At the frontiers of mathematical knowledge lie a number of extraordinary problems. With the advent of digital computing and the continued struggle of brilliant minds, some of these problems have been recently solved, including:

Fermat's Last Theorem: If an integer n is greater than 2, then the equation aⁿ  + bⁿ = cⁿ has no solutions in non-zero integers a, b, and c. This theorem was solved by Andrew Wiles in 2003 but had resisted proof since 1637. Indeed, many wondered if the conjecture was true at all, imagining that a solution with large values of a, b, c and n might soon be found. The theorem is named after Pierre de Fermat (1601-1665).

Large Mersenne Primes: Prime numbers in the form 2^p-1, where p is a prime, are called Mersenne primes, named after Marin Mersenne. So the first four Mersenne primes are 3 (=2²-1), 7 (=2³-1), 31 (=2⁵-1) and 127 (=2⁷-1) although they rapidly increase as the value of p increases further. In December 2008, the largest known prime was expressed as 213,466,917-1, containing 4,053,946 decimal digits, was discovered by researchers in the GIMPS (Great Internet Mersenne Prime Search) project.  It would take more than three weeks to write out longhand and the race is on to find a prime number with p containing at least ten million digits.

Calculation of E8: In March, 2007 it was announced by Brian Conrey, Director of AIM (the American Institute of Mathematics in California), that group theorists at the institute had solved one of the toughest problems in mathematics. Namely, performing a calculation of the symmetry of a 248-dimensional object known as the Lie group E8. The solution is so large that it would take days to download over a standard Internet connection. Lie groups were introduced in the 19th century by the Norwegian mathematician Sophus Lie [pronounced "lee"], to express the symmetry of three-dimensional objects like spheres, cones and cylinders.

Counting Euler circuits in K_n: In graph theory, K_n represents a complete graph on n vertices in which all vertices are interconnected and it is known that the number of Hamilton circuits (starting at a given vertex and travelling through the other vertices exactly once, returning to the start vertex) is given by n! (i.e. factorial n). However, the number of Euler circuits (starting at a given vertex and travelling through the edges exactly once, returning to the start vertex) is much larger than the number of Hamilton circuits for given n. An asymptotic approximation of Euler circuits in K_n was given in April 2008 by John Dwyer. Euler circuits were introduced in 1736 by Swiss mathematician Leonhard Euler.

Solutions of famous equations in physics: The Dirac field equation is as central to theoretical physics as the Yang-Mills and Einstein field equations. Paul Dirac is regarded as the founder of quantum electrodynamics, being the first to use that term. Since his original study of the equation in 1928, several researchers have investigated its solutions, including Wathek Talebaoui in June, 2005.

Enormath is devoted to the calculations of solutions to enormous problems like these. We are usually concerned with either large values approaching infinity or infinitesimal values approaching zero, such as those encountered in boundary value differential equation problems. Some well-known problems even attract prizes for their solvers. These problems include the so-called Millenium Prize problems:

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