An inert-prime lifting criterion for the Cornelissen K3 equation

Definitions and mathematical statement

Question 13 (Cornelissen) asks to determine all $(A,B,X,Y)\in{\mathbb{Z}}^4$ satisfying \[ (A^2+B^2)(A^2+11B^2)=3^2\cdot5^2\,(X^2-5Y^2)^2. \] No positivity, coprimality, or nonzero restrictions are imposed in the source. It connects the equation to defining ${\mathbb{Z}}$ in ${\mathbb{Q}}$ by a $\Sigma_3^+$-formula and cites Cornelissen--Zahidi. The exact syntactic convention for this formula class is not supplied and is not replaced here by a different definability claim.

For the aid's partial reduction, put $F(a,b)=(a^2+b^2)(a^2+11b^2)$. A primitive square point means $a,b\in{\mathbb{Z}}$, $\gcd(a,b)=1$, and $c\in{\mathbb{Z}}_{>0}$ with $F(a,b)=c^2$. The aid asserts that such a point has $a$ odd and $b$ even, and satisfies exactly one of the two systems \[ a^2+b^2=d u^2,\qquad a^2+11b^2=d v^2,\qquad d\in\{1,5\},\quad u,v\in{\mathbb{Z}}. \] The uniqueness concerns $d$, not the signs of $u,v$.

Call a rational prime $p$ inert if it remains prime in ${\mathbb{Q}}(\sqrt5)$. The supplied norm assertion is that a nonzero integer $n$ has the form $X^2-5Y^2$ with $X,Y\in{\mathbb{Z}}$ exactly when $v_p(n)$ is even for every inert prime $p$, where $v_p(n)=v_p(|n|)$. Zero is represented by $(X,Y)=(0,0)$ separately. On that basis, the aid gives the scale-invariant criterion \[ \begin{split} &\exists g\in{\mathbb{Z}}_{>0}\ \exists X,Y\in{\mathbb{Z}}:\quad F(ga,gb)=225(X^2-5Y^2)^2\\ &\quad\Longleftrightarrow\quad v_3(c)\text{ is odd and }v_p(c)\text{ is even for every inert }p\ne3. \end{split} \] When this holds, it asserts that $g=1$ suffices if $5\mid c$, and $g=5$ otherwise suffices. The norm to be represented is $\pm g^2c/15$; these scale choices clear the denominator. This criterion concerns existence at some scale and does not assert that every positive scale works.

For $(a,b)=(1,2)$ one has $c=15$, giving the Pell-type family $A=g$, $B=2g$ and $X^2-5Y^2=\pm g^2$; for example $X+Y\sqrt5=g(2+\sqrt5)^k$ with $g\geq1$, $k\geq0$ gives solutions. In contrast, $(a,b)=(35,12)$ has $c=1961=37\cdot53$ and cannot lift at any scale by the stated criterion. The zero case $A=B=0$ forces $X=Y=0$. Every other original solution has a primitive pair obtained by dividing $A,B$ by their positive gcd, but the aid does not classify all primitive square points. Its summary reports the full K3 rational-point classification open in the literature checked, without a dated check or precise surface model.

Short English statement

The source asks for every integer solution of a quartic equation related to low-quantifier definability of the integers in the rationals. The supplied partial result decides which primitive square points can lift after scaling, using inert-prime valuations. It gives a Pell family and a non-liftable square point, while leaving the full classification unresolved as reported.

Sources

• Local UnsolvedMath registry, permanent ID $20002350$, catalogue entry; aim-logic-notes.json, notes[125].
• American Institute of Mathematics, Extensions of Hilbert's tenth problem, Question 13 (Cornelissen), https://aimath.org/WWN/hilberts10th/hilberts10th.pdf.
• Cornelissen--Zahidi, arXiv identifier math.NT/0412473, supplied for the $\Sigma_3^+$ definability connection. No title, theorem number, or publication details are supplied.
• Research attempt 1: partial result, problem open at run, rigor label "proved," novelty confidence low. Registry classification partial progress, status partially solved, statement status exact. The verification field only says "The repository OCR reads" and supplies no further transcription.