Prime Time Part 2

prime numbers 2,3,5,7,11Prime numbers have fascinated mathematicians for centuries. Euclid (“The Father of Geometry”) was the first to prove that there is no greatest prime in what is a simple and easy-to-follow proof, beginning with the assumption that there IS a greatest prime. In the mode of Indirect Proofs, he then – in a few short steps – showed that such an assumption leads to a contradiction of the given, leaving the only other choice– that primes are infinite – to be the correct one.

Goldbach, an 18th century Prussian-born mathematician, intuited the following: Every even number greater than 2 can be expressed as the sum of two prime numbers. This assumption is still known as Goldbach’s Conjecture, rather than his Theorem, because it remains unproven to this day. He did not assert the uniqueness of the statement because it is easy to see that the sum can be expressed in many different ways. For example, 46 (an even number greater than 2) = 17 + 29 (the sum of two primes.) But, here are two other solutions for the same number: 46 = 23 + 23 and 46 = 5 + 41. There are more.

The study of Twin Primes has also intrigued many. A twin prime pair differs by 2, for example, 17 and 19; or 59 and prime table61. It has been conjectured, unproven as well, that the pairs of twin primes are infinite.

Sophie Germain (1776 – 1831) (one of the few female mathematicians) also did a lot of work with primes. In fact, we call a prime (p) a Germain Prime if 2p + 1 is also a prime. The first few Germain primes, therefore, are 2, 3, 5, 11, 23, 29…. Sophie’s parents did all they could to discourage her from studying what was then considered a man’s realm. In spite of this, she learned much and corresponded, using a male pseudonym, with two brilliant mathematicians of her day, LaGrange and Gauss. Lagrange was amazed that the author of the work sent to him was actually a female, but, recognizing her abilities, he became her mentor. With a male to introduce her, You can do itSophie was then free to enter the circle of scientists and mathematicians forbidden to her before. She went on to prove a theorem that was a major breakthrough in proving Fermat’s Last Theorem (not fully proven until the end of last century.)

A Mersenne prime is one that is one less than a power of 2 (2^n – 1), for example, 31 is a Mersenne prime because it can be expressed as 2^5 – 1. The Great Internet Mersenne Prime Search (GIMPS) consists of a large group of volunteers using super computers to find such numbers for monetary awards. The largest currently known prime of this sort is 2^43,112,609– 1 and was found by electrical engineer Hans-Michael Elvenich on Sept. 6, 2008. It has 12,978,189 digits. If each digit was .25 inches wide and took one second to write,  it would take approximMersenneately 150 days and 51 miles to write it all!

6 Comments

  1. I really like and appreciate your article.Really thank you! Want more.

    • Enjoyed them all….stay tune for a future post on a puzzler dealing with the power of 2.

      • Takes 2 to Tango? 😀

  2. That’s some prime information!

    Speaking of female mathematicians, two of the great names in my profession (computer science) are Lady Ada Lovelace (after whom the programming language Ada is named) and Rear Admiral Grace Hopper (who coined the term computer “bug”–after a real moth that had gotten into the works).

    And then there is Emmy Noether, perhaps one of the greatest of them all. Her work lays the basis for the connection between symmetry and physical conservation laws upon which most of modern physics is based.

    Florence Nightingale was also a mathematician… Anyone who says women can’t do math needs to write these each of names (and many others) 1024 times on a blackboard….

    • You are right about those women…..when I was teaching, I always assigned a Mathematician’s Biography to my students to get them to understand that there really were PEOPLE who “made” these things up, not a magic fairy to torture them. I specifically mentioned those you referenced, as I wanted the girls to see the female’s contributions for possible future inspiration.

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