Seeing Numbers

In 1890 William James listed several “elementary mental categories” that he postulated as having a natural origin. Among them, alongside the ideas of time and space, he also listed the idea of number. A symptomatic feature of Informatics as well as Cognitive Science today is the tendency not to talk so much about ideas as about their representations, either in the computer or in the brain. Taking up somewhat different perspective I will discuss the way natural numbers, viewed as counts of real or imagined objects, may be experienced phenomenally. I put forth even some speculative ideas about mental number processing by numerical savants

Flat primes and thin primes

A number is called upper (lower) flat if its shift by +1 ( −1) is a power of 2 times a squarefree number. If the squarefree number is 1 or a single odd prime then the original number is called upper (lower) thin. Upper flat numbers which are primes arise in the study of multi-perfect numbers. Here we show that the lower or upper flat primes have asymptotic density relative to that of the full set of primes given by twice Artin’s constant, that more than 53% of the primes are both lower and upper flat, and that the series of reciprocals of the lower or the upper thin primes converges

The twin prime conjecture

This is an exposition of recent developments in the theory of bounded differences between primes. Readers are expected to be beginners of analytic number theory. The present text is a substantially improved and augmented version of the one that I had prepared for my talk which I delivered at the Annual Meeting of the Mathematical Society of Japan on 15 March 2014.Comment: 17 page

A computation of modular forms of weight one and small level

We report on a computation of holomorphic cuspidal modular forms of weight one and small level (currently level at most 1500) and classification of them according to the projective image of their attached Artin representations. The data we have gathered, such as Fourier expansions and projective images of Hecke newforms and dimensions of space of forms, is available in both Magma and Sage readable formats on a webpage created in support of this project

Associative and repetition priming with the repeated masked prime technique: No priming found

Wentura and Frings (2005) reported evidence of subliminal categorical priming on a lexical decision task, using a new method of visual masking in which the prime string consisted of the prime word flanked by random consonants and random letter masks alternated with the prime string on successive refresh cycles. We investigated associative and repetition priming on lexical decision, using the same method of visual masking. Three experiments failed to show any evidence of associative priming, (1) when the prime string was fixed at 10 characters (three to six flanking letters) and (2) when the number of flanking letters were reduced or absent. In all cases, prime detection was at chance level. Strong associative priming was observed with visible unmasked primes, but the addition of flanking letters restricted priming even though prime detection was still high. With repetition priming, no priming effects were found with the repeated masked technique, and prime detection was poor but just above chance levels. We conclude that with repeated masked primes, there is effective visual masking but that associative priming and repetition priming do not occur with experiment-unique prime-target pairs. Explanations for this apparent discrepancy across priming paradigms are discussed. The priming stimuli and prime-target pairs used in this study may be downloaded as supplemental materials from mc.psychonomic-journals.org/content/supplemental. © 2009 The Psychonomic Society, Inc