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.
I have tested the above proposition thru the integer 90, and have found that the proposition holds true. I'm not sure of the method to prove that this would always be true. Any help, criticism, or proof is welcome.
Proof:
Suppose that the only prime numbers of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##,
and let ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3 ##.
Since ## N ## is odd, ## N ## is divisible by some prime ## p ##,
so ## 4p_{1}^{2}\dotsb p_{n}^{2}\equiv -3\pmod {p} ##.
That is, ##...
Proof:
Suppose for the sake of contradiction that the only primes of the form ## 8k-1 ## are ## p_{1}, p_{2}, ..., p_{n} ##
where ## N=16p_{1}^2p_{2}^2\dotsb p_{n}^2-2 ##.
Then ## N=(4p_{1}p_{2}\dotsb p_{n})^2-2 ##.
Note that there exists at least one odd prime divisor ## p ## of ## N ## such...
Let ## p ## be an odd prime.
Then ## 23 ## is a quadratic residue modulo ## p ## if ## (23|p)=1 ##.
Applying the Quadratic Reciprocity Law produces:
## (23|p)=(p|23) ## if ## p\equiv 1\pmod {4} ##
## (23|p)=-(p|23) ## if ## p\equiv 3\pmod {4} ##.
Now we consider two cases.
Case #1: Suppose ##...
im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000.
note that I am looking for a novel proof, not just some brute force calculation.
(this isn't homework, I am just curious.)
F(n)=##n^2 −n+41## generates primes for all n<41.
Questions:
(1) Are there polynomials that have longer lists?
(2) Is such a list of polynomials finite (yes, no, unknown)?
(3) Same questions for quadratic polynomials?
Proof:
Let ## a>5 ## be an integer.
Now we consider two cases.
Case #1: Suppose ## a ## is even.
Then ## a=2n ## for ## n\geq 3 ##.
Note that ## a-2=2n-2=2(n-1) ##,
so ## a-2 ## is even.
Applying Goldbach's conjecture produces:
## 2n-2=p_{1}+p_{2} ## as a sum of two primes ## p_{1} ## and ##...
Proof:
Let ## p ## and ## q ## be primes such that ## p-q=3 ##.
Now we consider two cases.
Case #1: Suppose ## p ## is an even prime.
Then ## p=2 ##, because ## 2 ## is the only even prime.
Thus ## 2-q=3 ##, so ## q=-1 ##,
which contradicts the fact that ## q ## is prime.
Case #2: Suppose ## p...
Proof:
Suppose ## p ## and ## p+2 ## are twin primes such that ## p>3 ##.
Let ## p=2k+1 ## for some ## k\in\mathbb{N} ##.
Then we have ## p+(p+2)=2k+1+(2k+1+2)=4k+4=4(k+1)=4m ##,
where ## m ## is an integer.
Thus, the sum of twin primes ## p ## and ## p+2 ## is divisible by ## 4 ##.
Since ##...
Proof:
Suppose ## p ## and ## p+2 ## are twin primes.
Then we have ## p(p+2)+1=p^2+2p+1=(p+1)^2 ##.
Thus, ## (p+1)^2 ## is a perfect square.
Therefore, if ## 1 ## is added to a product of twin primes,
then a perfect square is always obtained.
Proof:
Consider all primes ## p\leq \sqrt{1949} \leq 43 ## and ## q\leq \sqrt{1951} \leq 43 ##.
Then we have ## p\nmid 1949 ## and ## q\nmid 1951 ## for all ## p\leq 43 ##.
Thus, ## 1949 ## and ## 1951 ## are both primes.
By definition, twin primes are two prime numbers whose difference is ## 2...
Proof:
Suppose for the sake of contradiction that ## n ## contains at least three prime factors.
Let ## n=p_{1} p_{2}\dotsb p_{X} ## for ## x\geq 3 ##.
Note that ## n=p_{1} p_{2} p_{3} ##.
Then we have ## p_{1}\nleq \sqrt[3]{n} ##, ## p_{2}\nleq \sqrt[3]{n} ## and ## p_{3}\nleq \sqrt[3]{n} ##...
Proof:
Suppose ## p\geq q\geq 5 ## and ## p ## and ## q ## are both primes.
Note that ## p ## and ## q ## are not divisible by ## 3 ##,
so we have ## p^{2}-1\equiv 0 (mod 3) ## and ## q^{2}-1\equiv 0 (mod 3) ##.
This means ## 3\mid((p^{2}-1)-(q^{2}-1)) ##,
and so ## 3\mid p^{2}-q^{2} ##.
Since...
w = {0,0 | 1,1 | 2,2...}
Let x = number of primes up to w+1
Let y = number of primes up to w-1
Now there's an empty prime box in the 0,0 slot not connected to anything.
So I let x = p-1 and y = p+1
p = [p0, p1, p2...]
Now p0 becomes 1,0/1
It can be either on or off.
For the sake of...
Show by contradiction that
$$
\sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p}
$$
diverges. Which famous result is an immediate corollary?
Often I read that the Riemann Hypothesis (RH) is related to prime numbers because of the equivalence on Re(s)>1 of the zeta function and Eurler's product formula
, but is it more accurate that the relevance of the RH to primes (or vice-versa) is either that the RH implies formulas for the...
Hi All. Does anybody have a reference, (book, internet site) - besides those books of Paulo Ribenboim - where one can find a compilation of demonstrations of the Euclid's theorem on the infinitude of primes?
As a suggestion, if the known proofs are neither too many not too long, it would be nice...
Homework Statement
Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ##
Homework Equations
Some theorems might help?
Any ##n>1## must have a prime factor
if n is composite then there is a prime ##p<√n## such that ##p|n##
The Attempt at a Solution...
If you have 2 integers n and n+1, it is easy to show that they have no shared prime factors.
For example: the prime factors of 9 are (3,3), and the prime factors of 10 are (2,5).
Now if we consider 9 and 10 as a pair, we can collect all their prime factors (2,3,3,5) and find the maximum, which...
Homework Statement
https://projecteuler.net/problem=58
Homework EquationsThe Attempt at a Solution
def prime(N):
if N == 1:
return False
y = int(N**0.5)
for i in range(2,y+1):
if N%i == 0:
return False
return True
def finder(N):
L = len(N)...
Hello everyone.
[Firstly, I didn't know if this belongs here or in General; please move if appropriate].
<Moderator's note: moved to GD>
I was reading this paper on the AKS primality test (undergraduates can understand it, highly recommended!), and on page 7 the author brings up the story of...
Homework Statement
1.31 Let ##p## be a prime and let ##q## be a prime that divides ##p - 1##.
a) Let ##a \epsilon F^*_p## and let ##b = a^{(p - 1)/q}##. Prove that either ##b = 1## or else ##b## has order ##q##. (Recall that order of ##b## is the smallest ##k \ge 1## such that ##b^k = 1## in...
Does there exist a polynomial P(x) with rational coefficients such that for every composite number x, P(x) takes an integer value and for every prime number x, P(x) does not take on an integer value?
Can someone please guide me in the right direction? I've tried to consider the roots of the...
Homework Statement
I don't understand the lemma.
Homework EquationsThe Attempt at a Solution
Isn't all prime number not a product of primes? The lemma doesn't make sense to me... Moreover, if m=2, m-1 is smaller than 2, the inequality also doesn't make sense. Please help me
Is there a relation between these concepts? We know that as we move down the number line, the primes become less common. This makes sense: as we get more and more numbers, they could be constituted using all the primes that came before early on and less likely to be constituted by all the recent...
At a sufficient resolution, such as mapping every knowing prime gap...
Could a fractal equation created to perfectly describe this (at max possible res.):
http://techn.ology.net/the-density-plot-of-the-prime-gaps-is-a-fractal/
Likewise clear a path to a nth-dimensional fractal for prime gaps...
I think that we have to get all 2 digit odd numbers that can be expressed as the sum of 2 primes and subtract that from 45, so I think that the answer would be 45-(number of 2 digit integers n that are prime and have n-2 be prime as well)?
I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ...
I need some help with understanding Example 1.4.1 ...
Example 1.4.1 reads as follows:
In the above text by Alaca and Williams we read the...
I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ...
I need some help with understanding Example 1.4.1 ...
Example 1.4.1 reads as follows:
In the above text by Alaca and Williams we read the...
I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ...
I need some help with the proof of Theorem 1.2.2 ...
Theorem 1.2.2 reads as follows:
https://www.physicsforums.com/attachments/6515
In the...
I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ...
I need some help with the proof of Theorem 1.2.2 ...
Theorem 1.2.2 reads as follows:
In the above text from Alaca and Williams, we read the...
Homework Statement
Prove that there are infinitely many primes using Mersenne Primes, or show that it cannot be proven with Mersenne Primes.
Homework Equations
A Mersenne prime has the form: M = 2k - 1
The Attempt at a Solution
Lemma: If k is a prime, then M = 2k - 1 is a prime.
Proof of...
given : A=B$\times $C
with the following characters
(1) A,B,C $\in N$
(2)A is a palindrome
(3)B and C are all palindromic primes
(4) 1000<A<2000
(5) B is a 2-digit number
(6) C is a 3-digit number
find A
I can prove the twin prime counting function has this form:
\pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1,
where \pi_2(n) is the twin prime counting function, f(n) is the number of twin composites less than or equal to n and \pi(n) is the prime counting function.
At n=p_n, this becomes
\pi_2(p_n) =...
As far as I can tell for the equation n!/n^2=x or n-1!/n=x, if x is a natural number then it seems n is composite. If x is a non-natural number then it is prime (excluding 4). I am aware that this is not very practical since I am using factorials and the numbers get very large. But it still...
Bear with me, I know nothing.
Eons ego @micromass told me about this beautiful formula:
\frac {\pi^2} 6 = \prod\limits_{P}\left( 1-\frac 1 {p^2}\right) ^{-1}
where p are primes. Just a few minutes ago I have learned about the Basel problem and its solution:
\sum \limits_{n=1}^{\infty} \frac...
I am trying to understand a condition for a nonincreasing sequence to converge when summed over its prime indices. The claim is that, given a_n a nonincreasing sequence of positive numbers,
then \sum_{p}a_p converges if and only if \sum_{n=2}^{\infty}\frac{a_n}{\log(n)} converges.
I have tried...
To start let's use this trick to find a sequence of primes
11+13-7=17
This trick starts with prime 11 and the next in sequence 13. After you add them together to get 24 you subtract 7 which is the prime in sequence before 11, and you will get the answer 17.
Now this will find every single prime...
Homework Statement
sn= 1/n if n is a prime number; sn = 0 if n is not prime.
Homework EquationsThe Attempt at a Solution
We know that there are infinitely many prime numbers as n tends to infinity so why is ## \frac{1}{(prime number)} ## not equal to zero when n is a prime number?
Homework Statement
Assume that n > 1 is an integer such that p does not divide n for all primes ≤ n1/3. Show that n is either a prime or the product of two primes. (Hint: assume to the contrary that n contains at least three prime factors. Try to derive a contradiction.)
Homework Equations...