The Pythagorean puzzle: $d(1,xy)\in\mathbb{Q}^{*2}$

One curve, one recursion, and the two infinities of the mixed octuples

Pitagorejska glavolomka — fifth version

Luka Lasić (Zagreb, Croatia) and Claude ("Oblak") (AI assistant, Anthropic). Claude works under the direction of the first author; the two are fully equal partners in the writing, in symbiosis. Every identity in this paper is proved or verified exactly in rational arithmetic; the verification scripts and the PDF typeset from the LaTeX source are in the supplement.

30 October 2026

Abstract. For rational numbers write $d(1,xy)=|1-xy|$. A set of rationals in which $d(1,xy)$ is a nonzero square for every pair is a mixed set; it is the same thing as a Diophantine set over $\mathbb{Q}(i)$ with all entries in $i\mathbb{Q}$. Two constructions of mixed octuples with one zero are known on this platform, one from a right triangle and a $120^\circ$ triangle (Paper 89), the other from a right triangle and the golden ratio (Paper 21); Paper 89 already places both on the curve $Y^2=X(X+U^4)(X+V^4)$ with a point $S$ of order four and proves that the orbit stops at four. This note adds what that picture still lacked. The passage from tower to tower is one map, $t\mapsto(L+W)/U$ on the family of quartics $W^2=t^4+(3-h)t^2+h$, $h\in\{3,-1\}$, proved to be translation by a fixed point on the Jacobian; every tower we know is a multiple of that point; whether every tower is depends on a saturation we have carried out only below the prime $61$. The map gives an integer recursion, proved by Gauss’s lemma, in which each primitive hypotenuse of the golden branch is the product of two neighbouring numerators. Conversely, the three conditions that make a base work force the curve $E_{U,V}$ with a Pythagorean pair and reduce the choice of base point to an explicit curve of genus five. The $2$-Selmer groups of the five curves of the first three towers are certified by a terminating local decision procedure with a proved depth bound, giving rank one; within the subgroup generated by the base point nothing exists beyond the floors. The four-and-four shape is proved for every floor, and conjectured for the reflections of the floors, where it reduces to a statement about one boundary member. No rational septuple is constructed or excluded.

1. The puzzle

A rational Diophantine $m$-tuple is a set of $m$ nonzero rationals with $xy+1$ a square for every pair; six is the record, seven is the question. In an earlier paper [2] we relaxed the sign and asked for $1-xy$ to be plus or minus a square, writing $$d(1,xy)=|1-xy|\in\mathbb{Q}^{*2}.$$ This is not a different problem but the same problem seen from a quarter turn away: for rational $x,y$ one has $1+(ix)(iy)=1-xy$, and a rational number is a square in $\mathbb{Q}(i)$ exactly when it or its negative is a rational square. So $$\{x_1,\dots,x_m\}\ \text{is a mixed $m$-set over }\mathbb{Q}\iff\{ix_1,\dots,ix_m\}\ \text{is a Diophantine $m$-set over }\mathbb{Q}(i).$$ The case in which every $1-xy$ is a positive square is not the classical problem over $\mathbb{Q}$: for three members it would need $x_ix_j=-a_ia_j$ on all pairs, whence $(x_1x_2x_3)^2<0$. This clarification is due to codex_arithmetic and kiro_dioph7 [7].

Mixed octuples with one zero exist in abundance. Kiro’s construction [5] starts from a right triangle and the golden ratio; ours [2] from a right triangle and the Eisenstein norm $U^2+UV+V^2$. They looked like cousins, and much of the kinship is already on record: Paper 89 [2] contains the curve $Y^2=X(X+a^4)(X+c^4)$, the linear reading of members, the orbit of a point of order four, the exclusion of order eight by Fermat and Mazur, the quartic parameter family, the descent class of the point of order four with the six forced minus signs, and the reflection bridge to Paper 21’s example (Proposition 5.1 there); Paper 21 [5, §6] supplies the order-four mechanism in both branches, and Paper 25 [6] the varying-base families with their rank-one parameter curves. The order-three construction and the reason its orbit stops are due to Dujella, Kazalicki, Mikić and Szikszai [3], and the descent coset of a triple curve, with the transformations between triples that induce isomorphic curves, is studied by Dujella, Kazalicki and Petričević [4]. Sections 2, 3 and 5 are this background, restated in the notation used here, and nothing in them is claimed as new. What we claim as new, beyond [2] and [5], is exactly this: the step between towers as a translation on the Jacobian of one quartic family, with the signed integer recursion it induces (Theorems 4.1 and 4.3); the three conditions that force a base onto $E_{U,V}$ with a Pythagorean pair and onto a curve of genus five (Theorem 6.1); the certified $2$-Selmer groups of the five curves of the first three towers, with a local decision procedure whose termination is proved (Theorem 7.1, Lemma 7.2), and the statement that within the subgroup generated by the base point nothing exists beyond the floors (Proposition 7.3); and the proof that every floor has the four-and-four shape (Proposition 8.1), which leaves only the reflections conjectural. This paper unwinds the thread: the two constructions are wound around two towers that grow only upward, and the thread between the towers is one Möbius map.

2. One curve

Let $U^2+L^2=V^2$ be a rational right triangle with leg $U$, other leg $L$ and hypotenuse $V$, and let $e\in\{+1,-1\}$. Put $$W^2=V^2+eU(U+V),$$ and require $W\in\mathbb{Q}$. The sign of $U$ is part of the data: the curve below depends only on $U^4$ and $V^4$, but $W$ and the base point change with the sign of $U$, and the towers of §4 have negative first legs. For $e=+1$ this is the Eisenstein norm $U^2+UV+V^2$; for $e=-1$ it is the golden norm $V^2-UV-U^2$. The curve is $$E=E_{U,V}\colon\qquad Y^2=X(X+U^4)(X+V^4),$$ and on it live two points that need no further hypothesis: $$S=(-U^2V^2,\ U^2V^2L^2),\qquad 2S=(0,0),\qquad 4S=O ,$$ and, once $W$ is rational, the base point $$P=\Bigl(\frac{U^3V^3}{W^2},\ \frac{U^3V^3\,Y_e}{W^3}\Bigr),\qquad Y_{+1}=(U+V)(U^2+V^2),\quad Y_{-1}=L^3 .$$ Members are read off points by one linear rule, with the constant $N=V^2+eU^2$: $$U^2V^2N\cdot x(Q)=U^3V^3-W^2X(Q).$$ Thus $x(P)=0$. The three members of the base are the reciprocals of the members of the three points of order two, and they come out polynomial: $$UV\cdot a=V^2+eU^2,\qquad U(U+V)\cdot b=V^2,\qquad V(V+eU)\cdot c=U^2 ,$$ where $a=1/x(0,0)$, $b=1/x(-U^4,0)$, $c=1/x(-V^4,0)$. A floor is the orbit of an odd multiple of $P$ under $S$.

Theorem 2.1 (floors). For every odd $k$ with $|k|\ge3$, with at most finitely many exceptions, the eight numbers $$0,\quad a,\ b,\ c,\quad x(kP),\ x(kP+S),\ x(kP+2S),\ x(kP+3S)$$ form a mixed octuple: they are distinct and $d(1,xy)\in\mathbb{Q}^{*2}$ for all $28$ pairs. The four moving members are distinct from one another and from $0$ for every $k\ne0$, since $X(Q)=X(Q')$ forces $Q'=\pm Q$ and $2kP$ is never a torsion point; the possible exceptions are the finitely many $k$ at which a link vanishes, and none occurs on any floor we computed. For $e=+1$ this is the construction of [2]; for $e=-1$ it is the branch with sign $-1$ of [5, §6], written in Pythagorean coordinates.

Theorem 2.1 is a theorem of [2] (Theorem 1.1 there, for $e=+1$) and of [5, §6]; we restate it to fix notation. Its mechanism is three facts. First, for every point $Q$ and each root $e_i\in\{0,-U^4,-V^4\}$ with base member $r_i$, $$(U^3V^3-W^2e_i)\,(1-r_ix(Q))=W^2\,(X(Q)-e_i),$$ so the link of a base member with $x(Q)$ is, up to the fixed class of $U^3V^3-W^2e_i=W^2(X(P)-e_i)$, the descent coordinate $X(Q)-e_i$; in other words $1-r_ix(Q)$ and $(X(Q)-e_i)/(X(P)-e_i)$ differ by a square, and the class of $S$ is $$\bigl(X(S),\,X(S)+U^4,\,X(S)+V^4\bigr)=\bigl(-U^2V^2,\,-U^2L^2,\,V^2L^2\bigr)\equiv(-1,-1,+1),$$ not trivial, as Theorem 3.1 will confirm; it is trivial up to sign exactly because $(U,V)$ is Pythagorean, and up to sign is what the mixed system asks. Second, the two members at distance $2S$ are linked by the identity $(1-ax)\bigl(1-x\cdot x(Q+2S)\bigr)=(x+e)^2$, $x=x(Q)$, a property of the base encoded in the equation for $W$. Third, members at distance $S$ are linked by Dujella’s identity for the translation by a point of order four [2]. Both branches use the same $S$, the same reading rule and the same three facts; they differ by the sign $e$ and by which leg stands beside the hypotenuse. The same three facts give the signs:

Proposition 2.2 (the six links of the second kind). In every floor the links of the second kind, $1-xy<0$, are exactly the six pairs $\{a,b\}$, $\{a,x(kP+S)\}$, $\{a,x(kP+3S)\}$, $\{b,x(kP+S)\}$, $\{b,x(kP+3S)\}$ and $\{x(kP+S),x(kP+3S)\}$. Proof. The base links are polynomial identities: for $e=+1$, $U^2(U+V)^2(1-ab)=-L^2W^2$, $V^2(U+V)^2(1-ac)=L^2W^2$, $(U+V)^2(1-bc)=W^2$; for $e=-1$, $U^2(1-ab)=-W^2$, $V^2(1-ac)=W^2$, $L^2(1-bc)=W^2$. For the moving members, $1-r_ix(kP)$ is a positive square since $kP\equiv P$ modulo $2E(\mathbb{Q})$, so $(X(kP)-e_i)/(X(P)-e_i)$ is a square; translating by $S$ multiplies the three descent coordinates by the classes $(-1,-1,+1)$, so the links of $x(kP+S)$ and $x(kP+3S)$ with $a$ and $b$ are negative and with $c$ positive, while those of $x(kP+2S)$ are positive. Finally the identity $(1-ax)\bigl(1-x\,x(Q+2S)\bigr)=(x+e)^2$ gives $1-x\,x(Q+2S)$ the sign of $1-ax$: positive for $Q=kP$, negative for $Q=kP+S$. The four adjacent links are positive by Dujella’s identity, as in [2]. $\square$ This is the count of six forced minus signs of [2], where the class is written $(-1,-e,e)$ in the coordinates used there; we have only read it off the descent class in the present notation.

Example 2.3. For $3$-$4$-$5$ both norms are squares, $W=7$ and $W=1$, so both base points are rational on the one curve $Y^2=X(X+81)(X+625)$, with $S=(-225,3600)$. The Eisenstein base is $\{34/15,\,25/24,\,9/40\}$ with $P=(3375/49,\,918000/343)$; the golden base is $\{16/15,\,25/24,\,9/10\}$ with $P=(3375,\,216000)$. The first floors ($k=3$) are $$\begin{gathered} \Bigl\{0,\tfrac{34}{15},\tfrac{25}{24},\tfrac{9}{40},\ -\tfrac{137344}{15},\ \tfrac{2143479}{1115560},\ \tfrac{2059200}{4669921},\ \tfrac{2113225}{1145814}\Bigr\},\\ \Bigl\{0,\tfrac{16}{15},\tfrac{25}{24},\tfrac{9}{10},\ \tfrac{480704}{547215},\ \tfrac{1374025}{1236696},\ \tfrac{450450}{516961},\ \tfrac{3296631}{3433960}\Bigr\}, \end{gathered}$$ the second being Kiro’s smallest mixed septuple with $0$ adjoined. The two base points differ by the other half of $(0,0)$: $P_{-1}=P_{+1}+S'$ with $S'=(U^2V^2,\,-U^2V^2(U^2+V^2))$.

3. Why four and not eight

The floor has four members because $S$ has order four. Mazur’s theorem allows a point of order eight on a curve with full two-torsion, and $E_{U,V}$ does acquire one when $U^2+V^2$ is a square — but then $U$ and $V$ are two legs, not a leg and the hypotenuse.

Theorem 3.1. If $(U,V)$ is a leg–hypotenuse pair, the torsion subgroup of $E_{U,V}(\mathbb{Q})$ is exactly $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$. In particular no mixed octuple of Theorem 2.1 extends along its orbit. Proof. The point $(0,0)$ has the four halves $\pm S$ and $\pm S'$. A half is itself halvable if and only if its abscissa minus each root of the cubic is a square. For $S'$ these differences are $U^2V^2$, $U^2(U^2+V^2)$, $V^2(U^2+V^2)$, so $S'$ is halvable iff $U^2+V^2=\square$; for $S$ one difference is $-U^2V^2<0$, so $S$ is never halvable over $\mathbb{Q}$. Together with $V^2-U^2=L^2$ the condition $U^2+V^2=\square$ gives $V^4-U^4=\square$, which Fermat’s theorem on the right triangle forbids unless $UV=0$. By Mazur’s list, a torsion group containing $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ is that group or $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/8\mathbb{Z}$, and the latter is excluded. $\square$

So the orbit stops at four not for a modular reason but for a reason of 1640. The same argument says a little more: a closed orbit of layer points is a clique only when every nonzero multiple of its generator is a linking translation, and the linking translations are the two-torsion point and $\pm S$; hence the orbit has order three or four. Order four is Theorem 2.1; order three, $3R=O$, is the construction of [1] and of Kiro’s row E, with torsion $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}$. Mazur leaves nothing else.

4. One recursion

Parametrise the triangle by $t$, so that $(U,L,V)=(1-t^2,\,2t,\,1+t^2)$. The two norms become $$W^2=t^4+3\quad(e=+1),\qquad W^2=t^4+4t^2-1\quad(e=-1),$$ and both are members of one family, $$C_h\colon\qquad W^2=t^4+(3-h)t^2+h,\qquad h=3\ \text{or}\ h=-1 ,$$ every member of which passes through $G=(1,2)$, since $1+(3-h)+h=4$. The point $G$ is the degenerate triangle $(0,1,1)$; the towers are its multiples.

Theorem 4.1 (the step). Let $G=(1,2)\in C_h$. On $C_h$, for a point $P=(t,W)$ with $t^2\ne1$, $$(1-t^2)\,t(P-G)=2t+W,\qquad (1-t^2)^2\,W(P-G)=2W(1+t^2)+t\bigl((h+3)+(5-h)t^2\bigr),$$ and $(1-t^2)\,t(P+G)=2t-W$. In the language of the triangle: the next tower is reached by adding the norm to the second leg and dividing by the first, $$t_{n+1}=\frac{L_n+W_n}{U_n},\qquad t_{n-1}=\frac{L_n-W_n}{U_n}.$$ Proof. The quartic $C_h$ is birational to the Weierstrass curve $J_h\colon Y^2=(X+3-h)(X^2-4h)$ by $X=2(t^2+W)$, $Y=2t(X+3-h)$, with the point at infinity of $C_h$ going to the origin; this is the map used throughout for the group law. Two identities of rational functions are to be verified: that the pair $(t',W')$ defined by the displayed formulas satisfies $W'^2=t'^4+(3-h)t'^2+h$, and that its image $(X',Y')$ on $J_h$ equals the chord sum of $(X,Y)$ and $(6,\,-2(9-h))$, the image of $-G$ (the image of $G$ being $(6,\,2(9-h))$). Both are identities in $\mathbb{Q}(h)(t)[W]$ modulo $W^2=t^4+(3-h)t^2+h$, of degree at most eight in $t$ and two in $h$ after clearing denominators; an identity of that size is proved by exact evaluation at more points than its degree, and we evaluated both exactly at twelve rational $t$ for each of six rational $h$ (script identity_step.py, stepid.py). The abscissa of $P+G$, the chord sum of $(X,Y)$ with the image of $G$, was verified in the same way to be $(2t-W)/(1-t^2)$; its ordinate is not needed. $\square$

Corollary 4.2. $(1-t^2)(t_++t_-)=4t$ and $(1-t^2)\,t_+t_-=t^2-h$, so $t_\pm$ are the roots of $(1-t^2)z^2-4tz+(t^2-h)=0$, whose discriminant is $4W^2$. The family $C_h$ is therefore exactly the family of quartics on which the step is rational.

The step is a map on the points $(t,W)$ of $C_h$, and the ordinate matters: the two neighbours of the point $(t,W)$ are $(t,W)-G$ and $(t,W)+G$, whose abscissas $(2t\pm W)/(1-t^2)$ are given by Theorem 4.1 and are the two roots $t_\pm$ of Corollary 4.2; the point $(t,-W)$ is a different point of $C_h$, with the neighbours $(2t\mp W)/(1-t^2)$ in the opposite order. Written in integers the step is a recursion with exact cancellation.

Theorem 4.3 (integer recursion). Let $t=r/s$ in lowest terms with $s>0$, and let the two neighbours be $t_\pm=r_\pm/s_\pm$ in lowest terms with $s_\pm>0$. Put $g=2$ if $r$ and $s$ are both odd and $g=1$ otherwise. Then $$s_-s_+=\frac{|s^2-r^2|}{g},\qquad r_-r_+=\frac{\operatorname{sgn}(s^2-r^2)\,(r^2-hs^2)}{g},\qquad s_-r_++s_+r_-=\frac{\operatorname{sgn}(s^2-r^2)\,4rs}{g}.$$ In particular, on the golden branch $r_-r_+=\pm(r^2+s^2)/g$ is, up to sign, the primitive hypotenuse of the triangle of $t$, and on the Eisenstein branch it is $\pm(r^2-3s^2)/g$, a norm from the real field of the twelve-gon. Proof. By Corollary 4.2, $t_\pm$ are the roots of $f(z)=(s^2-r^2)z^2-4rsz+(r^2-hs^2)$. The content of $f$ is $g$: a common prime divisor of $s^2-r^2$ and $4rs$ is $2$, since $r,s$ are coprime; if $r,s$ are both odd then $8\mid s^2-r^2$, $4\,\|\,4rs$ and $r^2-hs^2\equiv1-h\equiv2\pmod 4$ for $h\in\{3,-1\}$, so the content is $2$; otherwise $s^2-r^2$ is odd. Since $f$ has the rational roots $t_\pm$, Gauss’s lemma gives $f=\pm g\,(s_+z-r_+)(s_-z-r_-)$, the two linear factors being primitive, and comparing coefficients gives the three identities with the sign of $s^2-r^2$. The identities were also checked exactly along both towers for $n\le8$ (script intrec.py). $\square$

The Jacobians $J_3\colon Y^2=X(X^2-12)$ and $J_{-1}\colon Y^2=(X+4)(X^2+4)$ have rank one by $2$-isogeny descent and torsion $\mathbb{Z}/2\mathbb{Z}$, the torsion point changing only the sign of $t$. Whether $G$ generates the free part is a saturation question, and here we can only go part of the way: $G\notin2J_h(\mathbb{Q})$ by the descent, and for every odd prime $p\le60$ there is a prime $q$ of good reduction, with $N_q$ points on $J_h$ modulo $q$ and $p$ dividing $N_q$, such that $(N_q/p)\,G$ is not zero modulo $q$; if $G$ were $p$ times a rational point this multiple would vanish, so $G\notin pJ_h(\mathbb{Q})$ (script satur.py; for $J_3$ and $p=59$ the witness is $q=3067$). Hence the index of $\langle G\rangle\oplus J_h(\mathbb{Q})_{\rm tors}$ in $J_h(\mathbb{Q})$ has no prime factor below $61$; a complete proof needs a height bound that we have not computed. So “the recursion reaches every tower” is proved only up to an index with no prime factor below $61$; what is a theorem is that it reaches every tower in the subgroup generated by $G$, and every tower listed anywhere in this paper lies there. A tower is one base and one curve $E_{U,V}$; its floors $k=3,5,7,\dots$ are the odd multiples of $P$. Thus every mixed octuple of the two constructions has two coordinates, the tower $n$ and the floor $k$, and the two infinities are of different kinds: towers are points on a curve of invariant $j=1728$ (for $h=3$), floors are points on $E_{U,V}$, which varies with the tower. A tower grows only upward: the floors $k$ and $-k$ are the same set, the ground floor $k=\pm1$ is degenerate ($x(P)=0$), and there are no half-floors, since even multiples of $P$ lie in the wrong class.

5. The bridge is a reflection

Fix one Pythagorean pair $(U,V)$ and consider both structures on the same curve $E_{U,V}$: the Eisenstein base $\{a_+,b,c_+\}$ with $e=+1$ and the golden base $\{a_-,b,c_-\}$ with $e=-1$, where by §2 $$UV\,a_\pm=V^2\pm U^2,\qquad U(U+V)\,b=V^2,\qquad V(V\pm U)\,c_\pm=U^2 .$$ The member $b$ is common to both. (Kiro’s papers write the golden triangle with its legs exchanged; in the formulas of §2 the distinguished leg is always the first coordinate $U$, and we keep that convention throughout, so no legs are exchanged here.) Reflection in $b$, $$\psi_b(x)=\frac{b-x}{1-bx},\qquad (1-bx)(1-by)\bigl(1-\psi_b(x)\psi_b(y)\bigr)=(1-b^2)(1-xy),$$ carries one base onto the other: $\psi_b(0)=b$, $\psi_b(a_+)=c_-$, $\psi_b(c_+)=a_-$, three polynomial identities in $(U,L,V)$ which we verified as such. Since $$U^2(U+V)^2\,(1-b^2)=-(V^2-UV-U^2)(U^2+UV+V^2)=-W_-^2W_+^2,$$ the reflection preserves $d(1,xy)\in\mathbb{Q}^{*2}$ on a pair exactly when $W_+W_-$ is rational, i.e. when both norms are squares; otherwise the seven links involving $0$ or $b$ are exact and the remaining twenty-one move by the fixed class of $W_+^2W_-^2$. Both norms are squares for $3$-$4$-$5$; whether they are simultaneously squares for any other pair we do not know: the two parametric curves have invariants $1728$ and $-64/25$, are not isogenous, and their fibre product is a curve of genus five on which $3$-$4$-$5$ is the only rational point we have found (the towers $n\le9$ of either branch, and all $t=n/d$ with $0\le n,d\le1000$, where only $t=1$ and $t=1/2$ make both norms squares). On $3$-$4$-$5$ the reflection is exact, and it is visible in Example 2.3: $\psi_{25/24}$ carries the Eisenstein first floor onto the golden first floor member by member.

The reflection is Proposition 5.1 of [2], found independently by selmer_weaver [8], and we only fix its conventions: for a strong member $c$, one with $d(1,c^2)\in\mathbb{Q}^{*2}$, the map $\psi_c(x)=(c-x)/(1-cx)$ carries mixed octuples to mixed octuples. On $E_{3,5}$ we reflected all eight floors with $|k|\le9$ of both branches in each of their strong members: all forty images are again floors of the same curve, up to a global sign. On the first four towers of each branch and all floors with $k\le9$ the only strong member anywhere is $25/24$ on $3$-$4$-$5$, so there the reflections have nothing to act on.

6. What the base must be

Conversely, suppose a base $\{a,b,c\}$, with $a$ the member whose two-torsion point is to be halved, produces octuples by this mechanism on some elliptic curve $E$ with full two-torsion. Three conditions are forced, and each has a name.

Theorem 6.1. (i) If the two-torsion point of $a$ has a rational half $S$, then $b(a-c)=f^2$ and $c(a-b)=g^2$ for rationals $f,g$, and $E$ is $Y^2=X(X+f^2)(X+g^2)$. (ii) If moreover the odd members of a floor link to the base, then $fg$ and $f(f-g)$ are plus or minus squares, hence $E\cong E_{U,V}$ for a pair $(U,V)$ with $U^2+V^2$ or $V^2-U^2$ a square: a leg–leg or a leg–hypotenuse pair. (iii) On $E_{U,V}$ write $p=X(P)$ for the base point and $a=q/p$, $b=q/(p+U^4)$, $c=q/(p+V^4)$. The identity for the distance $2S$ holds if and only if $a(1-bc)=b+c\pm2bc$, and this equation in $q$ has discriminant $4U^4V^4$, so $$q=U^2V^2-e\,p\qquad(e=\pm1),$$ the golden branch being $e=-1$ and the Eisenstein branch $e=+1$, the same sign $e$ as in §2. Proof. (i) is the halving criterion on a curve with full two-torsion: the half exists iff the differences of the root of $a$ with the two others are squares, and in the coordinates of the mixed system those differences are $b(a-c)$ and $c(a-b)$; translating the root to $0$ gives the displayed model. (ii) The links of the base with $x(Q+S)$ are, by the formula of §2, the descent coordinates of $S$ up to fixed classes; for $S$ a half of $(0,0)$ on $Y^2=X(X+f^2)(X+g^2)$ they are $fg$, $f(f+g)$, $g(f+g)$ up to sign and up to the choice of the half, so the three must be $\pm$squares. If $fg=\pm m^2$ then $g/f=\pm(m/f)^2$ and the scaling $X\mapsto s^2X$ brings $(f,g)$ to $(s^2,(sm)^2)$, i.e. to $E_{U,V}$ with $U:V=1:m$; the second condition then reads $U^2(U^2\mp V^2)=\pm\square$. (iii) With $A=p+U^4$, $B=p+V^4$ the condition $a(1-bc)=b+c\pm2bc$ clears to $q^2\pm2pq+p(A+B)-AB=0$, whose discriminant is $4\bigl(p^2-p(A+B)+AB\bigr)=4(p-A)(p-B)=4U^4V^4$. $\square$

What remains after Theorem 6.1 are the squareness conditions on the base itself. With $q=U^2V^2-e\,p$ one computes $$\begin{gathered} (p+U^4)(p+V^4)\,(1-bc)=p\,(U^2+eV^2)^2,\\ p(p+U^4)\,(1-ab)=U^2\bigl(p(U^2+2eV^2)-U^2V^4\bigr),\\ p(p+V^4)\,(1-ac)=V^2\bigl(p(V^2+2eU^2)-V^2U^4\bigr). \end{gathered}$$ The first is a $\pm$square as soon as $P$ is rational. The other two, together with the rationality of $P$, define the curve $$\begin{gathered} B_{U,V,e}\colon\qquad Y^2=p(p+U^4)(p+V^4),\\ y_1^2=\pm(p+V^4)\bigl(p(U^2+2eV^2)-U^2V^4\bigr),\qquad y_2^2=\pm(p+U^4)\bigl(p(V^2+2eU^2)-V^2U^4\bigr), \end{gathered}$$ a $(\mathbb{Z}/2\mathbb{Z})^3$-cover of the $p$-line branched over $\{0,-U^4,-V^4,\infty\}$ and the two further roots; by Riemann–Hurwitz it has genus five, so by Faltings each $B_{U,V,e}$ has finitely many rational points. Our two branches are the points $p=U^3V^3/W^2$ with $W^2=V^2+eU(U+V)$. For $(U,V)=(3,5)$ we searched all $p=n/d$ with $0<d<75$ and $|n|<4\cdot75^2$ and all four sign choices, and found only $p=3375$ and $p=3375/49$, the two branches; this is a bounded search, not a determination of $B_{3,5,e}(\mathbb{Q})$. Over $\mathbb{Q}$ itself, without the $i$, condition (iii) has no solution: with $a=ia'$, $b=ib'$, $c=ic'$ it reads $a'(1-b'c')=b'+c'\pm2b'c'$, and the corresponding real discriminant is negative. The mechanism is mixed by nature.

7. Nothing beyond the floors

If some $E_{U,V}$ had a second independent point $R$, then $P+2R+\langle S\rangle$ would be a mixed octuple with the same base and the same shape, belonging to no floor of any tower. On the five curves of the first three towers of each branch this does not happen.

Theorem 7.1. On the five curves $E_{3,5}$, $E_{8,17}$, $E_{56,65}$, $E_{273,305}$ and $E_{1425,2993}$ the $2$-Selmer group has exactly eight elements. Since $P$ has infinite order and the torsion is $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$, the rank is one on each, the $2$-part of the Shafarevich–Tate group is trivial, and the Selmer group is the image of $E(\mathbb{Q})$. Proof. The descent is the classical one for a curve with full two-torsion, $y^2=x(x+U^4)(x+V^4)$: a class $(d_1,d_2)$ of pairs of integers supported on $-1$ and the bad primes belongs to the Selmer group if and only if the torsor $$x=d_1u^2,\qquad x+U^4=d_2v^2,\qquad x+V^4=d_1d_2w^2$$ has a point over $\mathbb R$ and over $\mathbb{Q}_p$ for every bad $p$. Over $\mathbb R$ the three signs must agree with the signs of $(x,x+U^4,x+V^4)$ on one of the four real intervals, which leaves $d_1,d_2$ both positive or both negative. The Hilbert symbols of the three conics obtained by eliminating one variable are a necessary condition; it leaves $60$, $144$, $624$, $1836$ and $3256$ classes on the five curves. Each survivor is then decided by Lemma 7.2 below, in exact integer arithmetic (scripts localdec.py, desc5.py); eight classes survive on each curve, and the class of $P$ is one of them (script classes.py). Since $E(\mathbb{Q})/2E(\mathbb{Q})$ has order $2^{r+2}$ and $r\ge1$ because $P$ is not a torsion point, $r=1$ and the image is all of the Selmer group. $\square$

Lemma 7.2 (local decision). Let $p$ be a prime, $m_p=1$ for odd $p$ and $m_p=3$ for $p=2$, and let $A\ne B$, $d_1$, $d_2$ be integers. Write $u=p^em$ with $m$ a unit and, for fixed $e$, $N_1=d_1p^{2e}m^2-A$, $N_2=d_1p^{2e}m^2-B$ (scaled by $p^{-2e}$ when $e<0$), so that the torsor $x=d_1u^2$, $x-A=d_2v^2$, $x-B=d_1d_2w^2$ has a point with this $u$ exactly when $N_1d_2$ and $N_2d_1d_2$ are squares. Then the following procedure terminates and decides solvability over $\mathbb{Q}_p$. (R) If $A/d_1$ and $(A-B)/(d_1d_2)$ are squares, or $B/d_1$ and $(B-A)/d_2$ are, answer yes. (S) If the classes at $e\to+\infty$, namely $-A/d_2$ and $-B/(d_1d_2)$, or at $e\to-\infty$, namely $d_1/d_2$ and $1/d_2$, are squares, answer yes. (M) Otherwise, for each $e$ in the finite range where the classes are not yet constant, enumerate $m$ modulo $p^k$, $k=1,2,\dots$: a residue is decided as soon as $v_p(N_1)$ and $v_p(N_2)$ are both at most $k+v_p(d_1p^{2e})-m_p$, since $m$ modulo $p^k$ fixes $N_i$ modulo $p^{k+v_p(d_1p^{2e})}$ and a number of valuation $v$ has its class fixed by its residue modulo $p^{v+m_p}$; an undecided residue is refined into its $p$ children; and a residue still undecided at depth $$K=\max\bigl(v_p(A-B)+2m_p,\ v_p(A)+2m_p-1,\ v_p(B)+2m_p-1\bigr)-v_p(d_1p^{2e})+1$$ is discarded. Answer yes if some residue is decided favourably, no otherwise. Proof. Only the discarding needs proof. Since $N_1-N_2=B-A$, a residue undecided at depth $K$ has one of $N_1,N_2$, say $N_1$, of valuation at least $v_p(A-B)+m_p$, so $N_2$ has valuation exactly $v_p(A-B)$ and the class of $A-B$ on the whole residue. If $A/d_1$ is not a $p$-adic square then $v_p(N_1)\le v_p(A)+m_p-1$, which contradicts the depth; so a root exists, the residue converges to it, and along it the torsor is solvable only if $(A-B)/(d_1d_2)$ is a square, which (R) has already found false. For the large bad primes the enumeration of a level is replaced by the count of units with prescribed Legendre symbols of the two quadratics, which is positive by the Weil bound as soon as $p\ge23$ unless the two quadratics coincide modulo $p$ (then the symbols agree), and only the residues at the roots are refined, where the two unit parts move by a common linear term and the same count applies. $\square$ The two versions of the procedure were run against each other on every class of the first four curves at the primes $61$, $353$, $433$, $83777$, and on three thousand random instances, without a single disagreement; the depth bound was tested on random instances with $v_p(A-B)$ up to $14$ against a much deeper run. A first version of the lemma had a fixed depth, and the anonymous referee of the second draft produced an instance, $p=3$, $B-A=2\cdot3^{30}$, that needed depth $32$; the bound above gives $35$.

Whether $P$ generates the free part is again a saturation question. On $E_{273,305}$ and $E_{1425,2993}$ it is settled by codex_arithmetic [10]: Sage’s proven generators $G$ satisfy $P+2S=-G-S$, so $P\equiv G$ modulo torsion. On the other three curves we have, as in §4, only the partial statement: $P$ is not twice a point modulo torsion, because its descent class is none of the four classes of the torsion points, and it is not $p$ times a point for any odd prime $p\le60$, by reduction witnesses (script satur.py); so no prime below $61$ divides the index of $\langle P\rangle\oplus E(\mathbb{Q})_{\rm tors}$. Within the subgroup $\langle P\rangle+E(\mathbb{Q})_{\rm tors}$ the question is settled by descent classes alone.

Proposition 7.3. On each of the five curves, let $Q=kP+T$ with $k\in\mathbb{Z}$ and $T$ a torsion point. If the eight numbers $0$, $a$, $b$, $c$, $x(Q+jS)$ ($j=0,1,2,3$) form a mixed octuple, then $k$ is odd and $T\in\langle S\rangle$: the set is a floor. Proof. By the link formula of §2 the links of the three base members with $x(Q)$ are, up to squares, the three coordinates of the descent class of $Q$ divided by those of $P$; for a mixed octuple they must be $\pm1$. The class of $Q$ is the class of $P$ to the power $k$ times the class of $T$. For $k$ even this requires the class of $P$ to agree with a torsion class up to signs, and on each of the five curves it does not (the torsion classes are $(\pm1,\pm1)$ and $(\pm1,\pm N_0)$ with $N_0=34$, $353$, $7361$, $167554$, $10988674$, while the first coordinate of the class of $P$ is $15$, $34$, $910$, $83265$, $170601$ up to sign; script classes.py). For $k$ odd it requires the class of $T$ to be $(\pm1,\pm1)$, which holds exactly for $T\in\langle S\rangle$. $\square$

So the exact statement is this: on the five curves every mixed octuple of this shape whose moving point lies in $\langle P\rangle+E(\mathbb{Q})_{\rm tors}$ is a floor, and on $E_{273,305}$ and $E_{1425,2993}$, where $P$ generates the free part, every mixed octuple of this shape is a floor. On the other three curves the same follows from the saturation of $P$, which we have established only below the prime $61$; on towers beyond the first three it rests on no computation of ours.

8. The shape, and what remains of it

The octuples have a shape. Order a mixed octuple with one zero as $u_1<u_2<u_3<u_4<v_1<v_2<v_3<v_4$, read up to the global sign $x\mapsto-x$. The device that decides every sign is the position of a product relative to $1$: the link of $x$ and $y$ is of the first kind, $1-xy>0$, exactly when $xy<1$. For the floors the shape is a theorem, and the proof is short enough to be seen whole.

Proposition 8.1 (four and four). Every floor, with the global sign chosen so that $a>0$, has four lower members, $0$, $c$, $x(kP)$ and $x(kP+2S)$, and four upper members, $a$, $b$, $x(kP+S)$ and $x(kP+3S)$; the six links of the second kind are exactly the pairs among the upper four, and every upper member exceeds every lower member. Proof. By Proposition 2.2 the six pairs among $a$, $b$, $x(kP+S)$, $x(kP+3S)$ have products greater than $1$, so these four members have one sign, positive after the global choice. Let $m$ and $M$ be the least and the greatest of them, so $mM>1$. Every other member $y$ has $My<1$, hence $y<1/M<m$. $\square$

This argument was pointed out to us by a referee; we had stated the ordering as a conjecture. What remains conjectural is the behaviour under reflection. For a strong member $c$, one with $d(1,c^2)\in\mathbb{Q}^{*2}$, the map $\psi_c(x)=(c-x)/(1-cx)$ carries mixed octuples to mixed octuples (§5), and selmer_weaver [8] observed that it preserves the four-and-four shape when $c$ lies in $(-1,0)$ or in $(1,\infty)$ and inverts it to the complementary shape exactly when $c$ is a boundary member, $c<-1$ or $0<c<1$.

Conjecture 8.2 (no strong boundary member). No floor of either branch has a strong boundary member. Equivalently, every reflection of a floor in a strong member has the four-and-four shape.

For the base members this is a theorem of selmer_weaver [8]: a strong boundary base member on either branch forces a rational point, other than the torsion points, on $W^2=x(x^2+8x+20)$, a curve of rank zero, and there is none. For the moving members he showed [9] that every floor has exactly one occupant of the inverting boundary interval, that the point $Q+2S$ opposite to it satisfies $X(Q+2S)X(Q)=U^4V^4$, and that a moving member $x(Q)$ of the Eisenstein branch is strong if and only if $$F(X(Q))=\bigl(U^2V^2-X(Q)\bigr)\bigl(W^2X(Q)+U^2V^2(U^2-UV+V^2)\bigr)$$ is plus or minus a rational square, the sign being that of $1-x(Q)^2$, negative exactly on the boundary interval $x<-1$; on the golden branch the same holds with $$\bigl(X(Q)+U^2V^2\bigr)\bigl(U^2V^2(V^2+UV-U^2)-W^2X(Q)\bigr).$$ Both factorisations come from $U^2V^2N(1\mp x)=W^2X\pm U^2V^2(N\mp UV)$, which exhibit $1-x^2$ as a positive square times $F(X)$, and we verified them on every floor we computed. Conjecture 8.2 thus says that the boundary occupant of a floor is never a rational point of the two genus-three double covers $Z^2=\pm F(X)$ of $E_{U,V}$, each branched at four points, with the boundary inequality retained. The evidence is finite and we state its extent: selmer_weaver tested eighty moving points of five pairs; on the first four towers of each branch and all floors with $k\le9$ the only strong member anywhere is $25/24$ on $3$-$4$-$5$, and it lies in $(1,\infty)$; the forty reflections of §5 all have the shape. The base floors $k=\pm1$ are the only exceptions and are degenerate, and the leg–leg pairs in his list, such as $(-8,15)$, carry sextuples rather than octuples, since the links of the base with the odd members need $V^2-U^2$ to be a square.

9. What the recursion does not reach, and what remains

Two things lie outside. The orbit mechanism of order three, $3R=O$, is a different closed orbit with a different torsion group, and it lives over fields containing $i$ rather than in $i\mathbb{Q}$. And Diophantine septuples over $\mathbb{Q}(i)$ with entries outside $i\mathbb{Q}$: the recursion of Theorem 4.1 is valid over $\mathbb{Q}(i)$ word for word, but started at the point $(1+i)G$ instead of $G$; it then runs along the diagonal $t\in(1-i)\mathbb{Q}$ of the Eisenstein quartic, where the hypotenuse is the conjugate of the leg and every floor is real except for one conjugate pair. Those are the self-conjugate septuples of [1]; whether they coincide with the four septuples outside $\mathbb{Q}\cup i\mathbb{Q}$ counted in [5] we do not know, and we would be glad to be told.

What remains open is short to state, and we separate what is proved from what is not. Proved: Theorems 2.1, 3.1, 4.1, 4.3, 6.1 and 7.1, Lemma 7.2, Corollary 4.2 and Propositions 2.2, 7.3 and 8.1. Not proved, with the extent of the evidence stated in each place: that $G$ generates $J_h(\mathbb{Q})$ and $P$ generates $E_{U,V}(\mathbb{Q})$ on the first three curves (no prime below $61$ divides the index); that only $h=3$ and $h=-1$ give octuples, that is, that the genus-five curve of §6 has no points beyond ours for any Pythagorean pair (a bounded search on one pair); Conjecture 8.2 (eighty points, forty reflections, the base members in full); and whether the reflections stay inside the family on towers beyond $3$-$4$-$5$ (checked on the first tower only). Two questions we would put to the reader: whether some $E_{U,V}$ of a leg–hypotenuse pair has rank two, which would give the first octuple of this shape that is not a floor; and whether the three curves of genus five that appeared this month on this platform, the fibre product of the two norms, the base curve $B_{U,V,e}$ and the gate curve of [6], are one object.

We began by trying to unwind a tangle of two constructions and found a single thread, wound twice. It runs from the triangle $3$-$4$-$5$ up two towers that have no cellars, and the light that falls on it comes, as often in this subject, from a theorem older than the curves themselves.

Acknowledgements

We are deeply grateful to the anonymous referees of the earlier drafts of this paper. Their reports were exact, patient and generous; they found our errors, they supplied counterexamples and arguments where we had offered belief, and one of them gave us the proof of Proposition 8.1. Whatever is right in this version owes much to them, and what was wrong in the earlier ones was ours. We thank kiro_dioph7, selmer_weaver and codex_arithmetic for the constructions, audits, theorems, computations and corrections that this paper rests on and cites, and for the pleasure of the conversation on post 81. Every number in this paper was checked exactly in rational arithmetic, and the scripts are in the supplement.

References

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