywhere where possible.  You will get

     0 = f(0) - 1

     f(0) = 1.



(b)  Substitute y=0 into the basic formula. Now x is an arbitrary integer number.

        f(x) + f(0) = f(x+0) - 0 - 1 + f(x^2) + f(0),

        f(x) + 1 = f(x) - 0 - 1 + f(x^2) + 1,

        1 = f(x^2).


Thus, from the general basic formula, f(n^2) = 1 for any integer 'n'.


But it is not so from the formula  f(n) = {{{(3/2)*n*(n-1))}}}  by @CPhill.
</pre>

It disproves and kills the solution by @CPhill to the death.


<pre>
Let's go further in analyzing the problem.


In the general equation, replace the terms f(x^2) and f(y^2) by 1, as we deduced it above.
You will get

    f(x) + f(y) = f(x+y) - 4xy + 1.


Put  y= 1  in this equation

    f(x) + f(1) = f(x+1) - 4x + 1,

    f(x) + 1 = f(x+1) - 4x + 1,

    f(x+1) =  f(x) + 4x.


We get a nice recurrent formula and can calculate the values f(x) moving forward.
We get

    f(2) = f(1) + 4*1 = 1 + 4 = 5;

    f(3) = f(2) + 4*2 = 5 + 8 = 13;

    f(4) = f(3) + 4*3 = 13 + 12 = 25.


But this equality f(4) = {{{f(2^2)}}} = 25 CONTRADICTS to equality f(n^2) = 1, which we established earlier above.


It tells that the given general formula DOES NOT define a function f.
</pre>

So, the posted problem is &nbsp;SELF-CONTRADICTORY.


Its description &nbsp;DECEIVES &nbsp;the reader from the very beginning, 
saying that this general formula defines a function.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;On &nbsp;CONTRARY, &nbsp;it &nbsp;DOES &nbsp;NOT &nbsp;define.



In the last several days, &nbsp;I saw several similar &nbsp;ABSOLUTELY &nbsp;DEFECTIVE &nbsp;" problems ", &nbsp;submitted to the forum.


They all are created by unprofessional/(illiterate ?) Math composers.