Mathematical Projects

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testing out conjectures

The project "Computing Minimum Equal Sums of Like Power" is based on a conjecture by the mathematician Euler in 1772: (Meyrignac)

 "If a sum of n positive kth powers equals one kth power, then n >= k".

While that is now known to be a false statement, it provided the basis for the distributed computing project to calculate numbers (f(k,m)) to fit an equation, f(k,m+c) <= f(k,m)+c. There is a proof for this equation for those interested in the mathematics of it on the project's website, http://euler.free.fr/details.htm.

The users of this project churn out results almost every day. The solutions are given in the form (k,m,n), where k = power, m = left terms, and n= right terms. A description of the result showing equality generally follows. A short example of this is (2,1,2) 5=4+3. The power of this equation is 2 (k), so one would square all of the numbers on both sides of the equation. The 1 (m) denotes the number of left terms, and there is, indeed, just one term on the left side of the equation — the 5. The n represents the number of terms on the right side of the equation — here, two. This gets very complicated as numbers get larger; on a standard calculator one usually receives an error when trying to square a number with more than ten digits. Imagine trying to cube it or taking it to the fourth power!

more prime projects

Mathematician Fermat (wrongly) conjectured that numbers which have the form 2^2^n + 1 are prime. Thus, the "n" in that equation is dubbed a Fermat number. Fermat numbers aren't new in the world of distributed computing. Now a Distinguished Scientist at Apple, Richard Crandall and colleagues at NeXT (now owned by Apple) networked some NeXT machines and performed "the deepest computation ever performed" (Rheingold 180) by asking if the 24th Fermat number was prime. That number has more than 5 million digits and it took 100 quadrillion calculations (10^17), which, according to Crandall was, "about the same number of operations Pixar required to render a Bug's Life." (Rheingold 180)

There is now another similar distributed computing project in progress on the internet, called PrimeForm. PrimeForm is trying to find prime numbers using a related theory, the "Fermat Little Theorem.": "Let p be a prime which does not divide the integer a, then a^p^-1 = 1 (mod p)". (Caldwell: "Fermat's...") PrimeForm has a unique feature: the user chooses their own mathematical expression (such as p(k)=2n +1), parameters, and range of numbers in which to test a prime number. PrimeForm uses a test to factor the numbers, using N+1 and N-1.

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