History of Catalan numbers

We give a brief history of Catalan numbers, from their first discovery in the 18th century to modern times. This note will appear as an appendix in Richard Stanley's forthcoming book on Catalan numbers.Comment: 10 page

Secure Communication Protocols, Secret Sharing and Authentication Based on Goldbach Partitions

This thesis investigates the use of Goldbach partitions for secure communication protocols and for finding large prime numbers that are fundamental to these protocols. It is proposed that multiple third parties be employed in TLS/SSL and secure communication protocols to distribute the trust and eliminate dependency on a single third party, which decreases the probability of forging a digital certificate and enhances the overall security of the system. Two methods are presented in which the secret key is not compromised until all third parties involved in the process are compromised. A new scheme to distribute secret shares using two third parties in the piggy bank cryptographic paradigm is proposed. Conditions under which Goldbach partitions are efficient in finding large prime numbers are presented. A method is also devised to sieve prime numbers which uses less number of operations as compared to the Sieve of Eratosthenes.Electrical Engineerin

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

Explicit Interval Estimates for Prime Numbers

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta, x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - \Delta^{-1}), x]$.Comment: 15 pages, 3 tables, 1 figur

There is entanglement in the primes

Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this Prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the Prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the Prime state, which in turn produces new insights in the Hardy-Littlewood conjecture for the pairwise distribution of primes. The extension of these ideas to a Twin Prime state shows that this new state is even more entangled than the Prime state, obeying majorization relations. We further discuss the construction of quantum states that encompass relevant series of numbers and opens the possibility of applying quantum computation to Arithmetics in novel ways.Comment: 30 pages, 11 Figs. Addition of two references and correction of typo