Exploring the Relationship Between Primes and Their Products in Number Theory

In summary: The smallest prime is 2, since 2 is the smallest number that has exactly two divisors.- Warren##SolutionIn summary, the conversation discusses prime numbers and their products, specifically the series of products of primes and their limits. The conversation also mentions the primorial function and its use in number theory. The conversation includes a proof for the limit of the series involving the largest prime less than n. The conversation also mentions Mersenne-numbers and the Lucas-Lehmer Test for determining if they are prime. The conversation concludes with a discussion about a potential equation involving primes and their products.
  • #1
Loren Booda
3,125
4
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]

in analogy to phi of Fibonacci numbers?
 
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  • #2
This function is called the primorial function.
 
  • #3
Thanks for the helpful hyperlink, Hurkyl.

Do you or anyone else have a hint about the second series I mentioned, the prime Fibonacci analog, and its limit:

lim (2+3+5+7+11+13...pn-1)/(2+3+5+7+11+13...pn)=?
n-->[oo]
 
  • #4
I'm not sure why you call it a Fibonacci analog...

Anyways, your fraction can be rewritten as:

[tex]
\frac{2 + 3 + \dots + p_{n-1}}{2 + 3 + \dots + p_n}
= 1 - \frac{p_n}{2 + 3 + \dots + p_n}
= 1 - \frac{1}{1 + \frac{2 + 3 + \dots + p_{n-1}}{p_n} }
[/tex]

So solving your limit reduces to finding

[tex]
\lim_{n\rightarrow\infty}\frac{2 + 3 + \dots + p_{n-1}}{p_n}
[/tex]

No proof of any value for this leaps to mind, however.
 
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  • #5
I don't know how I managed not to read the big thread on TeX. :frown:

(PS you got some groupings wrong)
 
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  • #6
Originally posted by Hurkyl
I don't know how I managed not to read the big thread on TeX. :frown:

(PS you got some groupings wrong)
Whoops! Feel free to edit my posts to reflect the correct TeX, or delete them altogether if you wish.

Also, please note that I don't intend to coerce people into using TeX if they are already comfy and happy with basic HTML.

- Warren
 
  • #7
I need to learn LaTeX eventually anyways, might as do it here where I can get some practical benefit out of it. :smile: The only LaTeX I've written thus far was for writing a tutorial to use some code I had written, so I haven't gotten to play with any of the math stuff!
 
  • #8
All right, here goes.

Define

[tex]
S(n) := \frac{2 + 3 + ... + q_n}{n}
[/tex]

where [tex]q_n[/tex] is the largest prime less than [tex]n[/tex]. I aim to prove:

[tex]
\lim_{n\rightarrow\infty} S(n) = \infty
[/tex]

From which we can deduce

[tex]
\lim_{n\rightarrow\infty} \frac{2 + 3 + \dots + p_{n-1}}{p_n}
= \infty
[/tex]

and thus

[tex]
\lim_{n\rightarrow\infty} \frac{2 + 3 + \dots + p_{n-1}}{2 + 3 + \dots + p_n}} = 1
[/tex]


It's clear that

[tex]
\lim_{n\rightarrow\infty} S(n) = \lim_{\substack{ n\rightarrow\infty \\ n~{\it even}} } S(n)
[/tex]

So I will restrict my attention to the case where [tex]n[/tex] is even.


The general approach is to estimate the numerator of [tex]S(2n)[/tex] by just looking at the primes in the range [tex][n, 2n)[/tex], and underestimating [tex]S(2n)[/tex] by [tex]n[/tex] times the number of primes in this range. To do this, I will use Chebyshev's bound on the prime counting function:

[tex]
\frac{7}{8} < \frac{ \pi(n) }{ \frac{n}{ln~n} } < \frac{9}{8}
[/tex]

So here goes:

[tex]
\begin{equation*}
\begin{split}
S(2n) &= \frac{2 + 3 + \dots + q_{2n}}{2n} > \frac{1}{2n} (\pi(2n) - \pi(n)) n \\
&> \frac{1}{2} \left (\frac{7}{8} \frac{2n}{ln~2n} - \frac{9}{8} \frac{n}{ln~n} \right)
\end{split}
\end{equation*}
[/tex]

A bit of elementary calculus proves that

[tex]
\lim_{n\rightarrow\infty} \frac{1}{2} \left( \frac{7}{8} \frac{2n}{ln~2n} - \frac{9}{8} \frac{n}{ln~n} \right) = \infty
[/tex]

And we're done.
 
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  • #9
Hurkyl,

What experience do you have in math? You seem the most competant of a talented bunch here at PF. I hope you have seen the http://www.quantumdream.net at my website, my greatest accomplishment in mathematics.

LB
 
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  • #10
It's the LaTeX. It makes one look smarter. :smile:

I'm relatively fresh out of school, actually. I got my BS's in math and computer science two years ago, and started work this January. I've been hired as a mathematician, but my work thus far has been leaning more towards the programming.

However, math has been my hobby since I was little, so I have experienced a lot more than these credentials would suggest.
 
  • #11
Sorry.

But is this right then?

2p - 1 = p

2 - 1
4 - 1
8 - 1
32 - 1
128 - 1
2048 - 1


Am I wrong?
 
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  • #12
p*p

A multiple between two primes is always right in the middle of two primes.

1, 2*1, 3, 2*2, 5, 2*3, 7, /, 3*3, 2*5, 11, /, 13 etc.

3*3 is right in the middle of 7 and 11
2*5 is right in the middle of 7 and 13
etc.

(11*3 is right in the middle of 37 and 29)


Erik-Olof Wallman
 
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  • #13
But is this right then?

2<sup>p</sup> - 1 = p

2 - 1
4 - 1
8 - 1
32 - 1
128 - 1
2048 - 1


Am I wrong?

?? Yes, "2<sup>p</sup>-1= p" is wrong. In fact, the example you give show that :
4-1= 2<sup>2</sup>-1= 2 NOT 2.
8-1= 2<sup>3</sup>-1= 7 NOT 3, etc.

What exactly did you intend to say?
 
  • #14
I think he means to suggest that for every p, 2p-1 is a prime number? These are the so-called Mersenne-numbers and not all of them are prime. Although the largest prime numbers found to date are typically Mersenne-numbers, not every p generates a prime number. Simplest example: 211-1 is composite (23*89).
 
  • #15
Originally posted by suyver

That's what I, eh, he ment.

Best wishes Erik-Olof Wallman!
 
  • #16
I had never seen that before
 
  • #17
Originally posted by Loren Booda
I had never seen that before

The Mersenne-numbers are very interesting because of the so-called Lucas-Lehmer Test, which is a (relatively) easy method of deciding if any Mersenne-number is prime or composite.
 
  • #18
p1 = 1
p2 = 2
p3 = 3
p4 = 5
p5 = 7
p6 = 11

All numbers within the serie:
p1*p2*p3*p4*...*pn +/- 1 are primenumbers.

In this serie, pn can be raised to all primenumbers between 1 and p; the serie will still be giving primes.

All primes, that in the serie are raised to zero, multiplied with each other becomes the degree of conjugative.

I can prove that this equation is new:

p1*p2*p3*p4*...*pn = ( can't we call it p?, when n! = 1*2*3*...*n ? ).

It's to good to be old!
 
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  • #19
Two observations:

1) The smallest prime number is 2, not 1.

2) p1*p2*p3*p4*p5*p6*p7 = 2*3*5*7*11*13*17 = 510510
510510 - 1 = 510509 = 8369 * 61

Sorry, but if things were so simple...
 
  • #20
I'm sorry

Originally posted by suyver
Sorry, but if things were so simple... [/B]

I'm sorry. Thanks anyway. but... wait.. 510 510 is a double query.

Maybe it don't work on them? THANX!

by the way, 510 is 11111110 binary.

Maybe both sides don't have to be primes either? Or?
 
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  • #21
The smallest primenumber is one.

Originally posted by suyver
The smallest prime number is 2, not 1.

That's not true, though.

1 is the smallest primenumber.
 
  • #22
1 is not a prime number.
 
  • #23
Originally posted by Hurkyl
1 is not a prime number.

förlåt mig. I Sverige säger vi:

Primtal är alla tal som bara är delbara med 1 och sig självt.

sorry. In Sweden we say:

Primes are all numbers that you only can divide with 1 and itself.

1/1 = 1

1/1 = 1
 
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  • #24
Originally posted by Sariaht
In Sweden we say:

Primes are all number that you only can divide with 1 and itself.

This may be the case, but that is not the normally accepted definition of a prime number. Normally the smallest prime number is said to be 2...
 
  • #25
Originally posted by suyver
This may be the case, but that is not the normally accepted definition of a prime number. Normally the smallest prime number is said to be 2...

What about the equation?

What is your definition of a prime?
 
  • #26
Originally posted by Sariaht
What about the equation?

That makes no difference of course: 1*x = x for all x. So including your p1=1 doesn't change the fact that your equation is incorrect...

Originally posted by Sariaht
What is your definition of a prime?

The commonly accepted definition for a prime number is any number having no factor except itself and one. From this rule it follows that 1 is not a prime number, but 2 is.
 
  • #27
2*3 = 6 | +/- 1 | 5, 7

2*3*5 = 30 | +/- 1 | 29, 31

2*2*3 = 12 | +/- 1 | 11, 13

2*3*3 = 18 | +/- 1 | 17, 19

2*3*5*5 = 150 | +/- 1 | 149, 151

2*3*5*7*7 = 1470 | +/- 1 | 1469, 1471

allright.

Only one primesquare is aloud.

3*5*7*7 = 735 | +/- 2 | 733, 737
3*5*5*7 = 525 | +/- 2 | 523, 527
3*3*5*7 = 315 | +/- 2 | 313, 317
5*7*11 = 385 | +/- 2*3 | 379, 391
7*7 = 49 | +/- 30 | 19, 79

etc?
 
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  • #28
Originally posted by Sariaht
2*3 = 6 5 7

2*3*5 = 30 29 31

2*2*3 = 12 11 13

2*3*3 = 18 17 19

2*3*5*5 = 150 149 151

2*3*5*7*7 = 1470 1469 1471

allright.

Only one primesquare is aloud.

You should really learn to write your ideas (or whatever they are) more clearly. I have NO IDEA what you are trying to say...
 
  • #29
Originally posted by suyver
You should really learn to write your ideas (or whatever they are) more clearly. I have NO IDEA what you are trying to say...


Were does my equation error?
 
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  • #30
The prime numbers have a RANDOM distribution over the natural numbers. You will not succeed in finding such an (easy or not) algorithm to always generate a new prime number from a set of already known ones. However, the numbers tend to become large, making it difficult to see that they are not prime.

If you find a simple algorithm and PROVE that this algorithm works for at least the first 100 prime numbers, then I will look at it again. But like this it's becomming a waste of my time: you're just guessing new algorithms without any idea about why they should work in the first place...
 
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  • #31
Originally posted by suyver

(uip = un-included prime in serie)

Yes. I'll check. Sorry... (only one square is aloud).

No prime is aloud to have a negative conjugate.

.......uip.(pp)...prime or square
2......= 2...+-1...1 | 3
3......= 3...+-2...1 | 5
2*2......= 4...+-1...3 | 5
2*3......= 6...+-1...5 | 7
2*3*3......= 18...+-1...17 | 19
3*3......= 9...+-2...7 | 11
2*2*3......= 12...+-1...11 | 13
2*5......= 10...+-3...7 | 13
3*5......= 15...+-2...13 | 17
3*5*7......= 105...+-2...103 | 107
2*3*7......= 42...+-5...37 | 47
5*7*7......= 245...+-6...239 | 251
3*7*7......= 147...+-10...137 | 157
2*7*7......= 98...+-15...83 | 113
2*5*7......= 70...+-3...67 | 73
5*5*7 .....= 175...+-6..169(square) | 181
5*7......= 35...+-6...29 | 41
2*3*5*7*11...= 2310..+-1...2309 | 2311
The numbers in the serie must be a prime (3 5 7 11 are four numbers)
2*5*11......= 110...+-21...89 | 131
5*7*11......= 385...+-6...379 | 391

3*7*11......= 231...+-10...17*13(?) | 241
(maybe this has got something to do with that diff (3,7) = diff (7,11)
is not a prime, and that diff(x,y) cannot be a non-prime twice in a row?)

3*5*11......= 165...+-14...151 | 179
11*5.....= 55...+-42...13 | 97


Damit... Who cares if it works anyway: the permutations becomes to many.
 
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  • #32
Originally posted by suyver

In the middle of two conjugative primes is a third number oftenly divideable
with 6.

Can two such queries have the same factor-sum if the sum is a prime?

2*2 = 4 | 2 + 2 = 4

2*3 = 6 | 2 + 3 = 5

2*2*3 = 12 | 2 + 2 + 3 = 7

2*3*3 = 18 | 2 + 3 + 3 = 8

2*3*5 = 30 | 2 + 3 + 5 = 10

2*3*7 = 42 | 2 + 7 + 3 = 12

2*2*3*5 = 60 | 2 + 2 + 3 + 5 = 12
 
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  • #33
  • #34
Originally posted by suyver
Have you read this thread? You might find some of the contents interesting...


No, i have not read his thread. What does he mean?

But if this was true, you could find a lot higher primes a lot easier.
 
  • #35
No: this is indeed a function that generates all the primes. But it is a function with 26 parameters that can vary... Computationally very intensive!
 
<h2>1. What is Number Theory?</h2><p>Number Theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It involves studying patterns and structures in numbers and their properties.</p><h2>2. What are primes?</h2><p>Primes are positive integers that are only divisible by 1 and themselves. Examples of primes include 2, 3, 5, 7, 11, and so on. They play a fundamental role in number theory and have many interesting properties.</p><h2>3. What is the relationship between primes and their products?</h2><p>In number theory, the product of two or more prime numbers is known as a composite number. This means that the composite number can be broken down into its prime factors. For example, the composite number 12 can be broken down into 2 x 2 x 3, where 2 and 3 are prime numbers. The relationship between primes and their products is important in understanding the fundamental theorem of arithmetic, which states that every positive integer can be expressed as a unique product of primes.</p><h2>4. How is the relationship between primes and their products explored in number theory?</h2><p>In number theory, the relationship between primes and their products is explored through various methods such as prime factorization, prime number theorem, and Goldbach's conjecture. These methods help in understanding the distribution and properties of primes and their products, and have applications in cryptography, computer science, and other fields.</p><h2>5. What are some real-world applications of exploring the relationship between primes and their products in number theory?</h2><p>The study of primes and their products has many practical applications in fields such as cryptography, computer science, and data security. For example, the RSA encryption algorithm, which is widely used in secure communication, relies on the difficulty of factoring large composite numbers into their prime factors. Additionally, the study of primes and their products can also help in solving real-world problems, such as scheduling and optimization problems, by using number theory concepts and algorithms.</p>

1. What is Number Theory?

Number Theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It involves studying patterns and structures in numbers and their properties.

2. What are primes?

Primes are positive integers that are only divisible by 1 and themselves. Examples of primes include 2, 3, 5, 7, 11, and so on. They play a fundamental role in number theory and have many interesting properties.

3. What is the relationship between primes and their products?

In number theory, the product of two or more prime numbers is known as a composite number. This means that the composite number can be broken down into its prime factors. For example, the composite number 12 can be broken down into 2 x 2 x 3, where 2 and 3 are prime numbers. The relationship between primes and their products is important in understanding the fundamental theorem of arithmetic, which states that every positive integer can be expressed as a unique product of primes.

4. How is the relationship between primes and their products explored in number theory?

In number theory, the relationship between primes and their products is explored through various methods such as prime factorization, prime number theorem, and Goldbach's conjecture. These methods help in understanding the distribution and properties of primes and their products, and have applications in cryptography, computer science, and other fields.

5. What are some real-world applications of exploring the relationship between primes and their products in number theory?

The study of primes and their products has many practical applications in fields such as cryptography, computer science, and data security. For example, the RSA encryption algorithm, which is widely used in secure communication, relies on the difficulty of factoring large composite numbers into their prime factors. Additionally, the study of primes and their products can also help in solving real-world problems, such as scheduling and optimization problems, by using number theory concepts and algorithms.

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