A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.
However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
The property of being prime is called primality. A simple but slow method of checking the primality of a given number
n
{\displaystyle n}
, called trial division, tests whether
n
{\displaystyle n}
is a multiple of any integer between 2 and
n
{\displaystyle {\sqrt {n}}}
. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number is a Mersenne prime with 24,862,048 decimal digits.There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.
Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.
Hi,
Let's talk about Prime Numbers. Still an unsolved mystery, I don't understand why it's still unsolved. Has anyone discovered why its hard to find a pattern? Or is this a silly question?
Such as sqrt 5: (2.236067977...)
Start with the fractional seeds 2/1, 9/4,...
New members are generated (both numerators and denominators) by the rule new member = 4 times the current plus the previous.
Which generates the progrssion 2/1, 9/4, 38/17, 161/72, 682/305, 2889/1292...
I was told that all prime numbers (except 2 and 3) could be expressed as 6n +- 1 as long as the result divided by 5 is not another integer.
Is this true? Is there a proof for this (hopefully if possible not going much beyond basic calc, I am only in calc 1).
How can I prove that the sum of two odd primes will never result in a prime?
Would this be proof?:
Proof by contradiction:
The sum of two odd primes will sometimes result in a prime.
This is true because 2 + 3 = 5, which is a prime.
So since this is true, does this proof the...
I have to prove that if ab is divisible by the prime p, and a is not divisible by p, then b is divisible by p.
In order to prove this, I have to show (a,p)=1. I am not sure what this statement means.
Then I am supposed to use the fact that 1=sa + tp when s,t are elements of the set of...
Has anyone ever tried to make prime numbers into some kind of geometric equivalence? Such that prime numbers can be predicted through geometry?
I was thinking of a universe beginning with one 3D unit, and evolving from that unit. That all subsequent units would have a relation to the first...
Hey, check this out, the newest prime number found today!
http://story.news.yahoo.com/news?tmpl=story&cid=528&ncid=528&e=10&u=/ap/20031211/ap_on_hi_te/biggest_prime_number
This is strange... I can sort of proove this.
( n(1/2 + 1/3 + ... + 1/pa) - (1/3 + 2/5 + ... + (a-1)/pa) minus all whole queries ) <= ½
--> n = p
If it's true and I was the first to find the serie; can I name it after me?
In that case i would like to name my equation the...
Yes, ofcourse, a primenumber is a number that can only be divided with itself and 1.
But what do you call a query that has only got two different prime factors?
9 has got the factors 9, 3 and 1. But has only two different prime factors. 9 = 3*3, so the query has got the prime factors 3 and...
Consider all primes
2, 3, 5, 7, 11, 13...
and their products such that
2x3=6, 2x3x5=30, 2x3x5x7=210, 2x3x5x7x11=2310, 2x3x5x7x11x13=30030...
Is this latter series used in number theory?
Likewise, can one determine
lim (2+3+5+7+11+13...pn-1)/(2+3+5+7+11+13...pn)
n-->[oo]...
I've been Googleing for days now and haven't found a suitable answer to a question I have so I'll try it here. How exactly would knowing the distribution of prime numbers assist one in integer factorization?
Hi everyone, I'm new here. I have an interesting conjecture I have been trying to prove for some time. I've tried it out on a couple of forums, but perhaps some of you can help. Please make whatever comments you can (good or bad) and all input is welcome.
I will give this conjecture in...
Hi everyone, I'm new here. I have an interesting conjecture I have been trying to prove for some time. I've tried it out on a couple of forums, but perhaps some of you can help. Please make whatever comments you can (good or bad) and all input is welcome.
I will give this conjecture in...
Does exist any proof that prime numbers cannot be generated sequentially without jump across any one? And which is cardinality of prime numbers set? Is the set "the smallest" infinite set?
Problem: Prove that all odd numbers are prime.
Mathematician: 3 is prime, 5 is prime, 7 is prime, the rest follows from induction.
Physicist: 3 is prime, 5 is prime, 7 is prime, 9...that's probably an experimental error..., 11 is prime, 13 is prime. The law of nature in question is...
how can i proove or disproove that the sum of a prime numbers which equals to other prime numbers is a prime number?
i hope the question has been comprehended.
"A pair of mathematicians has made a breakthrough in
understanding so-called prime numbers, numbers that
can only be divided by themselves and one."
"Other mathematicians have described the advance as
the most important in the field in decades."
For the full BBC article...
SETI's search for alien contact includes detecting intelligent analog transmissions and also those carrying sequences of prime numbers signifying life elsewhere in the Galaxy.
What is the relative ease with which SETI can discern rational analog vs prime number transmissions? Is it a waste...