Possible webpage title: What are the Known Patterns in Prime Numbers?

In summary, in the conversation, the participants discuss the existence of patterns in prime numbers and how they can be represented by mathematical formulas. They also debate on the definition of a pattern and whether every sequence has a pattern. One participant mentions the Prime Number Theorem and another provides a formula for finding prime numbers.
  • #1
roger
318
0
Hi,

I just wondered what patterns are known about the prime numbers ?


cheers


roger
 
Mathematics news on Phys.org
  • #2
by pattern I mean any explicit formulas which give certain relations between primes.
 
  • #3
The Prime Number Theorem states that if you randomly select a number nearby some large number N, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime.

In other words, the prime numbers "thin out" as one looks at larger and larger numbers, and the prime number theorem gives a precise description of exactly how much they thin out.
 
  • #4
Well, one pattern I've noticed is they all have exactly 2 distinct divisors :shy:
 
  • #5
There is a pattern. We just don't know what it is yet.
 
  • #6
There is a pattern. We just don't know what it is yet.
Then what do you mean by "pattern"?

We know exactly when primes occur, and we have efficient methods for telling if a given number is a prime. We even have explicit formulas for the n-th prime number.
 
  • #7
Hurkyl said:
Then what do you mean by "pattern"?

We know exactly when primes occur, and we have efficient methods for telling if a given number is a prime. We even have explicit formulas for the n-th prime number.

By pattern, I mean formula.

Where is this explicit formula?
 
  • #8
JasonRox said:
There is a pattern. We just don't know what it is yet.

How do we know there is one?
 
  • #10
f(x) = px where pi is the ith prime.

Patterns are more to do with spotting simple occurrences in real life, not much to do with mathematics.
 
Last edited:
  • #11
Zurtex said:
Patterns are more to do with spotting simple occurrences in real life, not much to do with mathematics.

I don't agree with that statement.
 
  • #12
Helicobacter said:
How do we know there is one?

It was proven that if an algorithm can be made so that a computer can spit out the sequence, then a formula exists. Hence, a pattern.
 
  • #13
JasonRox said:
It was proven that if an algorithm can be made so that a computer can spit out the sequence, then a formula exists. Hence, a pattern.
That might not be very meaningful regarding the sequence of primes. For every algorithm there is an equivalent series of mathematical operations, or, if you combine these statements, a mathematical function. Hence, if an algorithm can "spit out" all the primes, then there is a mathematical function that produces that sequence. However this mathematical function can very well be infinite. This is the case with the sequence of primes. There is a very simple algorithm that outputs all the primes. Namely one that iterates through all the positive integers and divides each number by those that come before it (and are bigger than one), outputting a number when it is not divisable.
In order to output the sequence of all the primes this algorithm would need infinite time, an infinite sequence of operations and thus be equivalent to an infinite mathematical function.
What pattern exists there then?
If instead we wanted to output the first 100 primes, than the above algorithm would take less than 1000 operations, the equivalent mathematical function have in the order of 1000 operations. Would the sequence have a pattern?
For every sequence there is an algorithm that outputs all the elements in that sequence, it may require infinite memory and infinite time, but it will output all the elements in that sequence, does every sequence have a pattern then?

My definition of a pattern is something that is repeated. Yet, for every sequence of numbers without repetition, such as the primes, the digits of pi, etc, there is an algorithm capable of outputting that sequence.
I would say, if an algorithm can output any n elements of a sequence in O(n) time, with finite memory, then the sequence has a simple pattern. If it requires more then i don't know, but it certainly isn't a simple pattern.
 
Last edited:
  • #14
From..Wilson,s?..theorem we could find a formula to obtain primesifwe define:

[tex] g(x)=[\frac{(x-1)!+1}{x}]-\frac{(x-1)!+1}{x} [/tex]

where for the factorial we know the recurrence [tex] (x+1)!=xx! [/tex]

then every time g(x)=0 for an integer x we know that this x is prime...where [] is the "floor" function which is always an integer...
 
  • #15
-Job- said:
I don't agree with that statement.
Well define what a pattern is in mathematics for me then.

-Job- said:
. However this mathematical function can very well be infinite.
Lots of mathematical functions can be described as being 'infinitely long' take for example all taylor series, this does not make them less useful/

-Job- said:
This is the case with the sequence of primes. There is a very simple algorithm that outputs all the primes. Namely one that iterates through all the positive integers and divides each number by those that come before it (and are bigger than one), outputting a number when it is not divisable.
That's a particularly inefficient way but o.k.

-Job- said:
In order to output the sequence of all the primes this algorithm would need infinite time, an infinite sequence of operations and thus be equivalent to an infinite mathematical function.
That makes no sense, it takes an infinite amount of time to output all natural numbers i.e {1, 2, 3..) on a computer, would you describe the mathematical function f(x) = x as being infinite?

-Job- said:
What pattern exists there then?
If instead we wanted to output the first 100 primes, than the above algorithm would take less than 1000 operations, the equivalent mathematical function have in the order of 1000 operations. Would the sequence have a pattern?
For every sequence there is an algorithm that outputs all the elements in that sequence, it may require infinite memory and infinite time, but it will output all the elements in that sequence, does every sequence have a pattern then?

My definition of a pattern is something that is repeated. Yet, for every sequence of numbers without repetition, such as the primes, the digits of pi, etc, there is an algorithm capable of outputting that sequence.
I would say, if an algorithm can output any n elements of a sequence in O(n) time, with finite memory, then the sequence has a simple pattern. If it requires more then i don't know, but it certainly isn't a simple pattern.
Most of that is so much gibberish I can't really make sense of. Algorithms exist that calculate if a number is a prime or not, much faster than O(n).
 
Last edited:
  • #16
-Job- said:
That might not be very meaningful regarding the sequence of primes. For every algorithm there is an equivalent series of mathematical operations, or, if you combine these statements, a mathematical function. Hence, if an algorithm can "spit out" all the primes, then there is a mathematical function that produces that sequence. However this mathematical function can very well be infinite. This is the case with the sequence of primes. There is a very simple algorithm that outputs all the primes. Namely one that iterates through all the positive integers and divides each number by those that come before it (and are bigger than one), outputting a number when it is not divisable.
In order to output the sequence of all the primes this algorithm would need infinite time, an infinite sequence of operations and thus be equivalent to an infinite mathematical function.
What pattern exists there then?
If instead we wanted to output the first 100 primes, than the above algorithm would take less than 1000 operations, the equivalent mathematical function have in the order of 1000 operations. Would the sequence have a pattern?
For every sequence there is an algorithm that outputs all the elements in that sequence, it may require infinite memory and infinite time, but it will output all the elements in that sequence, does every sequence have a pattern then?

My definition of a pattern is something that is repeated. Yet, for every sequence of numbers without repetition, such as the primes, the digits of pi, etc, there is an algorithm capable of outputting that sequence.
I would say, if an algorithm can output any n elements of a sequence in O(n) time, with finite memory, then the sequence has a simple pattern. If it requires more then i don't know, but it certainly isn't a simple pattern.

I never said simple pattern.

Also, even if you give a computer the direct formula of a sequence, it requires more and more memory as n gets larger. So, your point is invalid.
 
  • #17
-Job- said:
I would say, if an algorithm can output any n elements of a sequence in O(n) time, with finite memory, then the sequence has a simple pattern.
Okay.

It's odd you would think that the sequence 1, 2, 3, 4, 5, 6, ... of positive integers, in order, doesn't have a simple pattern!

The list consisting of the first N integers has length [itex]\Omega (N \lg N)[/itex], so any algorithm for computing this sequence must take [itex]\Omega(N \lg N)[/itex] time.

If you don't allow the algorithm to review what it's already printed, it must require [itex]\Omega(\lg N)[/itex] memory to output the first N terms of this sequence. If you do allow the algorithm to review what it's already printed, I suspect it still takes [itex]\Omega(\lg \lg N)[/itex] memory, but I haven't worked out a proof yet.


-Job- said:
Yet, for every sequence of numbers without repetition, such as the primes, the digits of pi, etc, there is an algorithm capable of outputting that sequence.
That's not true! There are countably many algorithms, and uncountably many sequences. (Even if we only consisder sequences whose terms are either 0 or 1)
 
Last edited:
  • #18
Also, even if you give a computer the direct formula of a sequence, it requires more and more memory as n gets larger. So, your point is invalid.
You're misinterpreting me. I said if the algorithm takes a finite amount of memory, meaning if the algorithm can be stored in a finite amount of memory, meaning that if you had an algorithm you'd be able to store it in a hard drive with finite capacity, meaning that the algorithm isn't infinitely large. I didn't mean memory usage at run time. For that i would bound it at about O(n).

Hurkyl said:
Okay.

It's odd you would think that the sequence 1, 2, 3, 4, 5, 6, ... of positive integers, in order, doesn't have a simple pattern!

The list consisting of the first N integers has length [itex]\Omega (N \lg N)[/itex], so any algorithm for computing this sequence must take [itex]\Omega(N \lg N)[/itex] time.

My definition, where i mentioned O(n) wasn't with respect to the number of bits (that was patent in my statement, i thought). So a list composed of the first n integers would require O(n).

That's not true! There are countably many algorithms, and uncountably many sequences. (Even if we only consisder sequences whose terms are either 0 or 1)

But an algorithm doesn't have to create a sequence. The sequence might be embedded in the algorithm, hence to print a sequence all we have to do is:
mySequence = [2, 3, 5, 7, 11, 13, 17, 19 ...];
while(true){
trace("f(" + i + ") = " + mySequence);
}
For an infinite sequence the algorithm is infinitely large (requires infinite memory) and takes infinite time (has to of course). However, for every sequence, there is an algorithm that prints all the elements in that sequence. Unless we choose to define algorithms as something finite. Do you see what i mean?

To reiterate more precisely:
If an algorithm of finite size can output the first N elements of a sequence in O(N) time with O(N) memory, or, in other words, in O(2^n) time and O(2^n) memory where n=log(N), then that sequence has a simple pattern.

This was an impulsive statement, so if it isn't very adequate I'm not going to cry over it. :smile:
 
  • #19
As you haven't defined 'simple' your statement is vacuous or tautological.
 
  • #20
Let's make it the definition of "simple pattern" then, and work with that. :)
 
  • #21
Then it is tautological, so who can argue against it, but also a meaningless statement at the same time.
 
  • #22
Yes, or, you could argue that the definition of a simple pattern ought to encompass more, or less, than the definition above. Maybe argue why certain patterns should be classified as simple when the current definition doesn't classify them as such (like Hurkyl, when he proposed 1, 2, 3, 4...). A sort of a progressive discussion. I think it's acceptable.
 

1. What are prime numbers?

Prime numbers are positive integers that are only divisible by 1 and themselves. Examples of prime numbers include 2, 3, 5, 7, and 11.

2. How can we identify prime numbers?

One way to identify prime numbers is by using the Sieve of Eratosthenes, which is a method of systematically eliminating non-prime numbers to find all prime numbers up to a given limit. Another method is to use trial division, which involves checking if a number is divisible by any numbers other than 1 and itself.

3. Are there any patterns in prime numbers?

While prime numbers may seem random, there are some patterns and properties that have been identified. For example, prime numbers tend to become less frequent as they get larger, and there are certain patterns in the distribution of prime numbers on the number line.

4. Why are prime numbers important in mathematics?

Prime numbers play a crucial role in many areas of mathematics, from cryptography to number theory. They are also important in various real-world applications, such as in generating secure codes and in data encryption.

5. Can we predict the next prime number?

No, it is not possible to accurately predict the next prime number. While there are patterns and properties in prime numbers, they do not follow a specific sequence or formula that allows for prediction. Prime numbers are considered to be random and unpredictable.

Similar threads

Replies
8
Views
261
Replies
1
Views
741
  • General Math
2
Replies
56
Views
4K
Replies
1
Views
705
  • General Math
Replies
23
Views
3K
Replies
3
Views
378
Replies
3
Views
748
Replies
8
Views
1K
Replies
13
Views
1K
Replies
1
Views
695
Back
Top