On the Sum of Reciprocals of Amicable Numbers

Two numbers $m$ and $n$ are considered amicable if the sum of their proper divisors, $s(n)$ and $s(m)$, satisfy $s(n) = m$ and $s(m) = n$. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, $P$, is a constant. We obtain both a lower and an upper bound on the value of $P$.Comment: 18 pages. Paper version to appear in Integers; work first presented at Integers Conference 200

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

THE LUCAS-PRATT PRIMALITY TREE

Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log 2 p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than C log p for a set of primes p with relative asymptotic density 1. 1

Average Frobenius distributions for elliptic curves with nontrivial rational torsion

In this paper we consider the Lang-Trotter conjecture (Conjecture 1 below) for various families of elliptic curves with prescribed torsion structure. We prove that the Lang-Trotter conjecture holds in an average sense for these families of curves (see Theorem 3)

Sums over primitive sets with a fixed number of prime factors

A primitive set is one in which no element of the set divides another. Erdős conjectured that the sum f(A) :=∑n∈A 1/nlogn taken over any primitive set A would be greatest when A is the set of primes. More recently, Banks and Martin have generalized this conjecture to claim that, if we let Nk represent the set of integers with precisely k prime factors(counted with multiplicity), then we have f(N1) \u3e f(N2) \u3e f(N3) \u3e ···. The first of these inequalities was established by Zhang; we establish the second. Our methods involve explicit bounds on the density of integers with precisely k prime factors. In particular, we establish an explicit version of the Hardy-Ramanujan theorem on the density of integers with k prime factors