We present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes. The first method is based on an argument of Erdös which uses a sieve upper bound for prime twins to bound the density function for gaps between primes. The second method uses known results about the first three moments f...

In reaction time research, there has been an increasing appreciation that response-initiation processes are sensitive to recent experience and, in particular, the difficulty of previous trials. From this perspective, the authors propose an explanation for a perplexing property of masked priming: Although primes are not consciously identified, facilitation of target processing by a related prime...

orthographic and phonological encoding of visual words were contrastedin a patient with letter-by-letter reading using a priming paradigm. Lower caseprimes displayed for 100 msec preceded upper case target words the subject hadto name. Primes were the same word as the target, homophones to the target, orunrelated words. Relative to unrelated primes, same-word primes resulted in ...

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the technique...

Cryptographic algorithms often prescribe the use of primes whose length in bits is a power of 2. Recently, we proved that for m > 1, there is no prime number with 2 significant bits, exactly two of which are 0 bits. Here we generalize this theorem to impose many more restrictions on primes whose length in bits is a power of 2. No similar restrictions apply to primes of other lengths. We conside...

My research concerns explicit inequalities in elementary number theory. Specifically, I am interested in the distribution of twin primes, along with theoretical and computational techniques for putting explicit bounds on classes numbers such as twin primes. One important method I use to examine the density of twins is to look for bounds on Brun’s Constant. 1. Motivation and Background Twin prim...

We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.