## To make {(B-A),(C-A),(C-B),(D-A),(D-B),(D-C)} squares

To find four positive integers   $A, B, C, D$   such that
$B - A$,
$C - A$,     $C - B$,
$D - A$,     $D - B$,     $D - C$

are all squares.

Here are 4 sets of solutions:

$A \; = \; k_1$
$B \; = \; k_1 \; + \; 3404^2$
$C \; = \; k_1 \; + \; 3404^2 \; + \; 1680^2$
$D \; = \; k_1 \; + \; 6005^2$

$k_1 \; > \; 0$

$B \; - \; A \; = \; 3404^2$
$C \; - \; A \; = \; 3404^2 \; + \; 1680^2 \; = \; 3796^2$
$C \; - \; B \; = \; 1680^2$
$D \; - \; A \; = \; 6005^2$
$D \; - \; B \; = \; 6005^2 \; - \; 3404^2 \; = \; 4947^2$
$D \; - \; C \; = \; 6005^2 \; - \; (3404^2 + 1680^2) \; = \; 4653^2$

$A \; = \; k_2$
$B \; = \; k_2 \; + \; 9306^2$
$C \; = \; k_2 \; + \; 9306^2 \; + \; 3360^2$
$D \; = \; k_2 \; + \; 12010^2$

$k_2 \; > \; 0$

$B \; - A \; = \; 9306^2$
$C \; - A \; = \; 9306^2 \; + \; 3360^2 \; = \; 9894^2$
$C \; - B \; = \; 3360^2$
$D \; - A \; = \; 12010^2$
$D \; - B \; = \; 12010^2 \; - \; 9306^2 \; = \; 7592^2$
$D \; - C \; = \; 12010^2 \; - \; (9306^2 + 3360^2) \; = \; 6808^2$

$A \; = \; k_3$
$B \; = \; k_3 \; + \; 2576^2$
$C \; = \; k_3 \; + \; 2576^2 \; + \; 1332^2$
$D \; = \; k_3 \; + \; 8660^2$

$k_3 \; > \; 0$

$B \; - \; A \; = \; 2576^2$
$C \; - \; A \; = \; 2576^2 \; + \; 1332^2 \; = \; 2900^2$
$C \; - \; B \; = \; 1332^2$
$D \; - \; A \; = \; 8660^2$
$D \; - \; B \; = \; 8660^2 \; - \; 2576^2 \; = \; 8268^2$
$D \; - \; C \; = \; 8660^2 \; - \; (2576^2 + 1332^2) \; = \; 8160^2$

$A \; = \; k_4$
$B \; = \; k_4 \; + \; 831830^2$
$C \; = \; k_4 \; + \; 831830^2 \; + \; 742560^2$
$D \; = \; k_4 \; + \; 1116986^2$

$k_4 \; > \; 0$

$B \; - \; A \; = \; 831830^2$
$C \; - \; A \; = \; 831830^2 \; + \; 742560^2 \; = \; 1115050^2$
$C \; - \; B \; = \; 742560^2$
$D \; - \; A \; = \; 1116986^2$
$D \; - \; B \; = \; 1116986^2 \; - \; 831830^2 \; = \; 745464^2$
$D \; - \; C \; = \; 1116986^2 \; - \; (831830^2 + 742560^2) \; = \; 65736^2$

Can you find other types of solutions?