A set of $m$ non-zero ** rationals** {$a_1, a_2, ... , a_m$} is called a

*rational Diophantine $m$-tuple*if $a_i a_j+1$ is a square. It turns out an $m$-tuple can be extended to $m+2$ if it has certain properties. The problem is to generalize the relations below to $m=5$.

I.$m=2$. Given $a,b$:

$$ax_i+1 = \big(a\pm\sqrt{ab+1}\big)^2\tag1$$

then {$a,b,x_1$} is a triple for any choice of $x_i$ . However, {$a,b,x_1,x_2$} is a quadruple if,

$$2(a^2+b^2)-(a+b)^2-3 = y^2$$

Ex. From $a,b = \frac{1}{16},\frac{17}{4}$, we get $x_1,x_2 = \frac{33}{16},\frac{105}{16}$, a quadruple first found by Diophantus.

II.$m=3$. Given $a,b,c$:

$$ax_i+1 = \big(a\sqrt{bc+1}\pm\sqrt{(ab+1)(ac+1)}\big)^2\tag2$$

then {$a,b,c,x_1$} is a quadruple. However, {$a,b,c,x_1,x_2$} is a quintuple if,

$$2(a^2+b^2+c^2)-(a+b+c)^2-3 = y^2\,^{\color{red}\dagger}$$

Ex. From $a,b,c = \frac{28}{5},\frac{55}{16},\frac{1683}{80}$, we get $x_1,x_2 = \frac{3}{80},1680$.

III.$m=4$. Given $a,b,c,d$:

$$\small(ax_i+1)(abcd-1)^2 = \big(a\sqrt{(bc+1)(bd+1)(cd+1)}\pm\sqrt{(ab+1)(ac+1)(ad+1)}\big)^2\tag3$$

then {$a,b,c,d,x_1$} is a quintuple. However, {$a,b,c,d,x_1,x_2$} is a sextuple if,

$$2(a^2+b^2+c^2+d^2)-(a+b+c+d)^2-3-6abcd+(abcd)^2 = y^2\,^{\color{red}\dagger}$$

Ex. From $a,b,c,d = \frac{5}{4},\;\frac{5}{36},\;\frac{32}{9},\;\frac{189}{4}$, we get $x_1,x_2 = \frac{3213}{676},\;\frac{665}{1521}$, one of first sextuples found by Gibbs in 1999.

$^{\color{red}\dagger}$ These two can be satisfied by the parametric example in the variable $t$ in Dujella's website.

IV. Notes:

In general, an $n$-tuple can be extended to a $n+1$ (unconditional) and $n+2$ (conditional) for $n=2,3,4$. Also, one root $x_i$ is equal to zero if,

$$(a-b)^2 = 4\\ (a+b-c)^2 = 4(ab+1)\\ (a+b-c-d)^2 = 4(ab+1)(cd+1)$$

for relations $(1), (2), (3)$, respectively.

V. Question:

For $m=5$, given $a,b,c,d,e$:

$$\text{LHS}? = \text{RHS}?\tag4$$

If yes, then maybe we can use known $5$-tuples or $6$-tuples to generate $7$-tuples, of which there is yet no known example.*Can we find $(4)$, analogous to the first three?*- The pattern is suggestive. But, like quintics, is there a Galois-theoretic restriction on five variables $a,b,c,d,e$ that prevent generalization for $m>4$?