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$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.

  1. Is there elementary example where only $w$ is even and all four integers are close?

A non-example which is elementary is $w=2$ and $x=y=2k+1$ at a $k\in\mathbb N$ and $z=\frac{xy+1}2$ and it is a non-example because $w$ is a constant and so the integers are not close. However it appears there might be infinite number of examples where $w<z<2w$ and $x=y=2k+1$ holds. But there seems no obvious pattern.

  1. Is there a conjecture of sorts relating to squares and these unimodular examples?

Any references?

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  • $\begingroup$ Depends what you mean by "close". If you want $x,y,z,w$ to be within $O(1)$ of each other, then there will be only finitely many solutions, with the only exception being the one you gave (up to permutation of variables) $\endgroup$
    – Wojowu
    Oct 25, 2021 at 8:04
  • $\begingroup$ $|x-y|=O(1)$ and $|z|/|w|=O(1)$ perhaps is a reasonable bound. $\endgroup$
    – Turbo
    Oct 25, 2021 at 8:31
  • $\begingroup$ In that case you can have solutions coming from solutions to Pell's equations: if $a^2-2b^2=-1$, then $(x,y,z,w)=(a,a,b,2b)$ satisfies $wz-xy=1$. $\endgroup$
    – Wojowu
    Oct 25, 2021 at 8:36
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    $\begingroup$ Finding solutions to Pell equations is standard stuff. Also, $36\times25-31\times29=1$, $36\times49-43\times41=1$, $64\times49-57\times55=1$, and so on. $\endgroup$
    – Gerry Myerson
    Oct 25, 2021 at 12:01
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    $\begingroup$ Every intro Number Theory textbook treats Pell. Or just type Pell's equation into the internet and stand back while hundreds of links come flying at you. But my numerical examples have nothing to do with @Wojowu and Pell. $\{w,z\}=\{a^2,(a+1)^2\}$, $\{x,y\}=\{a^2+a+1,a^2+a-1\}$. $\endgroup$
    – Gerry Myerson
    Oct 26, 2021 at 12:16

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