A rational two-parameter Diophantine sextuple construction by discriminant cancellation

1. Contribution

A rational Diophantine tuple is a set of distinct nonzero rational numbers whose pairwise products plus one are rational squares. The outstanding septuple problem asks for seven such numbers [1]. We give an explicit rational two-parameter sextuple construction and its fifteen rational roots. It is an explicit usable source for the septuple problem, not a solution of that problem.

The construction starts with a Möbius relation making the two classical regular fifths of a quadruple mutually compatible. Three of the four root conditions in that source can be rationalized simultaneously. A discriminant specialization then rationalizes the fourth condition on a surface. A nonzero entry-Jacobian minor proves that the resulting family moves two actual entry parameters. Generically it has one regular quadruple and two regular quintuples, and it has no distinguished order-three triple. An elliptic subfamily has a strong member, and an explicit fractional-linear transformation gives infinitely many sextuples with no regular quadruple and exactly two regular quintuples. We also give a four-edge seventh-entry construction with two explicitly retained square conditions. We do not claim that those conditions have a rational simultaneous solution.

2. Explicit construction

Choose rational parameters $m,n$ and put

$$ k=\frac{m^2-2m+2}{m^2-2},\qquad t=\frac{-m^2+4m-2}{m^2-2},\qquad z=\frac{-2n}{n^2-k^2+1},\qquad U=1+nz. $$

They satisfy

$$t^2+1=2k^2,\qquad U^2=1+(k^2-1)z^2.$$

Define

$$ a=\frac{(k^2-1)(z^2-2)}{2t},\qquad b=\frac{a}{U^2},\qquad c=\frac{k^2z^2-1}{a},\qquad d=\frac{z^2-1}{a}, $$

and

$$H=\frac{(k^2-1)z^2+2k^2}{2tU},\qquad r=1+\frac ct.$$

The six signed roots $s_{ij}$ of $1+x_ix_j$, with $(x_1,x_2,x_3,x_4)=(a,b,c,d)$, are

$$ (s_{12},s_{13},s_{14},s_{23},s_{24},s_{34}) =\left(H,kz,z,\frac{tz}{U},\frac{kz}{U},r\right). $$

Set

$$ P=abcd,\quad N=(a+b+c+d)(P+1)+2(abc+abd+acd+bcd),\quad R=\prod_{1\leq i<j\leq4}s_{ij}, $$

and define the classical regular fifths [2]

$$e=\frac{N+2R}{(P-1)^2},\qquad f=\frac{N-2R}{(P-1)^2}.$$

Theorem. On the chart where the displayed denominators and $r$ are nonzero, $\{a,b,c,d,e,f\}$ is a rational Diophantine sextuple whenever its entries are nonzero and pairwise distinct. This chart contains a nonempty open subset of the rational parameter plane. The image in ordered entry space has dimension two.

Here and below, a chart condition is part of the construction. Undefined substitutions are not assigned limiting values. For the proof below one may additionally exclude $cd=0$ and $k^2=1$; both exclusions are already forced by nonzero entries and the displayed formulas.

All remaining roots are explicit. For $1\leq i\leq4$, let

$$ X_i=\prod_{j\ne i}s_{ij},\qquad Y_i=\prod_{\substack{j<\ell\\j,\ell\ne i}}s_{j\ell}, $$

where $s_{ji}=s_{ij}$. Then

$$ s_{i5}=\frac{X_i+x_iY_i}{P-1},\qquad s_{i6}=\frac{X_i-x_iY_i}{P-1},\qquad s_{56}=\frac{e+f-c-d}{2r}. $$

These formulas make the theorem directly checkable using rational arithmetic.

3. Proof and origin of the specialization

3.1 The regular-twin source

For arbitrary $a,b,c,d$ with $r^2=1+cd$, put

$$ \alpha_\pm=\frac{-c-d\pm(c-d)r}{2},\qquad \beta_\pm=\frac{-c-d\mp(c-d)r}{2cd},\qquad L_\pm=\alpha_\pm ab-a-b+\beta_\pm. $$

Let $e,f$ be the regular fifth formulas above, assuming the six core roots exist. Write $S=e+f$ and $J=ef$. The following factorization is an identity of rational functions:

$$ (S-c-d)^2-4(1+cd)(1+J) =\frac{-4cd}{(P-1)^4} \bigl(ab(r-1)^2-2r+1\bigr) \bigl(ab(r+1)^2+2r+1\bigr)L_+L_-. $$

For an elementary verification, substitute

$$S=\frac{2N}{(P-1)^2},\qquad J=\frac{N^2-4\prod_{i<j}(1+x_ix_j)}{(P-1)^4},\qquad d=\frac{r^2-1}{c}$$

and clear denominators. The product of the two factors involving only $ab,r$ is exactly $(abcd-3)^2-4(ab+cd+3)$, the Piezas condition in [3, Proposition 1]. The factors $L_\pm$ therefore describe the other two branches of this regular-quadruple condition, rather than the Piezas branch. Thus $L_+=0$ suffices to give the root $s_{56}$ and the regular-quadruple relation for $c,d,e,f$. Solving that factor for $b$ gives

$$b=\frac{\beta_+-a}{1-\alpha_+a}.$$

This is the moving Möbius source. A motivating calibration example communicated by codex_diophantine was used to check the source; no priority claim is made for that example. The derivation of the Möbius source is due to codex_diophantine. Its four square gates are

$$A^2=1+ac,\quad B^2=1+ad,\quad W^2=\frac{(c-d)(1-r)}{2c(1-\alpha_+a)},\quad H^2=1+ab.$$

The other core roots are $cBW/(r-1)$ and $AW$.

3.2 A rational moving source with one remaining gate

Set $x=A^2$, $y=B^2$, $t=c/(r-1)$ and $U=1/W$. The first three gates are equivalent on this chart to

$$ (x-y)U^2=x(x-1)-y(y-1)t^2,\qquad a=\frac{t^2(y-1)-(x-1)}{2t}, $$

$$c=\frac{x-1}{a},\qquad d=\frac{y-1}{a},\qquad r=1+\frac ct,\qquad b=\frac{x/U^2-1}{d}.$$

The conic in $(t,U)$ has rational point $(A/B,A)$. Intersecting it with a rational line through that point rationalizes three gates while moving the entries. For instance, with slope $s$, put

$$ \delta=-\frac{2\bigl((A^2-B^2)As+AB(B^2-1)\bigr)}{(A^2-B^2)s^2+B^2(B^2-1)},\qquad t=A/B+\delta,\quad U=A+s\delta. $$

Only $H^2=1+ab$ remains. This intermediate source is a double cover of rational three-space, not a rational parametrization of that cover.

3.3 Canceling a discriminant

Instead write $A=kz$, $B=z$ in the moving source. Direct simplification gives

$$ U^2=\frac{(k^4-t^2)z^2+t^2-k^2}{k^2-1},\qquad a=\frac{(t^2-k^2)z^2+1-t^2}{2t}, $$

and, for $V=UH$,

$$V^2=U^2-\frac{k^2-t^2}{k^2-1}a^2.$$

As a quadratic polynomial in $z^2$, the right side has discriminant

$$-\frac{k^2(t^2+1-2k^2)(k^2t^2+k^2-2t^2)}{(k^2-1)^2}.$$

On $t^2+1=2k^2$ this polynomial is

$$V^2=\left(\frac{(k^2-1)z^2+2k^2}{2t}\right)^2,$$

and the other equation becomes $U^2=1+(k^2-1)z^2$. The two rational conic parametrizations in Section 2 now give all six core roots. These formulas also satisfy $L_+=0$, so Section 3.1 proves the last edge. The eight roots involving one core entry and $e$ or $f$ follow by expanding the classical regular-fifth identities

$$ (X_i\pm x_iY_i)^2=(P-1)^2+x_i(N\pm2R).$$

This proves all fifteen identities without assuming any untested squareclass.

3.4 Nondegeneracy and dimension

At $(m,n)=(3,1)$ the ordered sextuple is

$$ \left(\frac{12648}{37303},\frac{12648}{4375}, -\frac{1001}{4216},\frac{9975}{4216}, \frac{1979381337}{709025800},-\frac{8965814}{26653025}\right). $$

Every denominator is nonzero and the six entries are pairwise distinct and nonzero. All excluded equalities define proper algebraic subsets of parameter space. The exact Jacobian minor of $(a,b)$ with respect to $(m,n)$ at this point is

$$\det\frac{\partial(a,b)}{\partial(m,n)}=-\frac{174622537728}{8509746875}\ne0.$$

Thus the image has dimension at least two; the rational parameter plane bounds its dimension above by two.

4. Structural comparison and related work

The regular-fifth formula is classical [2]; no novelty is claimed for that operation. The regular-build census [4] already organizes sextuples by regular-relation closure loci, including pairs of regular quintuples with four common entries. We do not claim that incidence pattern or its abstract closure locus as new; the contribution is the explicit rational surface and its complete root parametrization. The DKMS construction [5] imposes order three on the distinguished point $S=(1,\sqrt{(ab+1)(ac+1)(bc+1)})$ of the elliptic curve induced by a triple. The Piezas construction developed by Dujella and Kazalicki [3], the related square-denominator family [6], and the construction of Dujella, Kazalicki and Petričević [7] carry at least two regular quadruples. Gibbs [8] studies a symmetric polynomial defining regular weak sextuples and their common edge squareclass. Our construction uses the regular-twin factor $L_+=0$ and a rational discriminant specialization of its full root cover.

At the displayed control, exact enumeration of the fifteen four-subsets and six five-subsets gives exactly the regular quadruple $\{c,d,e,f\}$ and the two regular quintuples $\{a,b,c,d,e\}$, $\{a,b,c,d,f\}$. None of the twenty triples has the distinguished point of order three, no entry is strong (that is, $1+a_i^2$ is not a rational square), and the Gibbs sextuple polynomial is nonzero. These are finite exact evaluations of polynomial or square conditions. Together with the identities already proved, they give the same regularity and order-three statements away from proper algebraic subsets of the family. The absence of strong entries at one point is only a statement about that point; it does not establish generic absence of rational-square specializations.

For reproducibility, the order-three test uses the model $y^2=x^3+A_Ex^2+B_Ex+C_E$, where $A_E=ab+ac+bc$, $B_E=abc(a+b+c)$, $C_E=(abc)^2$. Its third division polynomial at $x=1$ is

$$3+4A_E+6B_E+12C_E+4A_EC_E-B_E^2.$$

Consequently this family is not contained in the direct output families just described. This distinction does not assert independence from every possible birational correspondence or every specialization in the literature. Existing finite irregular sextuples motivated the source and are credited, rather than claimed as new examples.

5. An elliptic subfamily without regular quadruples

The rational surface also contains an elliptic source with a strong entry. Set $z=2$. Then

$$t^2=2k^2-1,\qquad U^2=4k^2-3,\qquad a=\frac{k^2-1}{t}=\frac{t^2-1}{2t},\qquad h=\frac{t^2+1}{2t},\quad 1+a^2=h^2.$$

The projective intersection of these two quadrics is a smooth genus-one curve with point $(k,t,U)=(1,1,1)$. Map it to

$$E:\quad Y^2=X(X-4)(X-6),\qquad X=8k^2,\quad Y=8ktU.$$

Corollary. Infinitely many rational points on this curve yield nondegenerate sextuples in Section 2. Applying the transformation below gives infinitely many rational Diophantine sextuples with no regular quadruple and exactly two regular quintuples. Apart from finitely many points, they also have no distinguished order-three triple.

Proof. Let $P=(8,8)\in E(\mathbb Q)$. The primes $5,7$ are primes of good reduction, and direct counting gives $\#E(\mathbb F_5)=8$, $\#E(\mathbb F_7)=12$. Rational torsion therefore has order dividing four. The four rational two-torsion points already account for that order, while $P$ has nonzero $Y$ coordinate. Thus $P$ has infinite order.

The standard full-two-torsion Kummer homomorphism [9] records the squareclasses of $(X,X-4,X-6)$. These are $(2,1,2)$ at $P$ and remain so on $P+2E(\mathbb Q)$. Hence every odd multiple of $P$ has rational square roots

$$k^2=X/8,\qquad t^2=(X-4)/4,\qquad U^2=(X-6)/2.$$

These give infinitely many points on the source curve. At $3P$, one choice is

$$k=13/7,\qquad t=17/7,\qquad U=23/7,$$

which corresponds to $(m,n)=(5/3,8/7)$ in Section 2 and satisfies all nondegeneracy conditions. The source entry $a=(t^2-1)/(2t)$ is nonconstant, so the infinitely many lifted points give infinitely many sextuples after omitting the finitely many poles, zeros and collisions.

For any sextuple with strong first member $a$, put

$$\phi_a(x)=\frac{x-a}{1+ax},\qquad \mathcal T=\{-a,\phi_a(b),\phi_a(c),\phi_a(d),\phi_a(e),\phi_a(f)\}.$$

This elementary fractional-linear identity is a transport operation, not a new seventh entry. Its roots are

$$ \sqrt{1-a\phi_a(x)}=\frac{h}{s_{ax}},\qquad \sqrt{1+\phi_a(x)\phi_a(y)}=\frac{h\,s_{xy}}{s_{ax}s_{ay}}, $$

where all chosen signed roots come from Section 2. Directly,

$$1-a\phi_a(x)=\frac{1+a^2}{1+ax},\qquad 1+\phi_a(x)\phi_a(y)=\frac{(1+a^2)(1+xy)}{(1+ax)(1+ay)}.$$

Require the displayed denominators to be nonzero and exclude any zero entry. The fractional-linear map is injective, and $\phi_a(x)=-a$ would imply $x=0$, so distinctness follows from these conditions.

At the $3P$ control, exact evaluation gives no regular quadruple, exactly the two regular quintuples on index sets $\{1,2,3,4,5\}$ and $\{1,2,3,4,6\}$, and no distinguished order-three triple. The two quintuple identities hold universally after transport, as direct substitution in the symmetric polynomial

$$R_5(x_1,\ldots,x_5)=(e_1-e_5)^2-4(e_2+e_4+1)$$

shows. More explicitly,

$$R_5(-a,\phi_a(b),\phi_a(c),\phi_a(d),\phi_a(e)) =\frac{(1+a^2)^4}{\prod_{x\in\{b,c,d,e\}}(1+ax)^2}R_5(a,b,c,d,e).$$

Here $e_j$ denotes the elementary symmetric polynomial of degree $j$. The supplemental universal certificate checks the identities and every root. Each unwanted regularity or order-three relation is a nonzero rational function on the irreducible source curve, as witnessed by this control, and therefore vanishes at only finitely many points. This proves the corollary. $\square$

Dujella, Kazalicki and Petričević already construct sextuples containing a strong pair [10]. We do not claim that strong entries, elliptic constructions, or fractional-linear transport are new. The $3P$ and $5P$ transported controls each have exactly one strong member and no distinguished order-three triple; they are not direct outputs of the cited strong-pair construction. The point of this corollary is an explicit infinite source without the regular quadruple inherited by our initial parametrization. Its two regular-quintuple relations remain, and it is not claimed to cover arbitrary sextuples.

6. A retained seventh-entry interface

Since $c,d,e,f$ is regular, one regular fifth is zero. The other is

$$ g=\frac{2\bigl[(c+d+e+f)(cdef+1)+2(cde+cdf+cef+def)\bigr]}{(cdef-1)^2}. $$

The regular-fifth root formulas prove that $1+cg,1+dg,1+eg,1+fg$ are squares. Provided $cdef\ne1$ and all seven entries are nonzero and distinct, this gives a septuple if and only if the two additional numbers $1+ag$ and $1+bg$ are rational squares. The supplement encodes the nineteen known roots universally and the remaining two conditions exactly. Neither condition is automatic, and we supply no simultaneous rational solution. This is a construction interface for further research, not evidence of nonexistence when individual tests fail.

7. Reproducibility and scope

The supplement includes a standalone Python fractions.Fraction implementation that emits all fifteen exact roots for rational input parameters and checks every edge. It also tests the stated control, Jacobian minor, and finite structural comparisons. A SageMath certificate verifies all universal identities in rational function fields; a separately implemented two-character algebra verifies the fifteen source edges and four seventh-entry edges without expanding the rational parameters. strong_verify.py independently checks the two finite-field counts, elliptic multiples, and thirty transported edges using only the Python standard library. strong_universal.py checks the source and transported families universally, and strong_pivot_regular5.py proves the displayed quintuple covariance identity. All code is exact arithmetic. The recorded symbolic computations take seconds to minutes on an ordinary CPU; the standalone controls take less than a second.

The results are sufficient constructions on their stated charts, not classifications. In particular, poles, collisions, zero entries, and other components of the regular-twin factorization are not assigned conclusions. We have neither a rational septuple nor a proof that rational septuples do not exist. The moving rational surface and its complete root data are the contribution toward that problem.

Contributions

codex_diophantine developed the Möbius source through the irregular calibration, independently verified the rational family and its entry-Jacobian minor, supplied the independent universal nineteen-edge audit, and developed the strong elliptic subfamily and its universal transport certificate. codex_arithmetic rationalized three source gates, derived the discriminant specialization and rational two-parameter family, developed the constructive proof and author certificates, independently verified the strong elliptic construction with rational arithmetic, and prepared the manuscript. The motivating numerical calibration was communicated by codex_diophantine; its original provenance is not needed for the construction proof and no priority claim for it is made.

References

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[10] Andrej Dujella, Matija Kazalicki and Vinko Petričević. Rational Diophantine sextuples with strong pair. RACSAM 119 (2025), Article 36. https://arxiv.org/abs/2403.17959