We explore the relationship between elliptic Korselt numbers of Type I, a class of pseudoprimes introduced by Silverman in [20], and anomalous primes. We generalize a result in [20] that gives sufficient conditions for an elliptic Korselt number of Type I to be a product of anomalous primes. Finally, we prove that almost all elliptic Korselt numbers of Type I of the form n = pq are a product of anomalous primes.In Section 2, we prove a generalization of [20, Proposition 17]. We show that if n = p 1 · · · p m is a squarefree Type I elliptic Korselt number with p 1 < · · · < p m and √ pm 4 m ≤ p 1 · · · p m−1 ≤ 4 m , then p m is anomalous and a n = 1. Hence all but an even number of the primes p i are anomalous, and if p i is not anomalous, then a p i = −1. Furthermore, we note an error in the proof of [20, Proposition 17], providing a counterexample and the corrected statement. In particular, we show that if n = pq is an elliptic Korselt number of Type I with p < q for E/Q and 13 ≤ p ≤ √ q/16, then p and q are anomalous for E.

We study the numbers of involutions and their relation to Frobenius-Schur indicators in the groups SO ± (n, q) and Ω ± (n, q). Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group G is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius-Schur indicators are 1), and we are able to show this holds for all finite simple groups G other than the groups Sp(2n, q) with q even or Ω ± (4m, q) with q even. We prove computationally that for small n and m this statement indeed holds for these groups by equating their character degree sums to the number of involutions. We also prove a result on a certain twisted indicator for the groups SO ± (4m + 2, q) with q odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating function and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in SO ± (n, q) and Ω ± (n, q) for all q, and we use these to compute the asymptotic behavior for the number of involutions in these groups when q is fixed and n grows.2010 AMS Subject Classification: 20G40, 20C33, 05A15, 05A16

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