2007-01-01 03:19:45 · 22 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Theorem: There are infinitely many prime numbers.
Proof:
Suppose the opposite, that is, that there are a finite number of prime numbers. Call them p1, p2, p3, p4,....,pn. Now consider the number
(p1*p2*p3*...*pn)+1
Every prime number, when divided into this number, leaves a remainder of one. So this number has no prime factors (remember, by assumption, it's not prime itself). This is a contradiction. Thus there must, in fact, be infinitely many primes.
So, that proves that we'll never find all of the prime numbers because there's an infinite number of them. But that hasn't stopped mathematicians from looking for them, and for asking all kinds of neat questions about prime numbers.
2007-01-01 03:21:44 · answer #1 · answered by Wildamberhoney 6 · 5⤊ 0⤋
Yes, and others have linked to the proof of this. Now the trick is to find a formula which generates primes since they appear to be randomly distributed with no pattern as yet established. Here are two fascinating books and a website on the topic, which is the greatest unsolved problem in mathematics.
Also, a previous answer stated that the "distance between them is bigger". Actually, there is also infinite number of primes with a difference of only 2.
2007-01-01 04:01:24 · answer #2 · answered by Anonymous · 1⤊ 1⤋
This is an interesting question because it highlights a rarely taught aspect of mathematics, the fact that there are "different sizes" of infinity!
Think about it, if there are (and there clearly are) an infinite number of 'whole numbers' (because however big a number you have, you can always add 'one' to it), there are also an infinite number of 'even numbers'; but common sense tells you there are half as many "even numbers" as there are "whole numbers"!
Euclid came up with a final proof (see link), but common sense should also tell you that "prime numbers" are also infinite, and that infinite number must be smaller than the number of whole numbers or the number of even or odd numbers!
Now my head hurts...
2007-01-01 03:34:29 · answer #3 · answered by Anonymous · 1⤊ 0⤋
Yes, there are an infinite number of primes. Suppose that you have some prime number P which you suspect is the largest prime. You can construct the product of that number, multiplied by all smaller primes, and add 1. The result is a much larger prime.
2007-01-01 03:24:01 · answer #4 · answered by Anonymous · 2⤊ 1⤋
Yes. Let P be a prime number. Then P! +1 must also be prime because it leaves a remainder of 1 when divided by all number less than it.
Example:
7 is a prime number
So 7! =7*6*5*4*3*2 +1 always leaves a remainder of one when divided by all the numbers +< than 7.
So whatever prime number you find, you canalways find another prime greater than it.
2007-01-01 03:47:39 · answer #5 · answered by ironduke8159 7 · 2⤊ 1⤋
Prime numbers are infinite. I know for sure that some discovered a 17 digit prime number and there is no formula for testing it other than common sense.
2016-05-23 02:55:08 · answer #6 · answered by Anonymous · 0⤊ 0⤋
Yes, there are infinitely many primes.
Euclid showed that if one assumes there
is a largest prime, one can find a larger one.
Another proof of this can be done analytically.
It turns out that ∑1/p, where p ranges
over all primes, diverges. Thus there
must be infinitely many primes.
2007-01-01 04:26:19 · answer #7 · answered by steiner1745 7 · 1⤊ 0⤋
Yes
2007-01-01 08:38:22 · answer #8 · answered by confused 4 · 1⤊ 0⤋
Yes just like counting numbers are,would,nt like to list many over 97 though.
2007-01-01 03:28:07 · answer #9 · answered by Anonymous · 1⤊ 0⤋
Yes.
2007-01-01 03:21:08 · answer #10 · answered by Thomas K 6 · 2⤊ 0⤋