Infinitude of elliptic Carmichael numbers (Q2898886)

Elliptic Carmichael numbers are composite numbers which pass a primality test developed by \textit{D. M. Gordon} [Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 290--305 (1989; Zbl 0684.10006)]. The test is similar to the well-known test based on Fermat's little theorem, but is based on the arithmetic of elliptic curves with complex multiplication. The relevant fact is that, for any elliptic curve \(E\) defined over \(\mathbb Q\) with complex multiplication by \(\mathbb Q(\sqrt{-d})\), one has \(\# E(\mathbb F_p) = p+1\) for any prime \(p\) which does not divide \(6 \Delta_E\) and is inert in \(\mathbb Q(\sqrt{-d})\).NEWLINENEWLINEThe authors show that one can find elliptic Carmichael numbers through a criterion similar to Korselt's criterion for ordinary Carmichael numbers, and then prove that, assuming a conjecture on the least prime in arithmetic progression, there are infinitely many elliptic Carmichael numbers. The proof has similar arguments with [\textit{W. D. Banks} and \textit{C. Pomerance}, J. Aust. Math. Soc. 88, No. 3, 313--321 (2010; Zbl 1208.11109)] and [\textit{A. Ekstrom}, On the infinitude of elliptic Carmichael numbers. PhD Thesis, University of Arizona (1999)].