Conjecture on elliptic Korselt numbers of Type I and anomalous primes

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Let M≥7M\geq 7. Choose distinct primes p,qp,q uniformly at random subject to 5≤p,q≤M5\leq p,q\leq M, and set N=pqN=pq. Choose an elliptic curve E(Z/NZ)E(\mathbb{Z}/N\mathbb{Z}) uniformly at random, with good reduction at pp and qq, such that

#E(Z/pZ)=p+1−apand#E(Z/qZ)=q+1−aq\#E(\mathbb{Z}/p\mathbb{Z})=p+1-a_p\quad\text{and}\quad \#E(\mathbb{Z}/q\mathbb{Z})=q+1-a_q

both divide N+1−aNN+1-a_N. Elliptic Korselt–anomalous-prime conjecture. The probability that

#E(Z/NZ)=N+1−aN\#E(\mathbb{Z}/N\mathbb{Z})=N+1-a_N

tends to 11 as MM tends to infinity:

lim⁡M→∞Pr⁡[#E(Z/NZ)=N+1−aN]=1.\lim_{M\rightarrow\infty}\operatorname{Pr}\left[\#E(\mathbb{Z}/N\mathbb{Z})=N+1-a_N\right]=1.

This conjecture concerns how often an elliptic Korselt number of Type I is the product of distinct anomalous primes; the supplied text presents it as a conjecture from the cited earlier work, without providing evidence of resolution.

References

Primary source

L. Babinkostova, A. Hernández-Espiet and H. Kim, “On Types of Elliptic Pseudoprimes”, arXiv:1710.05264 (2021).

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