Regularity-Selected Generic Rank at Least Four in the Piezas Diophantine Sextuple
Audio version created with Paper2Audio.
C. R. Hill
Audio by Paper2Audio.
Abstract
The Piezas one-parameter rational Diophantine sextuple contains twenty three-element subfamilies. We show that its regularity structure selects six of them in a uniform way and that every selected triple induces an elliptic curve of Mordell–Weil rank at least four over Q(t) . The regular quadruples share the pair {e, f} ; after removing this pair, the remaining incidence graph is the path c - d - a - b . The two endpoints and the two distance-two pairs of this path select the six triples ace, acf, bde, bdf, bef, cef.
For each selected triple we construct four rational sections explicitly from the Diophantine square witnesses. Independence is proved by specializing all six families at the same good value t = 12/5 and certifying positive-definite canonical-height Gram matrices with rigorous interval arithmetic. The result is a generic lower-bound theorem. It does not claim exact rank four or a rank bound for the remaining fourteen triple classes.
Website: https://hillaxiom.com | Email: crhillresearch@gmail.com
1 Introduction
A rational Diophantine m-tuple is a set of distinct nonzero rational numbers such that the product of any two distinct elements, increased by 1, is a rational square. Rational Diophantine sextuples exist in infinite families. The particularly simple one-parameter sextuple used here was given by Piezas and subsequently studied by Dujella and Kazalicki.
A rational Diophantine triple {u, v, w} induces an elliptic curve
Math summary: This expression defines an elliptic curve induced by a rational Diophantine triple. The output is calculated by squaring a value and setting it equal to the product of three linear terms, each consisting of a weight from the triple multiplied by an input value plus one.
The use of rational Diophantine triples to construct elliptic curves of high rank has a substantial history. Dujella constructed induced families of generic rank at least four; Aguirre, Dujella, and Peral later obtained generic rank at least five; and Dujella and Peral constructed a family of generic rank at least seven. Thus generic rank at least four is not itself the new feature of the present result.
Regular substructures inside rational Diophantine sextuples have also been studied previously. Dujella, Kazalicki, and pee-treečević constructed infinite families containing regular quadruples and regular quintuples. More recently, Zeng studied regular extensions and compatibility for standard parametric rational Diophantine sextuples over Q (t) .
The contribution here is the link between the internal regularity structure of the Piezas sextuple and the ranks of elliptic curves induced by its three-element subfamilies. The regularity identities themselves are not claimed as new. What is proved here is that their incidence pattern selects six specific triple classes from the twenty available three-element subfamilies, and that each selected class carries four independent rational sections over Q(t) . Thus the theorem gives a regularity-indexed generic rank lower bound for these six induced elliptic-curve families.
The proof has two parts. First, five exact regularity identities produce a small incidence graph that selects six distinguished triples. Second, each selected triple is shown to carry four rational sections. Their independence is certified at the common good specialization
Math summary: This expression calculates the value of the variable t. The result is the fraction twelve over five.
by positive-definite canonical-height Gram matrices. Only persistence of a section relation under good specialization is used; injectivity of the full specialization homomorphism is not required.
2 The Piezas family and its regularity graph
Let
Math summary: These expressions define the functions that form the Piezas rational Diophantine sextuple. Each value is computed as a ratio of polynomials involving a variable input, with various combinations of squared terms and linear shifts in the numerators and denominators.
These functions form the Piezas rational Diophantine sextuple over the rational functions of t. Thus for every distinct i, j in the set a, b, c, d, e, f there exists rho i j in the rational functions of t satisfying
Math summary: This expression defines the square of a value for the Piezas rational Diophantine sextuple. It calculates this square by adding one to the product of two distinct input indices.
We choose one such square root for each unordered pair and keep that choice fixed.
We use the standard algebraic regularity tests. An ordered quadruple (p, q, r, s) is regular when
Equation 1 summary: This expression defines the condition for an ordered quadruple to be regular. It calculates the square of the result from subtracting the second and third values from the sum of the first and fourth values, then equates it to four times the product of the first value plus one and the second value times the third value plus one.
and an ordered quintuple p, q, r, s, z is regular when
Equation 2 summary: This expression defines a condition for a regular ordered quintuple by equating a squared term to a scaled product. It calculates the square of a sum involving five input values and their products and sets it equal to four times the product of four binomials combining those input values.
These identities are convenient here because they can be checked directly in the field of rational functions in t; no specialization is involved.
Proposition 1 (Regularity pattern). For the Piezas family, the identities (1) and (2) hold identically in the field of rational functions in t for the ordered quadruples
Math summary: This expression defines three ordered quadruples for the Piezas family. These sets consist of the input values a, b, e, and f, followed by a, d, e, and f, and finally c, d, e, and f.
and the ordered quintuples
Math summary: This expression identifies two specific ordered quintuples used for a rational function calculation. The first set contains the elements a, b, c, d, and e, while the second set contains a, b, c, d, and f.
Proof. The claim is an exact rational-function calculation. Substitute the six displayed Piezas functions into the left side minus the right side of (1) for
Math summary: This expression lists three sets of four variables each. The sets combine the input values identified as a, b, e, and f, then a, d, e, and f, and finally c, d, e, and f.
and into the corresponding residual of (2) for
: Math summary: This expression displays two sequences of five elements. Both sequences are identical except for the final element, which changes from one value to another.
After cancellation in the fraction field Q of t, all five residuals are zero.
For the combinatorial argument, we now forget the ordering and retain only the underlying label sets. The three regular quadruples share the pair
Math summary: This expression identifies a set containing two elements. These elements are the specific labels e and f.
Removing that pair leaves the two-element sets
Math summary: This expression defines a set of edges on a vertex set containing four elements. It lists three distinct pairs consisting of the members a and b, a and d, and c and d.
which we regard as edges on the vertex set a, b, c, d. Thus the resulting graph is the path
Equation 3 summary: This expression defines a sequence of connected vertices that forms a path. It lists the edges as the connections between vertex c and d, d and a, and a and b.
Its endpoints are b, c, and its distance-two pairs are ac, bd.
Definition 2 (Regularity-selected triples). The regularity-selected triples are obtained in either of two ways:
1. take the special pair e, f together with an endpoint of (3);
2. take one element of {e, f} together with a distance-two pair of (3). Thus
Equation 4 summary: This expression defines a set named S sub P containing six specific triples of characters. These triples are ace, acf, bde, bdf, bef, and cef.
There are exactly six such triples: two of the first type and four of the second. This is a selection rule coming from the regularity graph. It does not by itself say anything about the ranks of the other fourteen triple classes.
3 Four sections on each selected curve
For a Diophantine triple T equals the set containing u, v, and w, we use the standard equivalent model for elliptic curves induced by Diophantine triples,
Equation 5 summary: This expression defines an elliptic curve model induced by a Diophantine triple. The output value squared is equal to the product of three terms, each consisting of an input value added to a pair of elements from the triple.
It is obtained from
Math summary: This expression defines a relationship where a squared output value is set equal to the product of three linear terms. Each term consists of an input value scaled by a specific weight and then increased by one.
by the change of variables
Math summary: This operation performs a change of variables to define two new values. Each new value is calculated by multiplying three distinct rational functions by a corresponding input coordinate.
In the six cases considered below, u, v, w are nonzero and pairwise distinct as rational functions, so (5) is nonsingular over Q of t.
Write
Math summary: This expression defines three variables based on correlation coefficients. It assigns the values r, s, and q to represent the specific correlations between the pairs of variables u and v, u and w, and v and w respectively.
so that
: Math summary: This expression defines three equations as part of the standard Diophantine-triple construction. It determines the squares of the values r, s, and q by adding one to the products of different pairs of the variables u, v, and w.
The standard Diophantine-triple construction gives the rational point
Equation 6 summary: This expression defines the coordinates of a rational point for the standard Diophantine-triple construction. The point is represented as a three-dimensional vector consisting of zero and the product of three variables.
The same construction gives
Equation 7 summary: This expression defines a pair of values denoted as R sub T. The first value is the sum of the products of r and s, r and q, s and q, and the number one, while the second value is the product of the sums of r and s, r and q, and s and q.
Direct substitution using
Math summary: This expression defines three separate equations where squared values are set equal to the product of two variables plus one. Specifically, the square of r equals the product of u and v plus one, the square of s equals the product of u and w plus one, and the square of q equals the product of v and w plus one.
verifies that
Math summary: This expression states that an element called R sub T is a member of a specific set E sub T. This set consists of the rational points of E sub T defined over the field of rational functions in t.
Every fourth element z from the ambient sextuple gives another rational point
Equation 8 summary: This expression defines a rational point designated as E sub z within the set E sub T. The point is represented as a pair consisting of the product of four variables and the product of three variables multiplied by three correlation coefficients.
Indeed, substitution into (5) gives
Math summary: This expression represents an equality between a product of three linear terms and a product of three squared rho values. The first part multiplies three binomials consisting of input values and a common variable, while the second part multiplies the squares of three specific rho terms.
For the six selected triples we use the following four sections.
Table summary: The composition of six different T types, each consisting of four sections. Every T type includes P sub T and R sub T. The remaining two sections vary by type: ace contains E sub d and E sub f; acf contains E sub d and E sub e; bde contains E sub c and E sub f; bdf contains E sub c and E sub e; bef contains E sub c and E sub d; and cef contains E sub a and E sub b.
Proposition 3 (Section identities). For every T in S P, the four sections listed above lie in E T of the Q of t.
Proof. The point P T satisfies (5) immediately.
For R T, substitute (7) into (5) and use
Math summary: This expression defines three separate equations to solve for the variables r squared, s squared, and q squared. Each result is calculated by adding one to the product of two different input variables chosen from u, v, and w.
The resulting residual is zero.
For E z, equation (8) and the identities
Math summary: This expression defines three separate equations for rho values squared. Each result is calculated by adding one to the product of two specific variables, specifically for the pairs u and z, v and z, and w and z.
give the curve equation directly.
Thus all four listed sections have coordinates in Q of t and satisfy 5 identically.
4 Independence from one good specialization
It remains to prove that the four sections chosen for each of the six selected families are independent over the rational functions in t. We do this at one common good specialization.
Lemma 4 (Relation persistence). Let E over the rationals of t be an elliptic curve and let Q 1 through Q m be in E of the rationals of t. Suppose that t 0 in the rationals is a value at which E specializes to a nonsingular elliptic curve and all Q i specialize to finite rational points. If
Math summary: This expression lists a sequence of specialized points on an elliptic curve at a specific value. It represents the specialization of points from one to m at the value t zero.
are independent in E t 0 of the rationals, then
Math summary: This expression defines a sequence of quantities from one to m. It lists these elements to establish their independence within specific mathematical spaces.
are independent in E of the rational functions in t.
Proof. Any integral relation
Math summary: This expression represents an integral relation among generic sections. It calculates a weighted sum of quantities where each section is multiplied by an integer and summed to equal zero.
among the generic sections specializes at a good value t 0 to
Math summary: This expression calculates a weighted sum of several quantities at a specific time point that equals zero. The process combines multiple scaling factors with their corresponding values at time zero to achieve a total output of zero.
If the specialized points are independent, every n i must be zero.
We use the common specialization
: Math summary: This expression defines the value of a specific time constant. It assigns the result of twelve divided by five to this variable.
For each
Math summary: This expression defines the set of possible values for T. The values are listed as six specific three letter combinations: ace, acf, bde, bdf, bef, and cef.
the specialized curve is nonsingular and the four sections listed in Section 3 specialize to finite rational points.
Their exact coordinates and the specialized curves are recorded in Appendix A. Appendix B gives rigorous interval enclosures for the complete 4 times 4 canonical-height Gram matrix of each four-point set. The certification method is described in Appendix C.
For points Q 1 through Q 4, write
: Math summary: This expression defines a matrix named G T. It is constructed by calculating the Néron–Tate pairing for every possible pair of four points from Q one to Q four.
where inner product of dot and dot is the Néron–Tate pairing. The interval-Cholesky certificate in Appendix B has four strictly positive pivot intervals for every selected class. Hence each exact matrix G T is positive definite.
Proposition 5 (Specialized independence). At t=12/5, the four selected points are independent on each of the six curves
Math summary: This expression lists six specific curves identified by unique three letter labels. These labels represent the distinct curves upon which four selected points are independent at a specific time value of twelve fifths.
Proof. For each curve, Appendix B certifies that the exact canonical-height Gram matrix G T is positive definite. Therefore, for every nonzero n in the set of integers to the fourth power,
: Math summary: This expression calculates the canonical height of a sum of four weighted points. It determines that the result is a positive value produced by a quadratic form involving a vector of integers and the canonical height Gram matrix.
Thus no nontrivial integral relation exists among the four specialized points.
By Lemma 4, the corresponding four sections are therefore independent over Q of t.
This argument uses specialization only to rule out relations among the four explicitly constructed sections. It does not require injectivity of the full specialization homomorphism.
5 Main theorem
The regularity graph selects six triple classes from the Piezas sextuple. For each selected class, Section 3 constructs four rational sections, and Section 4 proves their independence by specialization.
Theorem 6 (Regularity-selected generic rank). Let a, b, c, d, e, f be elements of the field of rational functions in t be the Piezas sextuple defined above. For every
: Math summary: This expression defines a set of possible values for the variable T. The set consists of six specific three letter combinations: ace, acf, bde, bdf, bef, and cef.
let E T over Q open parenthesis t close parenthesis be the elliptic curve induced by T. Then
: Math summary: This expression establishes a lower bound for the rank of the elliptic curve induced by T. It states that the rank of this curve over the field of rational functions in t is greater than or equal to four.
More precisely, the four sections listed in Section 3 are independent in E T of the rational functions of t.
Proof. Proposition 3 gives four rational sections on each selected curve. Proposition 5 proves that these four sections specialize to independent rational points at the common good value
Math summary: This expression solves for the variable t. The result is defined as twelve fifths.
By Lemma 4, any integral relation among the generic sections would specialize to the same relation among those four specialized points. No such nontrivial relation exists. Therefore the four generic sections are independent, and
Math summary: This expression establishes a lower bound for the rank of the group of rational points on the elliptic curve E sub T over the field of rational functions. It states that this rank is greater than or equal to four.
for every selected triple T.
Corollary 7. The regularity pattern determined by the special pair {e, f} and the path
: Math summary: This expression calculates the difference between four distinct values. It subtracts the second, third, and fourth values from the first initial value.
selects six induced elliptic-curve families, each with generic rank at least four.
Scope. The theorem proves a generic rank lower bound for the six selected families. It does not prove that their generic rank is exactly four. It gives no rank upper bound for the remaining fourteen triple classes, and it does not assert that the regularity-selection condition is necessary for generic rank at least four.
6 Discussion
The result here is structural. The six rank greater than or equal to 4 families are not identified by a rank search alone. They are selected first by the regularity pattern of the parent sextuple, and the rank lower bound is proved afterward. That order matters.
The three regular quadruples share the pair {e, f} . Removing that pair leaves the path
Math summary: This expression calculates the difference between four variables. It subtracts the values of the second, third, and fourth variables from the first variable.
Its endpoints and distance-two pairs select
: Math summary: This expression lists a set of six specific three letter combinations. These combinations represent the endpoints and distance two pairs selected for the given system.
For each of these six selected triples, the theorem proves four independent rational sections over the rational functions in t.
The height calculation is used only to certify independence at one good specialization. It does not choose the six families. The selection comes from the regularity structure itself. This separates the structural part of the argument from the rank certification step.
This proves the forward direction only. The six selected families have generic rank at least four. The theorem does not show that their generic rank is exactly four, and it does not show that the other fourteen triple classes have generic rank at most three.
It also does not show that the regularity rule is necessary for generic rank at least four. Additional independent sections may exist on either side of the split.
The next test is to leave the Piezas family. If another parametric rational Diophantine sextuple has a different internal regularity pattern, does that pattern select a different rank stratification? If so, then the induced elliptic curves retain information from the parent Diophantine structure that is lost when the curves are studied one at a time. The present theorem proves one concrete case of that mechanism.
Use of language models
Large language models were used extensively for mathematical derivation, literature review, computational cross-checking, and manuscript editing. The research direction, structural ideas, and computational implementation were developed by the author. The author independently reviewed the code and verified the reported computational outputs, but does not claim independent expert verification of every mathematical derivation. All mathematical claims remain subject to independent review.
A Exact specialization data
This appendix records the exact rational data used in the specialization argument of Proposition 5. No floating-point arithmetic enters the construction of the curves or points.
We specialize throughout at
: Math summary: This expression solves for the initial time value. The result is defined as twelve fifths.
At this value the Piezas sextuple becomes
Math summary: This expression defines six specific constant values labeled a through f. These values are assigned as negative fractions and whole numbers, including negative six million eight hundred twenty five thousand eight hundred forty for c.
All fifteen Diophantine square conditions are exact. With the positive square roots chosen at this specialization, we have
Math summary: This expression calculates the square of a specific value labeled rho sub i j. The result is determined by multiplying two input indices and adding one.
with
Table summary: Correlation values rho ij for pairs of variables from a through f. The values vary significantly in form and magnitude, with some being whole numbers like 338 for pair ac and 57,122 for pair cd, while others are expressed as fractions. Examples of fractional values include 57,122 over 57,121 for pair ab and 28,561 over 239 for pair bc.
For a specialized triple T equals the set containing u, v, and w we use the model
Math summary: This expression defines a curve designated as E sub T. The process calculates the square of a vertical coordinate by multiplying the sums of an input value and the product of two distinct elements from a specialized triple.
The four points recorded below are precisely the specializations of the four rational sections used in Theorem 6. Direct substitution verifies each point on its stated curve.
The class ace
The specialized curve is
Math summary: This expression defines the equation for a specialized curve called E ace. It calculates the squared output value by multiplying three binomials consisting of an input value added to one hundred fourteen thousand two hundred forty three, one and point zero two eighty-five, and four hundred eighteen million one hundred sixty-one thousand six hundred.
The four points are
Math summary: This expression defines four specific points labeled P, R, E sub d, and E sub f. Each point is represented as a pair of numerical coordinates consisting of large fractions and integers.
The class acf
Math summary: This expression defines the equation for the class acf curve. It calculates the square of the output value by multiplying the sum of the input and one hundred fourteen thousand two hundred forty three, the sum of the input and three hundred fifty six million four hundred seventy six thousand two hundred forty one divided by three hundred sixty five million five hundred seventy four thousand four hundred, and the sum of the input and three hundred ninety seven million six hundred eighty three thousand three hundred sixty three.
The four points are
Math summary: This expression defines four specific points labeled P, R, E sub d, and E sub e. Each point is represented as a pair of coordinates consisting of precise fractions and whole numbers.
The class bde
The specialized curve is
Math summary: This expression defines the specialized curve known as E bde as an equation for y squared. The output is computed by multiplying three terms, each consisting of the input value x added to a specific constant: fifty seven thousand one hundred twenty three over fifty seven thousand one hundred twenty one, four hundred eighteen million one hundred sixty one thousand six hundred over three billion two hundred sixty two million eight hundred eight thousand six hundred forty one, and one billion six hundred fifty eight million seven hundred ninety four thousand seven hundred ninety seven over fifty six thousand six hundred forty four.
The four points are
Math summary: This expression defines four specific coordinate points labeled P, R, E sub c, and E sub f. Each point is represented as a pair of precise rational numbers consisting of large integer numerators and denominators.
The class bdf
The specialized curve is
Math summary: This expression defines a specialized curve named E bdf. It calculates the square of a value by multiplying three terms, each consisting of an input value added to a specific constant fraction.
The four points are
Math summary: This expression defines the coordinates for four specific points labeled P, R, E sub c, and E sub e. Each point is represented as a pair of precise rational numbers consisting of large integer numerators and denominators.
The class bef
Math summary: This expression defines the equation for the class bef. It calculates a value by multiplying the sum of an input and four hundred eighteen million one hundred sixty thousand one hundred divided by three billion two hundred sixty two million eight hundred eight thousand six hundred forty one, the sum of that same input and three hundred ninety seven million six hundred eighty three thousand three hundred sixty three divided by three billion two hundred sixty two million eight hundred eight thousand six hundred forty one, and the sum of that input and eight hundred fifteen million five hundred two thousand two hundred thirty seven divided by two hundred twenty eight thousand four hundred eighty four.
The four points are
Math summary: This expression defines the coordinates for four distinct points labeled P, R, Ec, and Ed. Each point consists of a pair of precise rational numbers representing its horizontal and vertical positions.
The class cef
The specialized curve is
Math summary: This expression defines the specialized curve E cef as an equation for the square of a vertical coordinate. The vertical coordinate squared equals the product of the horizontal coordinate plus four hundred eighteen million one hundred sixty one thousand six hundred, the horizontal coordinate plus three hundred ninety seven million six hundred eighty three thousand three hundred sixty three, and the horizontal coordinate plus the fraction eighty one million five hundred fifty thousand two hundred twenty three divided by two hundred twenty eight thousand four hundred eighty four.
The four points are
Math summary: This expression defines the coordinates for four specific points named P, R, E sub a, and E sub b. These points are represented as pairs of rational numbers, including values such as zero and large fractions like negative five trillion eight hundred twenty-two billion six hundred eighty-five million nine hundred seventy-two thousand one hundred eighty divided by two hundred thirty-nine.
The data above are exact. In particular, the independence calculation in Appendix B begins from rational curves and rational points; the interval arithmetic enters only when enclosing canonical heights.
B Certified canonical-height Gram data
This appendix records the complete interval Gram enclosures used to prove Proposition 5. The canonical-height pairing is
Math summary: This expression defines the canonical-height pairing between two points. It calculates one half of the difference between the height of the sum of two points and the sum of their individual heights.
For each selected curve, let
Math summary: This expression defines the Gram matrix G sub T. It consists of the inner products between pairs of four selected curves labeled from one to four.
Every entry below is an interval containing the corresponding exact entry of G T . The displayed decimal intervals have been enlarged outward to six decimal places from the intervals used in the proof. Thus the printed intervals remain rigorous enclosures.
The matrices are listed in the exact point order used by the fixed-order interval Cholesky calculation. After each matrix we give the four resulting pivot intervals. Strict positivity of the lower endpoint of every pivot proves positive definiteness of the exact Gram matrix.
The class ace
The point order is
Math summary: This expression defines the Cholesky pivots for a set of point orders. It specifies a four by four matrix named G ace where each entry consists of a numerical range between a minimum and maximum value.
The Cholesky pivots satisfy
Math summary: This expression defines the ranges for the Cholesky pivots. The first two values are between six point seven three one six nine and six point eight two eight zero zero, and one point eight two eight zero zero four and one point eight zero eight nine eight, while the last two are between six point four eight two four one two and six point five seven three six four, and four point four zero nine six three six and four point six four nine zero six.
The class acf
The point order is
Math summary: This expression defines the Cholesky pivots as a four by four matrix of value ranges. Each entry in the matrix consists of a lower and upper bound, such as six point six zero nine two five five to six point six five six four two nine for the first diagonal element.
The Cholesky pivots satisfy
Math summary: This expression defines the intervals for the Cholesky pivots. The first pivot is between six point six zero nine and six point six five six, the second is between one point five nine eight and one point six five two, the third is between six point one three eight and six point two two nine, and the fourth is between four point one five eight and four point three eight three.
The class bde
The point order is
Math summary: This expression defines a set of Cholesky pivots for a specific point order. It provides the G b d e matrix as a collection of four by four intervals, with diagonal values ranging approximately from six point five to six point nine and off-diagonal values spanning from negative two point two to zero point eight.
The Cholesky pivots satisfy
Math summary: This expression defines the ranges for the Cholesky pivots. The first pivot is between six point eight two zero six nine nine and six point eight seven two one nine, the second is between one point eight four one two and one point nine three six eight six two, the third is between six point one four zero four nine and six point two seven five eight four nine, and the fourth is between three point eight seven two four nine one and four point two zero one three seven.
The class bdf
The point order is
Math summary: This expression defines a set of parameters and a four by four matrix for the Cholesky pivots. The matrix contains specific numerical ranges for each entry, such as approximately six point nine three to six point nine eight in the top left cell and negative one point eight eight to negative one point eight zero in the third row of the first column.
The Cholesky pivots satisfy
Math summary: This expression defines the intervals for the Cholesky pivots. The values are specified as p one between six point nine three three eight zero nine and six point nine eight one three four five, p two between one point six six one two one zero and one point seven one two six four five, p three between five point eight four one five nine four and six point zero zero eight three two four, and p four between three point eight nine zero four nine zero and four point two two three one four nine.
The class bef
The point order is
Math summary: This expression defines the Cholesky pivots for a set of points. The result is a four by four matrix of numerical ranges, with diagonal values spanning from six point five six to nine point two six and off-diagonal ranges between negative zero point six and zero point six eight.
The Cholesky pivots satisfy
Math summary: This expression defines the ranges for the Cholesky pivots. Each of the four pivots is constrained within specific intervals, ranging from approximately one point four three to zero point five zero for the second pivot and up to nine point one eight for the fourth pivot.
The class cef
The point order is
Math summary: This expression defines a set of Cholesky pivots represented by a matrix named G cef. Each entry in this four by four matrix consists of a range between two decimal values, such as the top left value ranging from six point six two eight four nine to six point seven zero zero eight nine.
The Cholesky pivots satisfy
Math summary: This expression defines the specific numerical ranges for the Cholesky pivots. The first pivot is between six point six two eight four nine and six point seven zero zero eight nine, the second is between ten point seven zero four one nine seven and ten point seven four seven zero seven nine, the third is between five point eight five three six one nine and five point eight nine seven zero four, and the fourth is between one point two nine four one four two and one point three six five seven six.
Proposition 8 (Verification of specialized independence). For each
Math summary: This expression defines a set of possible values for T. The set contains six specific letter combinations: ace, acf, bde, bdf, bef, and cef.
the four rational points recorded in Appendix A are independent in E T of the rationals at t equals 12 over 5.
Proof. For every T, each lower endpoint of each fixed-order interval-Cholesky pivot is strictly positive. Hence every real symmetric matrix contained in the displayed interval enclosure is positive definite. The exact canonical-height Gram matrix is one such matrix. Therefore, for every nonzero n in Z to the power of 4,
Math summary: This expression calculates the canonical-height of a weighted sum of four points. It shows that the result is computed as a quadratic form using a vector of coefficients and a Gram matrix, resulting in a value strictly greater than zero.
Thus no nontrivial integral relation exists among the four points.
C Canonical-height certification method
This appendix describes the rigorous height enclosure used in Appendix B. The purpose is to identify the numerical step that carries the independence proof and to state the bounds used there.
Juel
Write a specialized curve in generalized Weierstrass form
Math summary: This expression defines a specialized curve in generalized Weierstrass form. It sets a combination of squared and linear output terms scaled by coefficients equal to a cubic input term combined with scaled squared and linear input terms and a constant.
Let b 2, b 4, b 6, b 8, c 4, c 6 denote the standard Weierstrass invariants. The exact change of variables
Math summary: This expression defines a change of variables used to map a curve to the short Weierstrass model. It calculates the new values X and Y by combining input values with constants and the standard Weierstrass invariants through specific scaling factors and additions.
maps E to the short Weierstrass model
Math summary: This expression defines the short Weierstrass model for an elliptic curve. It specifies that the square of the vertical coordinate equals the sum of the cube of the horizontal coordinate, a scaling factor times the horizontal coordinate, and a constant value.
where
Math summary: This expression defines the values for two constants named A and B. It calculates A as negative twenty seven times the fourth coefficient and B as negative fifty four times the sixth coefficient.
This model is used only for the height certificate. It is Q-isomorphic to the original curve.
For a projective x-coordinate (X colon Z), define
Math summary: This expression defines the function h sub x for a projective x coordinate. It calculates the logarithm of the maximum absolute value between the input components x and z.
after replacing X comma Z by a primitive integral pair.
On the short model, duplication of the x-coordinate is represented by the homo-jee-nee-us quartic forms
Equations 9, 10 summary: This computation defines two homogeneous quartic forms, called F and G, which represent the duplication of the x-coordinate on a short model. The functions are calculated using input values and coefficients by combining various powers of the inputs and constants into polynomial sums.
After clearing denominators and removing common content, F and G are taken to be primitive integral forms.
C.2 A uniform duplication-height bound
The height error is controlled by exact Bézout certificates for the two affine charts of the duplication map.
Define
Math summary: This expression defines two functions, f sub x of T and g sub x of T. It assigns the value of function F at one and T to the first function, and the value of function G at one and T to the second function.
and
Math summary: This expression defines two functions, f sub z of T and g sub z of T. These functions are calculated by evaluating the functions F and G using T and the constant one as their inputs.
Thus the x-chart is used when absolute value of X is greater than or equal to absolute value of Z, with T equals Z divided by X, and the z-chart is used when absolute value of Z is greater than or equal to absolute value of X, with T equals X divided by Z.
Exact extended Euclidean calculations give integer polynomials s x, t x, s z, t z and nonzero integers c x, c z satisfying
Math summary: This expression defines two equations based on exact extended Euclidean calculations. For both the x and z components, it computes a linear combination where input polynomials are multiplied by specific coefficients and summed to equal a nonzero integer constant.
Put
Math summary: This expression defines two constants, K sub x and K sub z. Each constant is computed by adding the L one norm of a starting vector to the L one norm of a target vector for the respective x and z dimensions.
and
Math summary: This expression calculates the certification constant. It determines the result by selecting the maximum value among one, the first norm of F, and the first norm of G.
The certification constant is
Equation 11 summary: This expression calculates the certification constant by identifying the maximum value among a set of five candidates. The resulting constant is the largest value chosen from one, the upper bound constant, and three different products involving constants and the absolute values of coordinate components.
All quantities entering (11) are obtained with exact integer arithmetic.
Lemma 9 (Uniform duplication-height bound). For every rational point Q on the short certification model,
Equation 12 summary: This expression defines a uniform duplication height bound for a rational point Q on the short certification model. It states that the absolute difference between four times the height of the point Q and the height of two times the point Q is less than or equal to the logarithm of the constant C E.
Proof. Let (X, Z) be the primitive integral projective x-coordinate of Q, and put
Math summary: This expression calculates the maximum value between two quantities. It determines the result by comparing the absolute values of the primitive integral projective x-coordinate and another value denoted as z.
Write
Math summary: This expression determines the greatest common divisor for two specific duplication forms. It identifies the values for the duplication forms based on input variables and then calculates the greatest common divisor of their absolute values.
The duplication forms have no common projective zero on a smooth elliptic curve, so F Q and G Q are not both zero. The primitive projective x-coordinate of 2.Q is therefore
:Math summary: This expression calculates the primitive projective x-coordinate of two times the point Q. It is expressed as a ratio between the components F sub Q and G sub Q, both of which are scaled by the divisor d.
For the upper bound, the coefficient norms give
Math summary: This expression establishes upper bounds for the absolute values of the quantities F sub Q and G sub Q. Each value is calculated as the product of a corresponding norm—either the norm of F or the norm of G—and the constant M raised to the fourth power.
Since d is greater than or equal to 1,
Math summary: This expression establishes an upper bound for the maximum absolute value of two scaled quantities. It states that the larger of the absolute ratios of the first quantity to the divisor or the second quantity to the divisor is less than or equal to an upper bound constant times the fourth power of a value, which is further bounded by an energy constant times that same fourth power.
Taking logarithms gives
Math summary: This expression defines an upper bound for a specific combination of function values. It states that the value of the function at two times the quantity minus four times the value of the function at that quantity is less than or equal to the logarithm of the constant C sub E.
For the reverse bound, first control the common divisor d. Homogenizing the two Bézout identities gives identities of the form
Math summary: This expression represents one of two homogenized Bézout identities. It computes a weighted sum of two functions by multiplying them by their respective coefficients and setting the result equal to a constant multiplied by the input value raised to the power of an index.
and
Math summary: This expression defines a relationship between two functions through a combination of polynomial multiplications. It sums the product of a polynomial and a function with the product of another polynomial and a second function to equal a constant multiplied by an input value raised to a positive integer power.
for suitable homo-jee-nee-us integer polynomials S x, T x, S z, T z and positive integers N x, N z. Since d divides both F Q and G Q,
Math summary: This expression establishes that a value d divides two separate mathematical terms. It specifies that d is a divisor of a constant multiplier times a variable raised to a positive integer power for both the x and z components.
Because X comma Z is primitive,
Math summary: This expression states that the greatest common divisor of the two values X and Z is equal to one. This computation confirms that the pair X and Z is primitive.
Prime by prime, the two divisibilities therefore imply
Math summary: This expression states that one value divides the product of two other values. Specifically, it describes a condition where the value d is a divisor of the product of the components c sub x and c sub z.
and hence
Math summary: This inequality defines an upper bound for the distance. The distance is less than or equal to the absolute product of the x-chart and z-chart coefficients.
Suppose first that absolute value of X equals M. Then absolute value of Z divided by X is less than or equal to 1. Evaluating the x-chart Bézout identity at T equals Z divided by X gives
Math summary: This expression establishes an upper bound for the absolute value of a coefficient based on the x-chart Bézout identity. It calculates this bound by multiplying the maximum absolute value of two specific functions by the sum of the absolute values of two other functions, all evaluated at the ratio of two sets.
Since absolute value of Z slash X is less than or equal to 1,
: Math summary: This expression establishes an upper bound for the sum of two absolute values related to a ratio. It states that the combined absolute values of the s and t components of the ratio of Z to X must be less than or equal to a constant K sub x.
Using
Math summary: This expression defines the quantities F sub Q and G sub Q. Each is calculated by multiplying a variable X raised to the fourth power by a function of the ratio between Z and X.
we obtain
Math summary: This expression establishes an inequality bound for the product of the absolute value of a coordinate and a constant raised to the fourth power. It states that this value is less than or equal to the product of a constant and the maximum of the absolute values of two quantities named F sub Q and G sub Q.
After division by d and using d less than or equal to absolute value of c x c z ,
Math summary: This expression determines if the maximum of two absolute ratios is greater than or equal to a specific threshold. The process compares the absolute values of the quantities F sub Q and G sub Q each divided by d against a value calculated by dividing M to the fourth power by the product of K sub x and the absolute value of c sub z.
If instead absolute value of Z equals M, the same argument on the reciprocal chart gives
Math summary: This expression determines if the maximum of two absolute ratios is greater than or equal to a specific threshold. It compares the larger of the absolute values of the quantity F sub Q divided by d and G sub Q divided by d against the value of M to the fourth power divided by the product of K sub z and the absolute value of c sub x.
Therefore, in both cases,
Math summary: This expression establishes an upper bound for the fourth power of a value called M. It computes this bound by multiplying the maximum of two scaled coefficients by the maximum of two scaled flux terms divided by a distance.
By (11),
Math summary: This expression states that a specific difference is less than or equal to the logarithm of a constant. It computes the result of subtracting the value of a function at twice the input from four times the function value at the original input.
Combining the two bounds proves (12).
We use the standard Néron–Tate canonical height and associated bilinear pairing. With the normalization used here,
Math summary: This expression defines the Néron Tate canonical height of a point. It is computed as the limit as n approaches infinity of the height of two to the power of n times the point, divided by two times four to the power of n.
Applying Lemma 9 along the duplication tower gives
: Equation 13 summary: This expression computes the absolute difference between an estimated value and a scaled version of a function. The result is constrained to be less than or equal to the logarithm of a constant divided by six times four to the power of n.
Indeed,
Math summary: This expression calculates an upper bound for the absolute difference between a predicted value and a scaled version of a function. The process sums the normalized differences across an infinite sequence, which simplifies to the final result of the logarithm of C E divided by six times four to the power of n.
For all six specialized curves, the maximum in (11) is attained by the same term:
:Math summary: This expression calculates a constant identified as the E constant. It is the result of multiplying a scaling factor K x by the absolute value of the variable c z.
The resulting exact integers are large, ranging from 195 to 275 decimal digits. Their sizes are:
: Table summary: The terms attaining C E for six different T values, all of which share the same term K x|c z|. The number of decimal digits of C E varies by T, with the highest value of 275 for bde and the lowest of 195 for bef. Other values include 265 for ace, 252 for acf, 254 for bdf, and 204 for cef.
The proof uses n = 6 throughout. For reference, the curve constants give the following outward-rounded upper bounds:
Thus, if
Table summary: Upper bounds for log C E and epsilon 6 across six different categories. The highest values are found in bde, with a log C E upper bound of 633.059132 and an epsilon 6 of 0.025760. The lowest values are in bef, with a log C E upper bound of 446.987750 and an epsilon 6 of 0.018188. Other categories include ace, acf, bdf, and cef, with epsilon 6 values ranging from approximately 0.019 to 0.024.
: Math summary: This expression calculates the certified height interval. It divides the height of a scaled input value by two times four raised to the power of the index.
then the certified height interval is
Equation 14 summary: This expression defines the certified height interval as a range between a lower and upper bound. The bounds are calculated by subtracting and adding a scaling factor from a midpoint, where the scaling factor is the logarithm of a constant divided by six times four raised to the power of n.
The lower endpoint may be replaced by zero because the canonical height is nonnegative.
The duplication tower itself is computed in exact rational/projective arithmetic. Floating-point arithmetic is used only to represent interval endpoints, and every elementary interval operation is rounded outward.
C.3 Pairing intervals
For rational points Q i, Q j, polarization gives
Math summary: This expression calculates the inner product of two rational points using the polarization method. It computes this value by taking the difference between the scaled evaluation of the sum of the points and the sum of their individual evaluations, then dividing the result by two.
If
Math summary: This expression defines intervals for two individual input values and their sum. It maps the first value to the range between a sub i and b sub i, the second to the range between a sub j and b sub j, and their combined sum to the range between a sub i j and b sub i j.
then an enclosing interval for the pairing is
Equation 15 summary: This expression defines an enclosing interval for the pairing. It computes two separate values by taking the difference between a matrix entry and the sum of two associated values, then dividing each result by two.
The diagonal entries are the intervals I h (Q i) . Applying and to the four points on each curve produces the six interval matrices displayed in Appendix B.
C.4 Interval Cholesky certificate
Let
Math summary: This expression defines a symmetric interval matrix named G. It specifies that the matrix is composed of individual interval elements denoted by their row and column indices.
be one of the symmetric interval matrices in Appendix B. Fixed-order interval Cholesky is performed with outward-rounded interval arithmetic.
At step i, suppose the preceding interval entries of the lower-triangular factor have been constructed. The next squared diagonal entry is enclosed by
Math summary: This expression calculates the squared diagonal entry for a lower triangular factor at step i. It computes the difference between a diagonal input value and the sum of the squares of all preceding interval entries in that row.
If
Math summary: This expression defines the infimum of the values for D i. It specifies that this lower bound is strictly greater than zero.
then square root of D i is defined and has strictly positive lower endpoint. The off-diagonal entries in column i are then enclosed by
Math summary: This expression calculates the off-diagonal entries for the interval Cholesky certificate. It subtracts the sum of the products of previous column entries from the input value and divides the result by the square root of the pivot interval.
Lemma 10 (Interval Cholesky certificate). Suppose the fixed-order interval Cholesky calculation for a symmetric interval matrix G produces pivot intervals
Math summary: This expression defines a sequence of pivot intervals. It identifies these intervals as the output of a fixed order interval Cholesky calculation for a symmetric interval matrix G.
such that
: Math summary: This expression establishes a condition where the minimum value of a set of terms is greater than zero. This requirement must hold true for every index in the sequence to ensure the matrix is positive definite.
Then every exact real symmetric matrix represented by G is positive definite.
Proof. Let G be an exact real symmetric matrix represented by G . Apply the ordinary Cholesky recursion to G in the same fixed order.
At the first step, the exact first squared diagonal entry lies in D 1 . Since
: Math summary: This expression establishes a lower bound for the first squared diagonal entry. It states that the infimum of this value is greater than zero.
it is positive. Its square root therefore exists and lies in square root of D 1 . Because the lower endpoint of square root of D 1 is strictly positive, the divisions defining the remaining entries in the first Cholesky column are valid. Outward-rounded interval arithmetic encloses all of those exact entries.
Assume inductively that the exact Cholesky entries through column i minus 1 lie in the intervals already constructed. Interval addition, subtraction, multiplication, squaring, square root, and division preserve inclusion whenever the denominator interval excludes zero. It follows that the exact i th squared diagonal entry lies in D i. Since
Math summary: This expression states that the infimum of the interval for the ith squared diagonal entry is greater than zero. This ensures that the diagonal entry is bounded away from zero to allow for the calculation of its square root.
that entry is positive, its square root lies in square root of D i, and square root of D i is bounded away from zero. The exact off-diagonal entries in column i therefore lie in the intervals produced by the stated recursion.
The induction continues through all columns. Every exact Cholesky diagonal entry of G is positive. Hence G admits a Cholesky factorization with positive diagonal and is positive definite. □
The six certificates in Appendix B have strictly positive lower endpoints at all four Cholesky steps. By Lemma 10, the exact canonical-height Gram matrix G T is positive definite for each of the six specialized curves. Therefore
: Math summary: This expression states that the canonical-height Gram matrix is positive definite. It performs a check where the product of a non-zero integer vector, the Gram matrix, and the transpose of that vector results in a value greater than zero to prove the independence of four specialized points.
which proves independence of the four specialized points.
C.5 Logical role of the computation
The certified computation is used only at the specialization
Math summary: This expression assigns a specific numerical value to the variable t. The value is defined as the fraction twelve over five.
The generic section identities are exact identities in Q of t and do not depend on numerical height calculations.
Suppose, for one of the six selected families, that the four generic sections satisfied a nontrivial relation
Math summary: This expression defines a nontrivial relation where a weighted sum of four generic sections equals zero. It specifies that the four coefficients must not all be equal to zero simultaneously.
All four sections and the curve have good specialization at t = 12/5 , so the same relation would hold among the four exact rational points listed in Appendix A. Appendix B proves that those four points are independent. Therefore the assumed nontrivial relation cannot exist, and the four generic sections are independent.
The computation therefore proves the independence required for the generic rank lower bound. It does not compute the exact generic rank, and it does not give a rank upper bound for any of the six families or for the remaining fourteen triple classes.
References
[1] Julián Aguirre, Andrej Dujella, and Juan Carlos Peral. On the rank of elliptic curves coming from rational diophantine triples. Rocky Mountain Journal of Mathematics, 42(6):1759–1776, 2012.
[2] Andrej Dujella. Diophantine triples and construction of high-rank elliptic curves over Q with three non-trivial 2-torsion points. *Rocky Mountain Journal of Mathematics*, 30(1):157–164, 2000.
[3] Andrej Dujella. On mordell-weil groups of elliptic curves induced by diophantine triples. Glasnik Matematički, 42(1):3–18, 2007.
[4] Andrej Dujella and Matija Kazalicki. More on diophantine sextuples. In Number Theory-Diophantine Problems, Uniform Distribution and Applications, pages 227–235. Springer, 2017.
[5] Andrej Dujella, Matija Kazalicki, Miljen Mikić, and Márton Szikszai. There are infinitely many rational diophantine sextuples. International Mathematics Research Notices, 2017(2):490–508, 2017.
[6] Andrej Dujella, Matija Kazalicki, and Vinko Petričević. Rational diophantine sextuples containing two regular quadruples and one regular quintuple. Acta Mathematica Spalatensia, 1:19–27, 2021.
[7] Andrej Dujella and Juan Carlos Peral. High rank elliptic curves induced by rational diophantine triples. Glasnik Matematički, 55(2):237–252, 2020.
[8] Tito Piezas III. Extending rational diophantine triples to sextuples, 2016. MathOverflow answer, March 18, 2016.
[9] Joseph H. Silverman. The Arithmetic of Elliptic Curves, volume 106 of Graduate Texts in Mathematics. Springer, New York, 2 edition, 2009.
[10] Zijian Zeng. The regular neighbourhood of the parametric rational diophantine sextuples, identically in $ \mathbb{Q}(t) $, 2026. Preprint, August 2026.