The number of solutions of pell equations x2 -ky2 = N and x2+xy- Ky2 = N over Fp

Let p be a prime number such that p equivalent to 1, 3(mod 4), let F-p, be a finite field, let N is an element of F-p* = F-p - {0} be a fixed. Let P-p(k) (N) : x(2) - ky(2) = N and (P) over tilde (k)(p)(N) : x(2) + xy - ky(2) = N be two Pell equations over F-p, where k = p-1/4 or k = p-3/4, respectively. Let P-p(k)(N)(F-p) and (P) over tilde (k)(p)(N)(F-p) denote the set of integer solutions of the Pell equations P-p(k)(N) and (P) over tilde (k)(p)(N), respectively. In the first section we give some preliminaries from general Pell equation x(2) - ky(2) = +/- N. In the second section, we determine the number of integer solutions of P-p(k)(N). We proved that P-p(k)(N)(F-p) = p+ 1 if p equivalent to 1(mod 4) or p equivalent to 7(mod 12) and P-p(k)(N)(F-p) = p - 1 if p equivalent to 11(mod 12). In the third section we consider the Pell equation (P) over tilde (k)(p)(N). We proved that (P) over tilde (k)(p)(N)(F-p) = 2p if p equivalent to 1(mod 4) and N is an element of Q(p); (P) over tilde (k)(p)(N)(F-p) = 0 if p equivalent to 1(mod 4) and N is not an element of Q(p) ; (P) over tilde (k)(p)(N)(F-p) = p + 1 if p equivalent to 3(mod 4)

Diophantine equations in positive characteristic

In this thesis three different topics in number theory are studied. The first part deals with exponential Diophantine equations in positive characteristic. In the second part various statistical properties of class groups are proven. In the final part we apply the circle method to count the number of representations of an odd integer n as the sum of three primes all with a fixed primitive root. Number theory, Algebra and Geometr

Nostratic Dictionary

A revised edition can be found at http://www.dspace.cam.ac.uk/handle/1810/244080.Aharon Dolgopolsky is the leading authority on the Nostratic macrofamily. His 'Nostratic Dictionary' presented here is, of course, something very much more than a dictionary. It is the most thorough and extensive demonstration and documentation so far of what may be termed the Nostratic hypothesis: that several of the world's best- known language families are related in their origin, their grammar and their lexicon, and that they belong together in a larger unit, of earlier origin, the Nostratic macrofamily. It should at once be noted that several elements of this enterprise are controversial. For while the Nostratic hypothesis has many supporters, it has been criticized on rather fundamental grounds by a number of distinguished linguists. The matter was reviewed some years ago in a symposium held at the McDonald Institute, and positions remain very much polarized. It was a result of that meeting that the decision was taken to invite Aharon Dolgopolsky to publish his Dictionary - a much more substantial treatise than any work hitherto undertaken on the subject - at the McDonald Institute. For it became clear that the diversities of view expressed at that symposium were not likely to be resolved by further polemical exchanges. Instead, a substantial body of data was required, whose examination and evaluation could subsequently lead to more mature judgments. Those data are presented here, and that more mature evaluation can now proceed.McDonald Institute for Archaeological Research Alfred P. Sloan Foundatio