<s>[INST] Every positive integer can be written as form of $ab-cd$? 
Show that:  every positive integer can be written as  form of $ab-cd$,where $a-b=2(c-d),a,b,c,d\in Z$

for example
(1):$a=1,b=1,c=d=0$ then we have
$\color{red}1=1\cdot 1-0\cdot 0,$ such $a-b=2(c-d)$
(2):$a=4,b=2,c=3,d=2$ then we have
$\color{red}2=4\cdot 2-3\cdot 2,a-b=2(c-d)$
(3) $a=7,b=1,c=4,d=1$
$\color{red}3=7\cdot 1-4\cdot 1,a-b=2(c-d)$
form this by some manipulation and by comparison we can obtain the first implication i don't know how to proceed further。
 [/INST] For every $n \in \Bbb N$ you can simply choose $a = 2n+1$, $c=n+1$
and $b = d = 1$.
Then $ab - cd = (2n+1) - (n+1) = n$ and $a-b = 2n = 2(c-d)$.
</s>