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Can you prove or disprove the existence of a positive integer P, such that P can convert the expression;

$6ab+a+b$ into the form $6xy+x-y$ by subtracting from it.

What I am trying to find is a positive integer P that makes Z, in the following equation,expressible in the form; $6xy+ x - y$

$Z=6ab+a+b-P$

for all $a$ and $b$ where $a,b,x,y,∈N $ , $P>0 $

Clarification:

Say if ;

$Z=2K+1-P$, and you want to find and expression P that would make Z be of the form $2a-1$

Then $P=4$ would be an example of such a number, Since then;

$Z=2k+1-2-2=2(k-1)-1$

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Can you supply some context or an example on this? It's hard to grasp what you really want to achieve here. –  Christoph Mar 2 at 15:11
1  
E.g. $P = 6ab + a + b - (6xy + x - y)$? –  hardmath Mar 2 at 15:14
    
P cannot involve the variables $a$ or $b$ –  user129967 Mar 2 at 15:16
    
Note that your example $2K + 1$ is already (by subtracting zero) of the form $2(K+1) - 1$. So I'm not sure how much subtracting something (if cannot involve $a$ or $b$) enlarges the possibilities. Probably what you mean is to subtract a constant, and then rewrite the resulting expression in a form $6xy + x - y$. –  hardmath Mar 2 at 15:20
    
I think that would make more sense. –  user129967 Mar 2 at 15:21

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