A polynomial f over a finite field F is called a permutat,on poIFnomial if the mapping F F defined by f is one-to-one. In this paper we consider the problem of characterizing permutation polynomials;â€¦ (More)

Shift registers are at the heart of cryptography and error-correction. In cryptography they are the main tool for generating long pseudorandom binary sequences which can be used as keys for twoâ€¦ (More)

Let K be a finite abelian extension of the rational field Q. If A is a central simple algebra over K then we let [A] denote the class of A in the Brauer group B(K) of K. The Schur subgrozcp S(K) ofâ€¦ (More)

We revisit the Diophantine equation of the title, and related equations , from new perspectives that add connections to continued fractions , fundamental units of real quadratic fields, Jacobi symbolâ€¦ (More)

In the late eighteenth century both Euler and Legendre noticed that n +n+41 is prime for n = 0, 1, 2 . . . 39, and remarked that there are few polynomials with such small degree and coefficients thatâ€¦ (More)

We use the theory of continued fractions in conjunction with ideal theory (often called the infrastructure) in real quadratic fields to give new class number 2 criteria and link this to a canonicalâ€¦ (More)

We present an infinite class of integers 2c, which turn out to be Richaud-Degert type radicands, for which 2x2 âˆ’ cy2 = âˆ’1 has no integer solutions, but for which 2x2 âˆ’ cy2 â‰¡ âˆ’1 (mod n) has integerâ€¦ (More)

A conjecture was related to this author in correspondence, some years ago, with Irving Kaplansky, which according to Professor Kaplansky, was inspired by the proof of [4, Theorem 6.5.9, p. 348]. Itâ€¦ (More)

In this column we review the following books. 1. Excellence Without a Soul: How a Great University Forgot Education by Harry Lewis. Review by William Gasarch. This book is about how Harvard hadâ€¦ (More)