Prime numbers...

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10 Mar 2009, 12:49 am

Though I really am very interested in Prime Numbers when it comes to Number Theory and things like the Ulam Spiral and Sacks Spiral.

Orwell
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10 Mar 2009, 1:02 am

DNForrest wrote:

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

As do I, with the exception of 2. I'm very OCD with 2's power series.

Same here, I counted by powers of two as a kid. 3 is nice too, 5 is just dull and bland. 7 and up is where they start to feel really grating.

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ruveyn
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10 Mar 2009, 1:52 am

makuranososhi wrote:

The remainder of the first hundred prime numbers, in the same format of 2A+B where A & B are both prime numbers as well.

M.

That does not prove the general statement. Enumerating particular cases will not prove the generality.

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10 Mar 2009, 1:54 am

twoshots wrote:

makuranososhi wrote:

While I understand that it is an exception... can you explain better to be why it is excluded from the group?

Y'know, off the top my head I'm really not sure why. The most commonly cited reason on the internet is that it would complicate the fundamental theorem of arithmetic: each number would not have a unique prime factorization if 1 was allowed to be prime, e.g. 6=2*3=1*2*3. This site looks like it might have some interesting stuff to chew on

That is exactly why 1 is not counted as a prime number. To have unique factorization with 1 occuring just once, the primes have to be greater than 1.

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10 Mar 2009, 2:16 am

ruveyn wrote:

makuranososhi wrote:

The remainder of the first hundred prime numbers, in the same format of 2A+B where A & B are both prime numbers as well.

M.

That does not prove the general statement. Enumerating particular cases will not prove the generality.

ruveyn

No, but a rudimentary study will indicate whether or not an idea is consistently right or wrong. Next is to take the total number of sums possible using this theory and see how much more efficient it would be to find subsequent primes by frequency using this method.

M.

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10 Mar 2009, 5:38 am

ruveyn wrote:

makuranososhi wrote:

The remainder of the first hundred prime numbers, in the same format of 2A+B where A & B are both prime numbers as well.

M.

That does not prove the general statement. Enumerating particular cases will not prove the generality.

ruveyn

But it can disprove a statement, so it is a good first step. Once you have more data building a theorem is typically easier.

- - -

Ive always held that we should use base 12. In {2..12} only {5,7,11} are relatively prime with 12, so for any other number on that range we can use the same sort of trick for divisibility as we do for 2's and 5's in base 10. Never mind ease of multiplication by 2, 3, 4, and 6.

It also benefits the metric system, as 1.2 meters is a commonly used unit in some applications (I believe lumber is sold in such lengths, but I use feet). It would allow the metric system to retain the simplicity of an exponential series and the ease of use of being highly factorable.

Similar consideration was given to the development of degrees, an arbitrary unit chosen to facilitate ease of use by hand.

- - -

As for efficiency of finding new primes, I believe the Sieve of Atkin is still considered optimized. I looked at it a few years ago when I thought of personally making a number sieve to investigate primes. If memory serves about how it works, and if I am not making an obvious mistakes in this nearly waking hell at 5:36AM, after 3 hours of sleep, it is superior to yours as the size of primes considered grow large.

ruveyn
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10 Mar 2009, 9:49 am

DNForrest wrote:

twoshots wrote:

DNForrest wrote:

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

As do I, with the exception of 2. I'm very OCD with 2's power series.

2's power series? What would that be....
2=2+x*0+x^2*0/2+x^3/6+...

I think I may have used the wrong term, I meant:
2^x
x = 0, 1, 2, 3, 4, ...
So 1, 2, 4, 8, 16, ...

Having a 1 in this sequence is rather odd, don't you think?

ruveyn

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10 Mar 2009, 10:25 am

DoP, thank you for the insight and information. Until I do a frequency analysis, I have no real idea of the efficacy of the process; until yesterday evening, I had no real basis to determine whether it was even applicable. What is the Sieve of Atkin, if I might ask?

M.

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10 Mar 2009, 10:54 am

http://en.wikipedia.org/wiki/Sieve_of_Atkin

Powers of 2:
-2 -1 0 1 2
1/4 1/2 1 2 4

and, most importantly:
http://en.wikipedia.org/wiki/Prime_number

Quote:

a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself.

The value "1" is not an exception to this rule, as it has but one divisor.
Note that "natural number" means positive, and may or may not include zero (which has an infinite number of divisors).

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DNForrest
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10 Mar 2009, 4:16 pm

ruveyn wrote:

Having a 1 in this sequence is rather odd, don't you think?

ruveyn

Not really, 2^0 = 1, how's that odd? And no, I don't go into two to the power of negative numbers since they result in fractions.

Besides, since when do OCD tendencies have to make sense to anyone besides the person with them?

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10 Mar 2009, 5:38 pm

If you are looking for primes that have a certain relationship to other numbers
check these out:

http://mathworld.wolfram.com/Prime-Gene ... omial.html
http://mathworld.wolfram.com/FibonacciPrime.html
http://mathworld.wolfram.com/FRACTRAN.html
http://en.wikipedia.org/wiki/Formulas_for_primes
http://en.wikipedia.org/wiki/FRACTRAN
http://en.wikipedia.org/wiki/Mersenne_Prime

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ruveyn
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10 Mar 2009, 7:41 pm

DNForrest wrote:

ruveyn wrote:

Having a 1 in this sequence is rather odd, don't you think?

ruveyn

Rather odd, rather odd. As in being the only odd number in the sequence. What is the matter? Are you an Aspie or something?

ruveyn

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10 Mar 2009, 7:51 pm

ruveyn wrote:

DNForrest wrote:

ruveyn wrote:

Having a 1 in this sequence is rather odd, don't you think?

ruveyn

Rather odd, rather odd. As in being the only odd number in the sequence. What is the matter? Are you an Aspie or something?

ruveyn

And of course, having 1 and 2 in the sequence is rather non-divisible-by-four, I'd say.

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10 Mar 2009, 8:26 pm

Death_of_Pathos wrote:

Ive always held that we should use base 12. In {2..12} only {5,7,11} are relatively prime with 12, so for any other number on that range we can use the same sort of trick for divisibility as we do for 2's and 5's in base 10. Never mind ease of multiplication by 2, 3, 4, and 6.

It also benefits the metric system, as 1.2 meters is a commonly used unit in some applications (I believe lumber is sold in such lengths, but I use feet). It would allow the metric system to retain the simplicity of an exponential series and the ease of use of being highly factorable.

Similar consideration was given to the development of degrees, an arbitrary unit chosen to facilitate ease of use by hand.

Being an IT student myself, I think that we should use either base 2 (binary) or base 16 (hexidecimal)... those are the two most commonly used base systems in the computer world... Hexidecimal would probably be preferable, since you avoid the incredibly long numbers that binary can give you (turning the decimal number 255 into binary requires 8 digits), but retain the versatility of binary...

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10 Mar 2009, 8:57 pm

I don't know where people get "Hexidecimal" from. Base 16 is hexadecimal. Anyway, for practical purposes, I much prefer Sexagesimal.

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Orwell
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10 Mar 2009, 9:22 pm

lau wrote:

I don't know where people get "Hexidecimal" from. Base 16 is hexadecimal. Anyway, for practical purposes, I much prefer Sexagesimal.

Sexagesimal is ideal, but I think it would be too unwieldy. Dozenal gets my vote for practical purposes.

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