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Prime numbers
03-22-2016, 07:49 PM (This post was last modified: 03-22-2016 07:50 PM by Tugdual.)
Post: #1
Prime numbers
Do you think this is important?
I'm not sure demonstrating that prime numbers somehow do not follow the distribution of random numbers is a discovery: they are not random numbers, they are prime numbers which can be divided by 1 and themselves.
Also what answers are we ultimately seeking regarding prime numbers? I would say essentially finding a function f(n) that gives prime #n. Are we approaching a solution with the discovery? I doubt.

What do you think?
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03-23-2016, 10:46 PM
Post: #2
RE: Prime numbers
.
Hi,
(03-22-2016 07:49 PM)Tugdual Wrote:  Do you think this is important?

Maybe. ArXiv.org is a repository of electronic preprints so the documents there usually haven't been subject to full peer review and thus it would be wise to wait and hold expectations till they actually have.

Quote:I'm not sure demonstrating that prime numbers somehow do not follow the distribution of random numbers is a discovery:

"the distribution of random numbers" is utterly vague, random numbers may come in many distributions: uniform distribution, Gaussian (normal) distribution, etc., etc.

Quote: they are not random numbers, they are prime numbers which can be divided by 1 and themselves.

In that sense, the decimal digits of Pi aren't random numbers either, they are perfectly determined by virtue of being the decimal representation of Pi. However, when subjected to statistical analysis they seem to behave as being random digits uniformly distributed (yet still formally unproven).

Quote:Also what answers are we ultimately seeking regarding prime numbers? I would say essentially finding a function f(n) that gives prime #n.

There are many such functions (formulas), with various degrees of usefulness. You can see a number of them here:

http://mathworld.wolfram.com/PrimeFormulas.html

V.
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03-23-2016, 11:17 PM
Post: #3
RE: Prime numbers
Just to say: Hello Valentin!

Greetings,
    Massimo

-+×÷ ↔ left is right and right is wrong
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03-23-2016, 11:51 PM
Post: #4
RE: Prime numbers
What is the meaning in them that they are all composition a*2 + b*3, where a,b are integers and can be that a=b. On the other hand so is any integer number above 3.

So could we in theory build a computer without rounding error if it would be constructed to use hybrid base 2 & 3.
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03-24-2016, 07:15 AM
Post: #5
RE: Prime numbers
(03-23-2016 11:51 PM)Vtile Wrote:  So could we in theory build a computer without rounding error if it would be constructed to use hybrid base 2 & 3.

Perhaps I misunderstand, but if you did that, you still couldn't exactly represent 1/5 in place value with a finite number of digits after the radix mark, so you'd still have to round it.
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03-24-2016, 08:07 AM
Post: #6
RE: Prime numbers
(03-24-2016 07:15 AM)brouhaha Wrote:  Perhaps I misunderstand, but if you did that, you still couldn't exactly represent 1/5 in place value with a finite number of digits after the radix mark, so you'd still have to round it.

This was my first thought too....

Using a factorial base would avoid most rounding errors but not all of them. Transcendental functions would produce non-rational results which would require rounding.


Pauli
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03-24-2016, 11:59 AM
Post: #7
RE: Prime numbers
Regarding the original topic, I believe that considering digits is never a good idea. Digits are just the letters in a certain base that form words to represent a number. The number as a whole means something while the first digit (base 10) is the number modulo 10.
To go further, let's consider the prime numbers expressed in base 2. First digit would always be 1 otherwise it would be an even number, hence not prime. OMG 100% of the prime numbers start with a 1!! Shall we conclude anything? Obviously no, so studying the digits of a number seem pointless to me.
Always intersted to get your feedback.
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03-24-2016, 01:23 PM
Post: #8
RE: Prime numbers
(03-24-2016 11:59 AM)Tugdual Wrote:  studying the digits of a number seem pointless to me.

But consider the number 27412317. Can you tell if it is prime by looking at the digits? Yes! The sum of the digits is divisible by 3, which makes the number divisible by 3, which makes it non-prime.

A number is nothing without its digits!
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03-24-2016, 07:17 PM
Post: #9
RE: Prime numbers
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(03-23-2016 11:17 PM)Massimo Gnerucci Wrote:  Just to say: Hello Valentin!

Hello to you as well, Massimo ! ... 8-D

Best regards.
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03-24-2016, 07:51 PM
Post: #10
RE: Prime numbers
(03-24-2016 01:23 PM)Don Shepherd Wrote:  
(03-24-2016 11:59 AM)Tugdual Wrote:  studying the digits of a number seem pointless to me.

But consider the number 27412317. Can you tell if it is prime by looking at the digits? Yes! The sum of the digits is divisible by 3, which makes the number divisible by 3, which makes it non-prime.

A number is nothing without its digits!
A bit like ideas exist because of crosswords?
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03-25-2016, 12:24 PM
Post: #11
RE: Prime numbers
(03-24-2016 11:59 AM)Tugdual Wrote:  Regarding the original topic, I believe that considering digits is never a good idea. Digits are just the letters in a certain base that form words to represent a number. The number as a whole means something while the first digit (base 10) is the number modulo 10.
To go further, let's consider the prime numbers expressed in base 2. First digit would always be 1 otherwise it would be an even number, hence not prime. OMG 100% of the prime numbers start with a 1!! Shall we conclude anything? Obviously no, so studying the digits of a number seem pointless to me.
Always intersted to get your feedback.

Not related with primes but surprisingly (at least to me) there exist formulas to compute the \(n^{th}\) digit of \(\pi\) in isolation, not needing to compute the previous \(n-1\) digits. Such formulas work in base 16, so in some cases, studying the digits in certain bases can give really interesting results and, of course, insight.

Regards.

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03-25-2016, 12:41 PM
Post: #12
RE: Prime numbers
(03-24-2016 11:59 AM)Tugdual Wrote:  Regarding the original topic, I believe that considering digits is never a good idea. Digits are just the letters in a certain base that form words to represent a number. The number as a whole means something while the first digit (base 10) is the number modulo 10.
To go further, let's consider the prime numbers expressed in base 2. First digit would always be 1 otherwise it would be an even number, hence not prime. OMG 100% of the prime numbers start with a 1!! Shall we conclude anything? Obviously no, so studying the digits of a number seem pointless to me.
Always intersted to get your feedback.

Hehe, you looked at the digits to tell if the number was odd or even, didn't you? :-)
So you do get some useful information by looking at the digits. However, actively looking for more information in those digits seems like a waste of time nowadays.
I agree studying the statistical distribution of digits in prime numbers seems dumb, but then I'm an engineer, and mathematicians probably think designing things is dumb, but then find looking at the digits of Pi truly fascinating, to each his own.
Some people thought this was a discovery worth publishing a paper, so at least somebody thinks it's cool to know. Not my case though, and I don't think it brings mankind closer to anything.
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03-25-2016, 01:07 PM
Post: #13
RE: Prime numbers
(03-25-2016 12:41 PM)Claudio L. Wrote:  Some people thought this was a discovery worth publishing a paper, so at least somebody thinks it's cool to know. Not my case though, and I don't think it brings mankind closer to anything.
Well, this is the point with mathematics. You don’t know what can come out of a discovery and when. Just think about how encryption is used today, I don’t think that people working on prime factorization in the previous centuries could have envisioned this kind of application.
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03-25-2016, 07:29 PM
Post: #14
RE: Prime numbers
(03-25-2016 12:24 PM)emece67 Wrote:  
(03-24-2016 11:59 AM)Tugdual Wrote:  Regarding the original topic, I believe that considering digits is never a good idea. Digits are just the letters in a certain base that form words to represent a number. The number as a whole means something while the first digit (base 10) is the number modulo 10.
To go further, let's consider the prime numbers expressed in base 2. First digit would always be 1 otherwise it would be an even number, hence not prime. OMG 100% of the prime numbers start with a 1!! Shall we conclude anything? Obviously no, so studying the digits of a number seem pointless to me.
Always intersted to get your feedback.

Not related with primes but surprisingly (at least to me) there exist formulas to compute the \(n^{th}\) digit of \(\pi\) in isolation, not needing to compute the previous \(n-1\) digits. Such formulas work in base 16, so in some cases, studying the digits in certain bases can give really interesting results and, of course, insight.

Regards.

Yes that is the formula of Mr Plouffe. I read he failed to find a decimal version. I agree with you this is quite fascinating.
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