By Tom Kando
Although I am totally a non-mathematician, I have long been fascinated by prime numbers. I don’t know why. Here are some random thoughts about this subject:
A prime number is a whole number that can only be divided by 1 and by itself. Or put differently, prime numbers cannot be divided by any other whole number without leaving a fraction. The smallest 25 prime numbers (those under 100) are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,73, 79, 83, 89, 97.
As you move up the scale of natural numbers, prime numbers become less and less frequent, but there is no largest prime number. There is an infinite number of prime numbers. Euclid was the first to prove this.
The only practical use of prime numbers, so far, is in encryption and secret codes for the safety and protection of sensitive materials in the military, commerce, etc. It’s not clear whether nature and the physical world have any use for prime numbers, although some say that timing their life cycle through the use of prime numbers (e.g. emerging from the ground after 7, 11, 13, etc, years) gives some insects certain survival advantages...
One fascinating characteristic of prime numbers is that they seem to obey no other law than that of chance, and that nobody can predict what the next higher - and not yet discovered - prime number will be!
Over the centuries, mathematicians have discovered larger and larger prime numbers, and now with computers, the search has reached astronomical levels.
The unfathomable magnitude of the prime numbers which current researchers are looking for - and finding - totally blew my mind:
On October 22, 2009, the Greater Internet Mersenne Prime Search Research Project was awarded a $100,000 prize for discovering the largest known prime: 2 to the power: 43,112,609, minus 1
This is a number with almost 13 million decimal digits. I calculated that writing it out would require about 4 ½ thousand pages, or fourteen 300- page long books.
Researchers are currently looking for primes with 1 billion digits. If they discover one, writing out that number would require 350 thousand pages, or 1,200 books. (One average book page has about 3000 characters).
Years ago my children and I were bantering at the dinner table, and I told them what a Googol was: it is a “1" followed by one hundred zeros. There are other names for Googol, for example “ten thousand sexdecillion.” Whatever you call it, my children and I agreed, back then, that this was a lot.
But now? My God, a Googol is nothing, compared to a prime number that has a billion decimal digits! Wouldn’t such a number far exceed the total number of atoms in the Universe? Isn’t this dizzying? leave comment here
5 comments:
On the planet where I come from PN's are an integral part of our culture and religion. Yes, they are mysterious. Much like their cousins, the irrational numbers. The latter of course are never ending and non repeating decimals; such as, "pi" and "e". Don't try to understand them. Just accept them on Faith. Trust in the Grand Pi Wizard...............and don't make trouble!
Tom Bridges
Prime numbers are not without a pattern. Prof. Peter Plichta in Germany discovered that they all fall in rows on his prime number cross. http://www.ganderson.us/images/prime%20numbers.jpg The image shown is copyrighted by Adri de Groot
thanks for your insights, Tom and Gordon.
As Alice would say, it's getting curiouser and curiouser.
If the universe turns out to be "infinite," that is to say, without boundaries and with no limit to its "contents," then the number of prime numbers will be a smaller infinite number than the larger infinite number of atoms in the universe. This is what I understand from my high school reading of George Gamow's ONE, TWO, THREE...INFINITY, in which he explains that some infinite sequences are actually larger than others (for example, the number of all integers should be twice the size of the number of odd integers)....
Tom 1 to Tom 3 (Tom 2 being Tombridges):
You also make 2 good points. The last one (about odd integers being half of all integers) is indisputable.
On the other hand, I have long been told by cosmologists something which I have not been able to comprehend, namely that the Universe is NOT infinite. When I said that these gigantic numbers were larger than the total number of atoms, I was going by the standard, big- bang, expanding-contracting, finite universe concept.
But an infinite universe without boundaries makes much more common sense to me, as it seems to you, too. In that case, of course, there would be more atoms than prime numbers.
And to show my further ignorance, I am pleasantly surprised to learn that mathematicians recognize that some infinite numbers are larger than others.
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