Prime extension-degree quotients of elliptic-curve orders

Definitions and mathematical statement

Fix an elliptic curve $E/{\mathbb{F}}_q$, and for $n\ge1$ put \[ N_n=\#E({\mathbb{F}}_{q^n}),\qquad Q_n=N_n/N_1. \] The inclusion $E({\mathbb{F}}_q)\subseteq E({\mathbb{F}}_{q^n})$ makes $Q_n$ a positive integer. Prime means an ordinary positive rational prime; $Q_1=1$ is not prime. The proposed conclusion is \[ \forall B\ge1\ \exists n>B,\qquad Q_n\text{ is prime}. \] The source does not state whether the claim is for every fixed curve or only fixed curves satisfying an unstated admissibility condition.

For $t=q+1-N_1$, let $\alpha,\bar\alpha$ be the roots of $z^2-tz+q$. The usual order formula gives \[ Q_n=\frac{(1-\alpha^n)(1-\bar\alpha^n)}{(1-\alpha)(1-\bar\alpha)}. \] Exceptional Frobenius behavior matters. If $\alpha=\bar\alpha=a\in{\mathbb{Z}}$, then \[ Q_n=(1+a+\cdots+a^{n-1})^2, \] so none of these integers is prime. The scalar-Frobenius/supersingular case therefore cannot simply be included in a universal assertion. No condition excluding this or other systematic factorization obstructions is supplied, and an ordinary-curve restriction alone is not invented as a sufficient replacement.

The supplied research summary calls the intended problem a Koblitz/Mersenne-type prime-value conjecture and reports no proof. That assessment does not repair the absent curve hypotheses or certify a version for every $E$. The draft preserves the infinitude target while marking the admissible fixed-curve class for review.

Short English statement

For the intended fixed elliptic curve, are the extension-order quotients prime infinitely often? The curve class needs exclusions: scalar Frobenius makes every quotient a square, so the unrestricted every-curve reading cannot stand.

Sources

• Local UnsolvedMath registry, permanent ID $8800039$, exact source code catalogue entry, "Prime extension-degree quotients of elliptic-curve orders." Proposer Igor Shparlinski.
• IPAM Workshop on Number Theory and Cryptography --- Open Problems (2006), attachment 6224, recorded PDF page 20, "Holy Grail"; https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule. Recorded access 2026-07-29; pdf-and-legacy-ppt visual-audit extraction.
• Supplied OPEN-TRIAGE research summary compares the claim with Koblitz-type and Mersenne-prime questions but supplies no precise admissibility criterion or complete supporting reference.