TA DC Projects | Mathematics

Mathematics

ECMNET [ECMNET]

History. Richard Brent has predicted in 1985 in a paper entitled Some Integer Factorization Algorithms using Elliptic Curves that factors up to 50 digits could by found by the Elliptic Curve Method (ECM). Indeed, Peter Montgomery found in November 1995 a factor of 47 digits of 5^256+1, and Richard Brent set in October 1997 a new genuine record with a factor of 48 digits of 24^121+1.

Goal. The original purpose of the ECMNET project was to make Richard's prediction true, i.e. to find a factor of 50 digits or more by ECM. This goal was attained on September 14, 1998, when Conrad Curry found a 53-digit factor of 2^677-1 c150 using George Woltman's mprime program. The new goal of ECMNET is now to find other large factors by ecm, mainly by contributing to the Cunningham project, most likely the longest, ongoing computational project in history according to Bob Silverman. A new record was set by Nik Lygeros and Michel Mizony, who found in December 1999 a prime factor of 54 digits using GMP-ECM

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Euler [EULER]

Computing Minimal Equal Sums Of Like Powers. This project is dedicated to all those who are fascinated by powers and integers.

In 1967, Lander and Parkin found a fifth power equal to FOUR fifth powers : 1445=1335+1105+845+275. They also found a sixth power equal to SEVEN sixth powers: 11416=10776+8946+7026+4746+4026+2346+746. We are trying to find a sixth power that is equal to SIX sixth powers. When we will find a sixth power that is equal to six sixth powers, we will start searching a sixth power that is equal to five sixth powers!

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Seventeen or Bust [SB]

SB (Seventeen or Bust) is a distributed attack on the Sierpinski problem.

The Sierpinski problem itself deals with numbers of the form N = k * 2^n + 1, for any odd k and n > 1. Numbers of this form are called Proth numbers. If, for some specific value of k, every possible choice of n results in a composite (non-prime) Proth number N, then that k is called a Sierpinski number.

John Selfridge proved, 40 years ago, that k = 78,557 is a Sierpinski number. Most number theorists believe that this is the smallest, but it hasn't yet been proven. In order to prove it, we have to show that every single k less than 78,557 is not a Sierpinski number, and to do that, we have to find some n that makes k * 2^n + 1 prime. When Seventeen or Bust was started, this had already been done for all but 17 values of k; hence the name of the project. After 20 months of computation, we have eliminated 6 multipliers: six down, eleven to go..

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ZetaGrid [ZETA]

Help verify Riemann's hypothesis in ZetaGrid.
The verification of Riemann's Hypothesis (formulated in 1859) is considered to be one of modern mathematic's most important problems. The last 140 years did not bring its proof, but a considerable number of important mathematical theorems which depend on the Hypothesis being true, e.g. the fastest known primality test of Miller.

The Riemann zeta function is defined for Re(s)>1 by

and is extended to the rest of the complex plane (except for s=1) by analytic continuation.

The Riemann Hypothesis asserts that all nontrivial zeros of the zeta function are on the critical line (1/2+it where t is a real number).

To verify empirically the Riemann Hypothesis for certain regions and make it usable, in 1903 the first fifteen zeros of Riemann's zeta function

(s) on the critical line were calculated. Thus, the Riemann Hypothesis is true at least in the region |t| < 65.801.

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SZTAKI Desktop Grid [STKI]

The aim of the project is to find all the generalized binary number systems up to dimension 11. A detailed project description is available. The project completed its 10 dimensional project and began its 11 dimension project in August.

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