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Old 2014-12-11, 18:04   #1
jwaltos
 
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Default Primality question.

I came across a reference to -1 considered prime years ago.

http://mathforum.org/kb/message.jspa?messageID=1093172

Is there any theoretical utility in considering this number as such?
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Old 2014-12-11, 18:12   #2
R.D. Silverman
 
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Quote:
Originally Posted by jwaltos View Post
I came across a reference to -1 considered prime years ago.

http://mathforum.org/kb/message.jspa?messageID=1093172

Is there any theoretical utility in considering this number as such?
The utility is less than zero.
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Old 2014-12-11, 19:25   #3
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Originally Posted by R.D. Silverman View Post
The utility is less than zero.
Good pun and thank you for the reply.

My reason for posting is that all primes can be developed from (-1,7,11,13,17,19,23 +30*n) where 2,3,5 are not included. Riesel's 1984? book notes this in the initial first or second chapters. The sum of digits of every prime is one of 1,4,2,8,5,7 which is the cyclic reciprocal of 7 ( I have not found a proof of this and pointer would be appreciated). To me, this is a curiosity but an interesting one.

Last fiddled with by jwaltos on 2014-12-11 at 19:28
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Old 2014-12-11, 19:38   #4
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Quote:
Originally Posted by jwaltos View Post
The sum of digits of every prime is one of 1,4,2,8,5,7 which is the cyclic reciprocal of 7 ( I have not found a proof of this and pointer would be appreciated). To me, this is a curiosity but an interesting one.
Any number whose digits add up to 3 is divisible by 3. This is something I remember learning back in early grade school as a "trick" to knowing what something's divisors might be. This applies to any number whose digits add up to 6 also. This is also true for 9, but these numbers have the added bonus of being divisible by 9 as well.

That leaves a sum of 0 (mod 10, I imagine). 19 is a counter example of that, so I'm not sure why it was missed.
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Old 2014-12-11, 21:42   #5
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Thanks Mawn.

Last fiddled with by jwaltos on 2014-12-11 at 22:30
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Old 2014-12-11, 22:36   #6
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http://en.wikipedia.org/wiki/Fundame..._of_arithmetic

12=2^2*3
12=2^2*3*-1*-1
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Old 2014-12-12, 00:50   #7
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Originally Posted by legendarymudkip View Post
It amazes me how much the Ancient Greek managed to accomplish without algebra.

"4 measures 12" as in if you have 12 blocks, you could measure them if you had a stick that was 4 blocks long.
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Old 2014-12-12, 03:14   #8
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Originally Posted by TheMawn View Post
That leaves a sum of 0 (mod 10, I imagine). 19 is a counter example of that, so I'm not sure why it was missed.
19 = 1+9 = 10 = 1+0 = 1

Sum of digits = number mod 9, except, instead of 0, we will use 9 itself.
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Old 2014-12-12, 04:39   #9
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Originally Posted by axn View Post
19 = 1+9 = 10 = 1+0 = 1

Sum of digits = number mod 9, except, instead of 0, we will use 9 itself.
Of so you keep re-adding the digits until you're left with a 1-digit number.


The thing about the 3 still holds then, @OP. If a number's digits add up to something bigger than 9 and still a product of 3, it eventually decomposes back to 3 6 or 9.

396 - > 18 -> 9.

1299 -> 21 -> 3.

Etc
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Old 2014-12-12, 17:22   #10
Jens K Andersen
 
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Quote:
Originally Posted by jwaltos View Post
My reason for posting is that all primes can be developed from (-1,7,11,13,17,19,23 +30*n) where 2,3,5 are not included.
You are missing 1, and it's more common to use 29 than -1. The usual form is (1,7,11,13,17,19,23,29 + 30*n). In case you don't know, 30 = 2*3*5 (product of the first three primes), and 1,7,11,13,17,19,23,29 are the numbers below 30 and coprime to 30, so (1,7,11,13,17,19,23,29 + 30*n) is simply the numbers with no prime factor <=5. Its' trivial to make similar expressions for numbers with no prime factor <= p for other small primes p. p=2 gives that all primes above 2 are odd (2n+1). p=3 gives that all primes above 3 are of form 6n+1 or 6n+5, but in this case it's more common to use -1 in 6n+/-1. p=7 is also relatively easy by hand. After that I recommend using a computer (it's a trivial programming exercise, please don't post the long lists of coprime numbers).
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