New Bounds and Computations on Prime-Indexed Primes

In a 2009 article, Barnett and Broughan considered the set of prime-index primes. If the prime numbers are listed in increasing order (2, 3, 5, 7, 11, 13, 17, . . .), then the prime-index primes are those which occur in a prime-numbered position in the list (3, 5, 11, 17, . . .). Barnett and Broughan established a prime-indexed prime number theorem analogous to the standard prime number theorem and gave an asymptotic for the size of the n-th prime-indexed prime. We give explicit upper and lower bounds for π2(x), the number of prime-indexed primes up to x, as well as upper and lower bounds on the n-th prime-indexed prime, all improvements on the bounds from 2009. We also prove analogous results for higher iterates of the sequence of primes. We present empirical results on large gaps between prime-index primes, the sum of inverses of the prime-index primes, and an analog of Goldbach’s conjecture for prime-index primes

Tame and wild primes in direct products of commutative rings

In this article, new advances in understanding the structure of prime ideals of an infinite direct product of rings are obtained. We observe that in a direct product ring there are two types of prime ideals, tame primes and wild primes. Among the main results, we prove that the set of tame primes is an open subscheme of the prime spectrum, and this scheme is non-affine if and only if the index set is infinite. As an application, it is shown that a prime ideal is a wild prime if and only if it contains the direct sum ideal. Next we show that an uncountable number of wild primes (but not all) of an infinite direct product ring can be described in terms of the non-principal ultrafilters of the index set. As an application, it is shown that if a direct product ring has at least a wild prime, then the set of wild primes is infinite (uncountable). We also prove that if all of the factors of a direct product ring are local rings then this ring modulo the direct sum ideal is canonically isomorphic to a certain localization of the ring. Finally, the connected components of the prime spectrum of an infinite direct product ring are investigated. We observe that, like the prime ideals, there are two types of connected components, tame ones and wild ones.Comment: 14 page

Some identities for enumerators of circulant graphs

We establish analytically several new identities connecting enumerators of different types of circulant graphs of prime, twice prime and prime-squared orders. In particular, it is shown that the semi-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Keywords: circulant graph; cycle index; cyclic group; nearly doubled primes; Cunningham chain; self-complementary graph; tournament; mixed graphComment: 17 pages, 3 tables Categories: CO Combinatorics (NT Number Theory) Math Subject Class: 05C30; 05A19; 11A4

Prime divisors of sequences associated to elliptic curves

We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences