Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms IΔ0( ) + def( ), where (x) is the number of primes not exceeding x, IΔ0( ) denotes the theory of Δ0 induction for the language of arithmetic including the new function symbol , and def( ) is an axiom expressing the usual recursive definition of . We prove a modified...

In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zeta-function. 1. The prime number theorem, from the beginning By studying tables of primes, Gauss understood, as a boy of 15 or 16 (in 1792 or 1793), that the primes occ...

Finding mathematical proofs for easily observed properties of the distribution of prime numbers is a difficult and often humbling task, at least for the authors of this paper. The twin prime conjecture is a famous example of this, but we are concerned here with the much more modest problem of proving that there are arbitrarily large primes that are “unusually close ” together. Statistically thi...

The present study was an attempt to replicate and extend Brown's (1979) finding of inhibition in retrieval from semantic memory produced by presentation of semantically related primes. Subjects' speed and accuracy in answering generalknowledge questions (e.g., Who was the first man to walk on the moon?) were measured when the question was preceded by one of four different prime types: neutral (...

Two masked priming experiments examined behavioural and event-related potential responses to simplex target words (e.g., flex) preceded by briefly presented, masked, derived word primes (flexible-flex), complex nonword primes formed by an illegal combination of the target word and a real suffix (flexify-flex), and simplex nonword primes formed by adding a nonsuffix word ending to the target (fl...

The authors examined the effects of suboptimally presented primes (facial expressions) embedded into video messages on self-reported positive activation (PA) and negative activation (NA). The results showed that joyful facial primes elicited increased PA only for positively valenced messages, whereas angry facial primes prompted increased NA (fear) only for negatively valenced messages. In addi...

According to the direct parameter specification (DPS) account, reaction time effects of invisible primes depend on top-down control settings directed to targets (Neumann & Klotz, 1994). If this hypothesis holds, effects of invisible primes should decrease in dual-task as compared with single-task conditions: Prior to the primes control settings for the alternative task should be activated, whic...

The nth Ramanujan prime is the smallest positive integer Rn such that if x ≥ Rn, then the interval ( 1 2x, x ] contains at least n primes. We sharpen Laishram’s theorem that Rn < p3n by proving that the maximum of Rn/p3n is R5/p15 = 41/47. We give statistics on the length of the longest run of Ramanujan primes among all primes p < 10n, for n ≤ 9. We prove that if an upper twin prime is Ramanuja...

Are processes of figurative comparison and figurative categorization different? An experiment combining alternative-sense and matched-sense metaphor priming with a divided visual field assessment technique sought to isolate processes of comparison and categorization in the 2 cerebral hemispheres. For target metaphors presented in the right visual field/left cerebral hemisphere (RVF/LH), only ma...

We construct a finite 1-connected CW complex X such that H * (ΩX; Z) has p-torsion for the infinitely many primes satisfying p ≡ 5, 7, 17, 19 mod 24, but no p-torsion for the infinitely many primes satisfying p ≡ 13 or 23 mod 24. A homology torsion prime for a topological space Y is a prime p such that H * (Y ; Z) has p torsion. In [3] and [4] respectively, D. Anick and L. Avramov constructed s...