Elliptic-Curve Bijections and Additive Bases from Restricted Primes

Definitions and mathematical statement

The supplied record contains two separate questions.

1. With the conventional interpretation that $p$ denotes a prime, let $p>3$, let ${\mathbb{F}}_p$ be the field with $p$ elements, and write ${\mathbb{F}}_p^\times={\mathbb{F}}_p\setminus\{0\}$. For each $t\in{\mathbb{F}}_p^\times$, define \[ E_p(t)=\left\{(x,y)\in({\mathbb{F}}_p^\times)^2:x+x^{-1}+y+y^{-1}=t\right\}. \] All operations in this definition take place in ${\mathbb{F}}_p$. Does there exist, for every such $p$ and $t$, an explicit bijection \[ \Phi_{p,t}:E_p(t)\longrightarrow E_p(16t^{-1})? \] Here $16t^{-1}$ is a nonzero field element because $p>3$ and $t\ne0$. The source asks for an explicit bijection or a bijective proof, but does not specify what form of construction qualifies as explicit. Its background describes a point-count identity without supplying its proof or a construction of the requested bijection.

2. Define \[ S=\left\{n\in{\mathbb{Z}}_{>0}:\text{every prime }q\text{ dividing }n\text{ satisfies }q\equiv3\pmod4\right\}. \] Thus $1\in S$, since it has no prime divisors, and $0\notin S$. Is the following assertion true? \[ \exists N_0\in{\mathbb{Z}}_{>0}\ \forall N\in{\mathbb{Z}}_{>0},\qquad \left(N\ge N_0\ \land\ N\equiv0\pmod2\right) \Longrightarrow \exists a,b\in S\quad N=a+b. \] The two summands are not required to be distinct.

The supplied record labels the composite problem open. Its dated assessment specifically describes the second question as unresolved; this is supplied curation, not an independently verified conclusion about either question.

Short English statement

For every prime $p>3$ and nonzero $t\in{\mathbb{F}}_p$, construct an explicit bijection between the two specified sets of finite-field points, or determine whether such a construction exists. Separately, determine whether every sufficiently large positive even integer is a sum of two positive integers whose prime divisors are all congruent to $3$ modulo $4$. The required meaning of "explicit" is unspecified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003278$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposer field is empty; no individual proposer is identified. The recorded proposed year is 2016.
• Supplied report citation: Analytic Number Theory (2017), Oberwolfach Reports, pp. 2975--3030, DOI 10.4171/owr/2016/53. Supplied source link: https://doi.org/10.4171/owr/2016/53.
• Supplied literature source, labelled "OWR 53 report": https://publications.mfo.de/bitstream/handle/mfo/3557/OWR_2016_53.pdf?isAllowed=y&sequence=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the composite record open, describes a finite-field point-count identity, and says that the sufficiently-large-even-integer assertion remains unresolved. This is supplied dated curation and has not been independently verified here; time-sensitive status and cited claims require independent checking.
• The supplied statement status is corrected_verified. Its verification note reports that the official problem-session report was checked and that this record retains two source questions omitted from the overlapping composite extraction OWR-15174-010. These are supplied provenance claims, not verification performed for this draft.