Four arithmetic theorems

Arithmetic and the Incompleteness Theorems

Arithmetic Duality Theorems Second Edition

Addendum/Erratum for Arithmetic Duality Theorems

Then H need not be divisible, i.e., we need not have pH D H . We only know H is contained in pG. Moreover, for x in H , there need not exist an infinite sequence y1;y2;y3:::: such that py2 D y1, py3 D y2; ::. We only know that there exist arbitrarily long finite such sequences. There does exist a unique maximal divisible subgroup D of M , and D is a subgroup of H . Moreover, M DD ̊N where N is a...

Arithmetic Progressions of Four Squares

Suppose a, b, c, and d are rational numbers such that a2, b2, c2, and d2 form an arithmetic progression: the differences b2−a2, c2−b2, and d2−c2 are equal. One possibility is that the arithmetic progression is constant: a2, a2, a2, a2. Are there arithmetic progressions of four rational squares which are not constant? This question was first raised by Fermat in 1640. There are no such progressio...

Four Theorems on the Psychometric Function

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, Δx. This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient ...