Terence Jackson

Bio: Terence Jackson is an academic researcher from University of York. The author has contributed to research in topics: Symbolic computation & Square root. The author has an hindex of 2, co-authored 6 publications receiving 45 citations.

A Short Proof That Every Prime Is of the Form x 2 + 2y 2 .

Abstract: In 1984 Heath-Brown [1] gave an interesting new proof of the Girard-Fermat theorem that every prime congruent to 1 modulo 4 is a sum of two squares. This was subsequently presented for students and mathematics teachers in [2] and [3], and it inspired Zagier [4] to give a one-sentence proof in 1990. A similar idea can be used to give a direct proof of Euler's result that every prime p = 3 (mod 8) is of the form x 2 + 2y • Let p be an odd prime and S be the non-empty finite set {(x, y, z) E N3 : x 2 + 2yz = p}. We define a map with domain S as follows:

95.42 Irrational square roots of natural numbers — a geometrical approach

Abstract: 2 1(e2Y +2 e_ e +2 e-) Isin (x + iY)1 -----e + e-w eiw + e-iw and recalling that cosh w = 2 and cos w = 2 we see that the first equation in (2) has been proved. Replacing x by (x + in) and using the fact that cosx = sin (x + in) we get the second equation in (2) immediately. MARCUS WRIGHT and THOMAS J. OSLER Mathematics Department, Rowan University, Glassboro, NJ 08028 USA e-mails: Wright@rowan.edu and Osler@rowan.edu

From Polynomials to Sums of Squares

Abstract: This text and software describes a "journey" through algebra and number theory based on the central theme of factorization. It begins with basic knowledge of rational polynomials, gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The treatment is made very concrete throughout the main text with illustrations using specific examples. More abstract material is confined to appendices. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are required. The accompanying software allows the reader to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities, mostly involving the computer.

Hessian Elliptic Curves and Side-Channel Attacks

Abstract: Side-channel attacks are a recent class of attacks that have been revealed to be very powerful in practice. By measuring some side-channel information (running time, power consumption, ...), an attacker is able to recover some secret data from a carelessly implemented crypto-algorithm. This paper investigates the Hessian parameterization of an elliptic curve as a step towards resistance against such attacks in the context of elliptic curve cryptography. The idea is to use the same procedure to compute the addition, the doubling or the subtraction of points. As a result, this gives a 33% performance improvement as compared to the best reported methods and requires much less memory.

Topological Strings And (Almost) Modular Forms

Abstract: hep-th/0607100 arXiv:hep-th/0607100v2 4 May 2007 Topological Strings and (Almost) Modular Forms Mina Aganagic, 1 Vincent Bouchard, 2 Albrecht Klemm, 3 University of California, Berkeley, CA 94720, USA Mathematical Sciences Research Institute, Berkeley, CA 94720, USA University of Wisconsin, Madison, WI 53706, USA Abstract The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3 (X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi- Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP 2 amplitudes near different points in the moduli space, which we use to give predictions for Z Gromov-Witten invariants of the orbifold C 3 /Z 3 . July 2006 and IP 1 ×IP 1 . As a byproduct, we also obtain a simple way of relating the topological string

Mahler's Measure and Special Values of L-functions

Abstract: If P(Xl,… xn) is a polynomial with integer coefficients, the Mahler measureM(P) of P is defined to be the geometric mean of |P| over the n-torus Tn. For n = 1, M(P) is an algebraic integer, but for n > 1, there is reason to believe that M(P) is usually transcendental. For example, Smyth showed that log M(l + x +y) = L'(X–3, −1), where X–3 is the odd Dirichlet character of conductor 3. Here we will describe some examples for which it appearsthat log M(P(x, y)) = rL'(E, 0), where E is an elliptic curve and r is a rational number, often either an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not rigorously proved.

A search for Wieferich and Wilson primes

Abstract: An odd prime p is called a Wieferich prime if 2 P-1 = 1 (mod p 2 ) alternatively, a Wilson prime if (p - 1)|= -1 (mod p 2 ). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5,13, and 563. We report that there exist no new Wieferich primes p < 4 x 10 12 , and no new Wilson primes p < 5x 10 8 . It is elementary that both defining congruences above hold merely (mod p), and it is sometimes estimated on heuristic grounds that the probability that p is Wieferich (independently: that p is Wilson) is about 1/p. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod p).

Efficient solution of rational conics

Abstract: We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.