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PRIME NUMBERS, those divisible by no others than 1, are the building blocks of the natural number system, so let’s have some primal fun. In doing this, I celebrate the Greeks Eratosthenes and Euclid, the French Father Marin Mersenne and Édouard Lucas, and the American mathematician Frank Nelson Cole. Quite a quintet.
Eratosthenes of Cyrene was a general all-around scholar of the third century B.C. He was chief librarian of the Library of Alexandra, the most significant library of the ancient world. In drawing the first maps using parallels and meridians, he founded the science of geography.
Even today, the Sieve of Eratosthenes continues to be one of the most efficient means of identifying all prime numbers less than 10 million or so. Start with 2 and toss out each of its (evidently non-prime) multiples. Do the same with 3. Then 5 (as 4 has already gone through the sieve), then 7 and so on.
The numbers remaining in the sieve are primes.
How many primes are there?
Infinitely many. Euclid’s Theorem of the infinitude of primes appeared in his classic work, (see Byrne: Six Books of Euclid).
See http://wp.me/p2ETap-Fg for Euclid’s foundations of geometric thought. Other proofs of the infinitude of primes, making use of a variety of mathematical approaches, have appeared as recently as the 1950s.
Mersenne is known as the Father of Acoustics. One of his compositions is heard in Ottorino Respighi’s Ancient Airs and Dances. As a mathematician, he also has the distinction of having his work appear in postal metering.
A Mersenne Prime is a prime of the form 2n – 1. As shown below, the formula starts off productively. (If n is composite, then so is 2n – 1, so we only need consider 2p – 1 where p itself is prime.)
That is, the formula 2p – 1 generates lots of primes, but not without exception. 2047 = 23 x 89.
The search for Mersenne Primes has been going on since Mersenne published a paper about them in 1536. It’s not known, for example, whether the set of Mersenne Primes is finite or infinite. Thus far, particularly with large-scale computing techniques, larger and larger Mersenne Primes have been discovered.
There’s a collaborative project of volunteers, the Great Internet Mersenne Prime Search, that uses distributed computer power in its efforts. As of April, 2014, GIMPS has discovered 14 Mersenne Primes, 12 of which were the largest known prime numbers of any kind at their respective times of discovery.
Mersenne Numbers have proved useful in other mathematical areas too, including pseudorandom number generation (used in cryptography). The Mersenne Twister, developed in 1997, is the most widely used pseudorandom number generator, so named because its algorithm is based on Mersenne Prime 219937 – 1.
Other Mersenne candidates have fallen prey to further scrutiny. For a long time (in fact, dating from Mersenne’s 1536 publication), it was thought that 267 – 1 was prime. However, in 1876, French mathematician Édouard Lucas proved indirectly that it was not prime, though he didn’t furnish its factors. (See http://wp.me/p2ETap-26q for another Lucas achievement.)
Then, in 1903, in a presentation to the American Mathematical Society, Professor F.N. Cole went to a chalkboard and, in complete silence, performed the following calculations, all long-hand.
His two values agreed. Without a word, Professor Cole returned to his seat. He was rewarded with a standing ovation. ds
© Dennis Simanaitis, SimanaitisSays.com, 2014