## Set {a,b,c,d} such that the product of any two of them increased by 1 is a square — Part 3

$1 + a \,b = A^2$   ………………….   $1 + b \,c = D^2$
$1 + a \,c = B^2$   ………………….   $1 + b \,d = E^2$
$1 + a \,d = C^2$   ………………….   $1 + c \,d = F^2$

$a = 3$
$b = 3 \, n^2 - 4 \, n + 1$
$c = 3 \, n^2 + 2 \, n$
$d = 4 \, (27 \, n^4 - 18 \, n^3 - 12 \, n^2 + 5 \, n + 2)$

$1 + a \,b = (3 n - 2)^2$   ……………………   $1 + b \,c = (3 \, n^2 - n - 1)^2$
$1 + a \,c = (3 \, n + 1)^2$   ……………………   $1 + b \,d = (18 \, n^3 - 18 \, n^2 - 2 \, n + 3)^2$
$1 + a \,d = (18 \, n^2 - 6 \, n - 5)^2$   ………..   $1 + c \,d = (18 \, n^3 - 8 \, n - 1)^2$

Or

$a = 3$
$b = 3 \, n^2 - 2 \, n$
$c = 3 \, n^2 + 4 \, n + 1$
$d = 4 \, (27 \, n^4 + 18 \, n^3 - 12 \, n^2 - 5 \, n + 2)$

$1 + a \,b = (3 \, n - 1)^2$   ……………………   $1 + b \,c = (3 \, n^2 + n - 1)^2$
$1 + a \,c = (3 \, n + 2)^2$   ……………………   $1 + b \,d = (18 \, n^3 - 8 \, n + 1)^2$
$1 + a \,d = (18 \, n^2 + 6 \, n - 5)^2$   ………..   $1 + c \,d = (18 \, n^3 + 18 \, n^2 - 2 \, n - 3)^2$

…………………………………………………………..

$a = 4$

$a = 4$
$b = n^2 + n$
$c = n^2 + 5 \, n + 6$
$d = 4 \, (4 \, n^4 + 24 \, n^3 + 45 \, n^2 + 27 \, n + 5)$

$1 + a \,b = (2 n + 1)^2$   ……………………………..   $1 + b \,c = (n^2 + 3 \, n + 1)^2$
$1 + a \,c = (2 n + 5)^2$   ……………………………..   $1 + b \,d = (4 \, n^3 + 14 \, n^2 + 10 \, n + 1)^2$
$1 + a \,d = (8 n^2 + 24 n + 9)^2$   ………………….   $1 + c \,d = (4 \, n^3 + 22 \, n^2 + 34 \, n + 11)^2$

…………………………………………………………..

$a = 5$

$a = 5$
$b = 5 \, n^2 - 8 \, n + 3$
$c = 5 \, n^2 + 2 \, n$
$d = 4 \, (125 \, n^4 - 150 \, n^3 + 27 \, n + 4)$

$1 + a \,b = (5 \, n - 4)^2$   ……………………….   $1 + b \,c = (5 \, n^2 - 3 \, n - 1)^2$
$1 + a \,c = (5 \, n + 1)^2$   ……………………….   $1 + b \,d = (50 \, n^3 - 70 \, n^2 + 14 \, n + 7)^2$
$1 + a \,d = (50 \, n^2 - 30 \, n - 9)^2$   ………….   $1 + c \,d = (50 \, n^3 - 20 \, n^2 - 16 \, n - 1)^2$

$10 \, (1 + b \,c) \; + \; 1 \; = \; 1 \; + \; a \,d$

Or

$a = 5$
$b = 5 \, n^2 - 2 \, n$
$c = 5 \, n^2 + 8 \, n + 3$
$d = 4 \, (125 \, n^4 + 150 \, n^3 - 27 \, n + 4)$

$1 + a \,b = (5 \, n - 1)^2$   ……………………….   $1 + b \,c = (5 \, n^2 + 3 \, n - 1)^2$
$1 + a \,c = (5 \, n + 4)^2$   ……………………….   $1 + b \,d = (50 \, n^3 + 20 \, n^2 - 16 \, n + 1)^2$
$1 + a \,d = (50 \, n^2 + 30 \, n - 9)^2$   ………….   $1 + c \,d = (50 \, n^3 + 70 \, n^2 + 14 \, n - 7)^2$

$10 \, (1 + b \,c) \; + \; 1 \; = \; 1 \; + \; a \,d$