Q: What is a “prime number” and how is it calculated?
J. Taylor, of Belleville
A: Boy, I’d love to know what prompted this question. I would think most people run from their final high school algebra class, crying with joy because they’ll never again have to think about prime numbers, polynomials and quadratic equations.
But since my curiosity is irrelevant here, let me provide what I hope will be a clear, easy answer. And before the rest of you blow off this question as boring, you might want to stick around for a few facts I’ll bet you find fascinating.
Sign Up and Save
Get six months of free digital access to Belleville News-Democrat
Most simply put, a prime number must fulfill three conditions:
First, it must be a number greater than 1. No negative numbers.
Second, it must be a “whole number” (also called an “integer”). No fractions allowed.
Finally — and most important — it must be a number that is evenly divisible only by itself and 1.
That’s all there is to them. Hence, such numbers as 2, 3, 5 and 7 are prime numbers because they can be divided only by themselves and 1 without anything “left over.” If you remember your basic division, yes, you can divide 5 by 2, but your answer would be 2 with a remainder of 1. In other words, it doesn’t come out even. That’s why 5 is a prime number as are 11, 13, 17, 19, 23, etc.
On the other hand, 4 is not a prime number because it is evenly divisible by 2. The same is true of 9, which is 3 times 3. Eventually, you wind up with numbers like 72, which is evenly divisible by 2, 3, 4, 6, 8, 9, 12, 18 and 36. Rather than being “prime,” these numbers are called “composite” numbers because they can be produced by multiplying together other positive whole numbers other than just themselves and 1.
And, before you ask, the number 1 itself is neither prime nor composite. Instead, it is generally just called a unit or unity. That’s why the list of prime numbers starts with 2.
2 The first in the list of prime numbers
17,425,170Number of digits in the largest prime number found so far
Now, you might be thinking that the list of prime numbers would be limited or finite. After all, no even number greater than 2 can be a prime number, so you’ve already eliminated half your possibilities. So, once you reach the sextillions and nonillions, wouldn’t every number be a product of numbers other than themselves and 1?
Nope, the list of primes is itself infinite, a fact established way back in about 300 B.C. by Euclid, a mathematician who led those scholarly Greeks in what is believed to be history’s first in-depth study of prime numbers. And now, with the use of the most powerful computers imaginable, mathematicians continue to prove him correct, leading to this piece of information that may amaze you:
On Jan. 25, 2013, math geeks discovered the largest prime number yet found. It’s the number 2 multiplied by itself 57,885,161 times minus 1, which produces a number with 17,425,170 digits. They found it on a computer used by Curtis Cooper at the University of Central Missouri, who is part of the Great Internet Mersenee Prime Search (or GIMPS). This search for prime numbers has been going on 18 years using combined worldwide computer resources with 360,000 CPUs that can do up to 150 trillion calculations per second. (Obviously, you can’t figure these out with a No. 2 pencil and paper.) And the search goes on, because there are infinitely more primes out there, although even at that unimaginable speed, it may take these computers years to find the next one.
By this time, however, you’re probably thinking, “Yes, that’s all very fascinating, but so what? Other than being a mathematical curiosity, what practical value do they have?”
Most important, their unique properties continue to make them a central element in the field of cryptography. Without going into terms that would make your eyes glaze over, let me just say that modern computer encryption used to keep your identity secure makes use of the prime factors of very large numbers because it would take the bad guys a long time to figure out the code. Primes also are valuable in more arcane areas of math that are far outside the scope of this column.
But even musicians and writers have found them intriguing. For example, French composer Olivier Messiaen used them to create ametrical music (music with no time signature or “beat”). He said these compositions were “inspired by the movements of nature, movements of free and unequal duration.” You also may remember in the novel and subsequent film “Contact,” renowned popular scientist Carl Sagan proposed that we might attract the attention of aliens on other worlds by transmitting prime numbers into space, thereby showing E.T. that intelligent beings inhabit our planet.
But perhaps it’s not just humans who have put prime numbers to constructive use. Some suggest Mother Nature has, too. For example, we constantly hear about the various types of cicadas, which have 7- or 13- or 17-year life cycles — all of which are prime numbers. Some suggest that this unusual span has evolved over eons to protect them against predators that may follow a more regular cycle.
Or, if you feel lazy, you can do what one blogger at www.stackexchange.com, a math website, tried. When he was 20 years old and living by himself he devised a schedule of chores based on prime numbers — for example, he washed the dishes every two days, watered the plants every three days, vacuumed every five, etc.
“It was a good system,” he concluded. “It made cleaning fun, it provided variety and structure at the same time, and I was obliged to devote the entire day to chores only once every 1,397.73 years.”
Ah, the joy of math.
What is a perfect number?
Answer to Monday’s trivia: “Out here in the fields, I fight for my meals. ...” That sloppy rhyme kicks off the first track of The Who’s “Who’s Next” album and was later used as the theme for “CSI: News York” in keeping with the TV franchise’s use of Who music. Some apparently think the song is entitled “Teenage Wasteland” after a critical phase used in the song. But the real title is “Baba O’Riley.” You never hear the name in the song, but lead guitarist Pete Townsend chose it because it combined the names of two of his biggest influences: Indian spiritual master Meher Baba and American composer Terry Riley.