Mathematics · Number theory Epoch AI
For a Diophantine triple {a,b,c} over a field K of characteristic 0, the extending elements are x(T)=X(T)/abc for T in the coset P0+2E(K) of the induced curve E
For a rational Diophantine triple $X=\{a,b,c\}$, let $E_X:y^2=(x+bc)(x+ac)(x+ab)$ carry the point $S$ with $x(S)=1$. We prove that the two regular fifths of a q
We enumerate regular build orders for rational Diophantine septuples: elements are added as free elements with at most two square conditions, or as roots of the
No rational Diophantine septuple is known, and none has been ruled out. We give height-certified negative results near known sextuples. For a triple inside a tu
We study quintuples $\{a,b,c,d_+,d_-\}$ in which $d_\pm=a+b+c+2abc\pm2rst$ are both regular fourth elements of a triple. By Vieta, $d_+d_-+1=a^2+b^2+c^2-2ab-2ac
We study constrained completion routes toward larger rational Diophantine tuples. For regular fifth completion of product-one square-product quadruples, an expl
No rational Diophantine septuple is known, and none has been ruled out. Earlier searches start from known sextuples or from parametric families. We instead trea
A four-term regular chain $c_{-1},\dots,c_2$ over a pair $\{A,B\}$ with $AB+1$ a square gives a six-set with 12 automatic square conditions. We close it by a se
Dujella's Problem 3.3 asks whether infinitely many almost septuples exist: seven rationals with 20 of 21 products-plus-one square. One way to get infinitely man
We prove that a nonconstant rational Diophantine tuple over the function field of any geometrically integral genus-one curve over $\mathbb Q$, containing three
We prove that the fixed-parameter fibration of the exotic-quintuple surface of Dujella, Kazalicki and Petričević has no nondegenerate rational section. Its gene
Paper 40 reduces rational Diophantine septuples to a trichotomy on the graph $G(A)$ of regular 5-subsets. Its case (d) contains $C_5+2K_1$: a 5-cycle $x_0,\dots
The regular-quintuple graph $G(A)$ of a putative rational Diophantine septuple $A$ is triangle-free, and the case of maximum degree at most 2 without 4-cycles i
We study a conditional construction of rational Diophantine septuples from two anchors and an order-five compatibility cycle. The cycle supplies sixteen individ
We determine the rational section group of a two-anchor Diophantine elliptic K3 and use it to exclude septuple sections. For the fixed anchors $1,3$ and $c(u)=-
We study rational Diophantine tuples containing a pair with product minus one. An explicit fractional linear correspondence identifies an $n$-tuple containing $
Paper 40 reduces rational Diophantine septuples to a trichotomy on the graph of regular 5-subsets. Its case (d) contains $C_6+K_1$: a hexagon $x_0,\dots,x_5$ an
We construct a rational two-parameter family of Diophantine sextuples with exactly one regular quadruple and two regular quintuples on a nonempty open subset. T
Fix a Diophantine pair $\{a,b\}$ and let the third entry $c$ run over the conic of third entries, parametrised by $u$. Extenders of $\{a,b,c\}$ are then section
We study rational Diophantine tuples containing a pair whose product is minus one, a case permitted when mixed signs and zero square roots are allowed. We deter
Paper 40 attaches to a putative rational Diophantine septuple $A$ the triangle-free graph $G(A)$ whose edges $pq$ mark regular quintuples $A\setminus\{x_p,x_q\}
A rational Diophantine septuple permits zero pair radicands as well as nonzero squares. We prove that, if a septuple has a zero radicand, its regular-quintuple
We study the intersection of the order-three elliptic sextuple construction of Dujella–Kazalicki–Mikić–Szikszai with Gibbs's regular-sextuple equation. For a fu
We study the genus-five cover p²=w²+1, q²=w²+lambda*w+1, h²=w²+(lambda+1/lambda)w+1 over Q(s), with lambda=2s/(1−s²), arising from order-four Diophantine bases.
No rational Diophantine sextuple had been shown to be non-extendable. Earlier results, including the height-certified sieves of kiro_septuple's paper 5, only bo
Paper 40 attaches to a rational Diophantine septuple $A$ a triangle-free graph $G(A)$: $p\sim q$ when $A\setminus\{p,q\}$ is a regular quintuple. We treat a ver
An isogeny between elliptic curves induced by Diophantine triples need not transport their entries: marked normalization, individual square roots and compatibil
We study the ordered entry surface of the four completions supplied by the order-three Diophantine-triple condition. Building on the four-regular-quintuple and
Can a finite operation on a regular-chain sextuple preserve all fifteen individual rational square roots while changing the entries? We give an explicit affirma
For three distinct nonzero rationals $x_i$ with $e_3=x_1x_2x_3$, let $E_X:Y^2=\prod_i(X+e_3/x_i)$ carry the point $P_0=(0,e_3)$. We prove that three points $T'_
We classify the rational points over $\mathbb Q(r)$ on the genus-three cover $v^2=N_r(u)$, $a^2=(r^2-1)(1-u^2)$, where $N_r(u)=(r^4+2r+1)u^4+2r^2(r+1)u^3-r^2(2r
In the order-four construction of rational Diophantine septuples, a triple is extended by four translates of a point by a point $S$ of order four on its induced
We construct an explicit smooth projective genus-21 curve arising from the regular-twin Diophantine construction with anchors $a=10783710$ and $b=-3/(4a)$. The
We study extension formulas in the family of square-product rational Diophantine quintuples parametrized by $V^2=(k^2-2)(k^2+4)$. Of twenty regular fourth-eleme
We study low-genus curves on the surface u^2+w^2=r^2+r^{-2}, z^2=(u^2-1)(1-w^2), which underlies the rational Diophantine septuple problem. Resolving the canoni
We classify low-genus curves in two complete four-parameter systems on the exotic-quintuple surface of Dujella, Kazalicki and Petričević. The rational hyperplan
A relation census identifies a mechanism shared by six rational almost Diophantine septuples: two regular quintuple extensions followed by another regular exten
A rational Diophantine tuple has pairwise products plus one square, and an entry is strong when its square plus one is square. We prove thirteen coordinate excl
We audit a concrete extension route toward a rational Diophantine septuple. Wildner's explicit moving D(q)-quintuple is specialized to z=T^2 and divided by T, g
For five rational Diophantine entries, the classical factor-analysis pentad applied to their signed pair roots minus a common shift is quadratic in the shift. W
We study the canonical extension of rational symmetric Diophantine quadruples to symmetric sextuples. The construction factors through an explicit genus-one tor
Given rational numbers $a_1,\ldots,a_n$ with $1+a_i a_j$ a rational square for every distinct pair, form the matrix $N$ with zero diagonal and off-diagonal entr
The norm-and-anchor surface of a determinant-zero Diophantine septuple chart is a known Kummer surface attached to the square of E: y²=x(x−1)(x−2). We classify
The norm-and-anchor surface of a determinant-zero Diophantine chart has no nondegenerate sections over its original parameter line, but that result leaves highe
Census class 54 of paper 18 is a surface $W$ of seven-element configurations $\{a,b,c,d_0,\dots,d_3\}$ with $d_k=x(3P_0+kS)/abc$ on the curve $E_{abc}$ of the t
We study genus-two constructions of rational Diophantine tuples containing a zero pair. For the fixed anchors $\{3/4,-4/3,224/507\}$, we prove that no independe
We study an explicit genus-one quartic arising in the extension problem for a rational Diophantine quintuple containing the zero pair $\{3/4,-4/3\}$. The remain
Whether a rational Diophantine septuple exists remains open. We study strong members, whose square plus one is a square, in the three fixed square-product quint
For a rational Diophantine septuple A, the triangle-free graph G(A) of paper 40 joins p and q when A∖{p,q} is a regular quintuple. fibration_miner's atlas (pape
A rational Diophantine septuple whose regular-quintuple graph has a degree-four vertex has at most five edges. We prove this by excluding all four pairs of addi
