Prime time news in the world of mathematics
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Published: June 29, 2005
Some interesting mathematical results regarding prime numbers have been in the news recently.
Recall that prime numbers are those whole numbers, greater than 1, that are evenly divisible only by one and themselves. The first few are 2, 3, 5, 7, 11 and so on. Primes are, in some sense, the "atoms'' of numbers and play an integral role in many mathematical problems.
Earlier this year it was announced that the world's largest prime had been found. The Great Internet Mersenne Prime Search has found the number 2 raised to the power 25964951 minus 1 to be prime, thus establishing the record. This record prime has more than 7.8 million digits! Mersenne primes are of the form 2 raised to a power minus 1 and are named after the French monk Marin Mersenne, who in the early 17th century published a list of these primes. The first few are 3, 7 and 31, with the latest record prime only the 42nd Mersenne prime.
GIMPS, started in 1996, has found the last 8 Mersenne primes in a joint project that harnesses the efforts of idle personal computers in the computational effort to discover the new record primes. Check out www.mersenneforum.org for details on the project and how you might join the prime effort.
Primes can also provide a source for simple problems that have baffled mathematicians for centuries. For example, the pair of primes 5, 7 and 11, 13 and 41, 43 are called twin primes. The Twin Prime conjecture states that there are infinitely many such pairs. No one has been able to verify whether it's really true. Results on the gaps between successive primes (not just twin primes) announced this past spring will perhaps provide insight to proving the twin prime conjecture.
Recent work by a graduate student at the University of Wisconsin sheds some light on the patterns associated with partitions of numbers — the ways to decompose a positive integer into a sum. For example, there are five partitions of 4: 1+1+1+1, 1+1+2, 1+3, 2+2, and 4. Thus we could write P(4) = 5. It is easy to verify that P(1) = 1, P(2) = 2, P(3) = 3, and somewhat more involved to see that P(5) = 7, P(6) = 11, P(7) = 15, P(8) = 22, and P(9) = 30. As you might guess, the number of partitions gets large rather quickly with P(22) = 1002 and P(200) almost 4 trillion.
Karl Mahlburg provides an explanation about patterns first observed by Indian mathematician Srinivasa Ramanujan (1887-1920) and partially explained by physicist Freeman Dyson within the sequence of the partition numbers (1, 2, 3, 5, 7, 11, 15, 22, 30 …).
For example, starting with P(4), every fifth partition number is divisible by 5, and starting with P(5) every seventh partition number is divisible by 7. Mahlburg's result gives a natural explanation (at least to a number theorist!) for this sort of divisibility or congruence relationship.
Freeman Dyson perhaps says it best in commenting on Mahlburg's work: "… the story as a whole is a sequence of episodes of rare beauty, a drama built out of nothing but numbers and imagination.''
Send your comments or questions to Dr. Jim Wright, Mathematics Department, Green Mountain College, Poultney VT 05764, or e-mail him at wrightj@greenmtn.edu.
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