dwheatley
dwheatley
  • Threads: 25
  • Posts: 1246
Joined: Nov 16, 2009
December 21st, 2013 at 1:08:01 PM permalink
Quote: Greasyjohn

I completely understand that there's no largest prime. I was merely trying to continue the calculation. Perhaps there is no calculation--multiply any know primes and add one and there' s always a prime greater than ANY prime on our list.



Right. Do the multiplication, add one, and either

a) the result will be prime, or
b) a prime larger than one in the list will divide the result.

Maybe you can characterize when this will happen and will a Fields medal? ;)
Wisdom is the quality that keeps you out of situations where you would otherwise need it
Greasyjohn
Greasyjohn
  • Threads: 135
  • Posts: 2178
Joined: Dec 8, 2013
December 21st, 2013 at 1:08:10 PM permalink
Quote: Twirdman

Yes if you take a list of all the primes up to a certain value say 23 so you have 2*3*5*7*11*13*17*19*23+1 you will get a larger prime. Though you have to still find what that larger prime is by doing repeated divisions of the number and there are simply much easier ways to find incredibly large primes.



But you could have any primes in your calculation? Is 3x17x509 just as valid as any more complete list of primes?
dwheatley
dwheatley
  • Threads: 25
  • Posts: 1246
Joined: Nov 16, 2009
December 21st, 2013 at 1:09:22 PM permalink
If you skip primes the result will be either:

a) the number is prime, or
b) a prime not in your list divides the result.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
Twirdman
Twirdman
  • Threads: 0
  • Posts: 1004
Joined: Jun 5, 2013
December 21st, 2013 at 1:14:32 PM permalink
Quote: Greasyjohn

But you could have any primes in your calculation? Is 3x17x509 just as valid as any more complete list of primes?



Yeah that is a valid list but you won't necessarily get a bigger prime then just one not on the list. Like I showed with 3*5+1=16 that is 2^4 so 2 is the new prime you got but its not bigger than 3 or 5 its just not on your list.
AxiomOfChoice
AxiomOfChoice
  • Threads: 32
  • Posts: 5761
Joined: Sep 12, 2012
December 21st, 2013 at 1:14:45 PM permalink
Quote: Twirdman

Yes if you take a list of all the primes up to a certain value say 23 so you have 2*3*5*7*11*13*17*19*23+1 you will get a larger prime.



No, that's incorrect. There's already been a counter-example given in this thread which shows that that's wrong (2*3*5*7*11*13 + 1 = 59 * 509)
Twirdman
Twirdman
  • Threads: 0
  • Posts: 1004
Joined: Jun 5, 2013
December 21st, 2013 at 1:28:31 PM permalink
Quote: AxiomOfChoice

No, that's incorrect. There's already been a counter-example given in this thread which shows that that's wrong (2*3*5*7*11*13 + 1 = 59 * 509)



And 59 is a larger prime then any on the list. I could have worded it better I guess but I was just saying it will prove the existence of a larger prime which you can find by either a simplistic algorithm of just dividing or some more advanced algorithms that I don't know since my area of math isn't number theory or algorithms.
Greasyjohn
Greasyjohn
  • Threads: 135
  • Posts: 2178
Joined: Dec 8, 2013
December 21st, 2013 at 1:32:57 PM permalink
Quote: AxiomOfChoice

No, that's incorrect. There's already been a counter-example given in this thread which shows that that's wrong (2*3*5*7*11*13 + 1 = 59 * 509)



I'm guessing that Twirdman means not "will get a larger prime" but that there IS a larger prime than is on his list.
Twirdman
Twirdman
  • Threads: 0
  • Posts: 1004
Joined: Jun 5, 2013
December 21st, 2013 at 1:35:26 PM permalink
Quote: Greasyjohn

I'm guessing that Twirdman means not "will get a larger prime" but that there IS a larger prime than is on his list.



Yeah I will admit I worded that badly and should have been more careful. I normally am but think I got lazy since I'm on break.
98Clubs
98Clubs
  • Threads: 52
  • Posts: 1728
Joined: Jun 3, 2010
December 24th, 2013 at 8:14:57 PM permalink
One of the ways to attack the problem of large primes is a rather simple rule that will eliminate 60% of all numbers larger than 5.

In a rather odd way, except for the number 2 and the number 5, all known primes have a final digit of either 1, 3, 7, or 9. However further testing is still needed: 21 is not prime as is 49 for example.
Some people need to reimagine their thinking.
ThatDonGuy
ThatDonGuy
  • Threads: 117
  • Posts: 6219
Joined: Jun 22, 2011
December 25th, 2013 at 10:37:42 AM permalink
Quote: 98Clubs

One of the ways to attack the problem of large primes is a rather simple rule that will eliminate 60% of all numbers larger than 5.

In a rather odd way, except for the number 2 and the number 5, all known primes have a final digit of either 1, 3, 7, or 9. However further testing is still needed: 21 is not prime as is 49 for example.


I'll do better than that; I will eliminate 2/3 of all numbers greater than 3.

If n > 3 and is prime, then either n+1 is a multiple of 6 or n-1 is a multiple of 6.
Proof: if n / 6 has a remainder of 0, 2, or 4, then it is a multiple of 2; if its remainder is 3, it is a multiple of 3; if it is 1, then n-1 is a multiple of 6; if it is 5, then n+1 is a multiple of 6.
Mosca
Mosca
  • Threads: 191
  • Posts: 4140
Joined: Dec 14, 2009
January 3rd, 2014 at 12:01:19 PM permalink
I was searching for something else and found this, I figure some folks would like it. Funny enough, I know what the Riemann-Zeta function is. Not the math, of course.

A falling knife has no handle.
Mosca
Mosca
  • Threads: 191
  • Posts: 4140
Joined: Dec 14, 2009
January 3rd, 2014 at 12:13:57 PM permalink
This was what I was looking for. I sort of get it, but could use a better explanation if someone feels like giving me one.


The weak twin primes conjecture states that there are infinitely many pairs of primes. The strong twin primes conjecture states that every prime p has a twin prime (p+2), although (p+2) may not look prime at first. The tautological prime conjecture states that the tautological prime conjecture is true.
A falling knife has no handle.
AxiomOfChoice
AxiomOfChoice
  • Threads: 32
  • Posts: 5761
Joined: Sep 12, 2012
January 3rd, 2014 at 1:24:25 PM permalink
What are you asking? To explain the joke?

IMO it's not that funny, but the two statements on the left are obviously true, and the two statements on the right are obviously false. The "Strong" and "Weak" are open problems (the Goldbach Conjecture) with "Strong" being a stronger statement than "Weak" (ie, Strong ==> Weak).

The joke is that there are several statements collectively known as "strong" or "weak" versions of the goldbach conjecture. When people say "The Goldbach Conjecture" they usually mean the one that is labelled "Strong" in the comic.

http://en.wikipedia.org/wiki/Goldbach_conjecture might help.
Mosca
Mosca
  • Threads: 191
  • Posts: 4140
Joined: Dec 14, 2009
January 3rd, 2014 at 2:30:01 PM permalink
Allowing for differences in humor, XKCD usually is at least worth a chuckle. There has to be some joke in the fact that the weaker the conjecture, the more obviously true, and the stronger the conjecture, the more obviously false.

edit: I found it.

Quote:

Randall is riffing on the relationship between "strong" and "weak" logical statements, which are an interplay between the boldness or usefulness of a statement and the ease with which it might be proven to be true. For example, if Goldbach's conjecture (given in the comic under the label "strong") could be proven to be true, it would automatically imply that Goldbach's weak conjecture (given in the comic under the label "weak") is also true, because any odd number greater than 5 can be expressed as 3 (a prime number) plus an even number greater than 2 (which, per the strong conjecture, would itself be the sum of two prime numbers), resulting in a way to express the original odd number as the sum of three prime numbers. The weak conjecture does not, however, imply the strong conjecture.

Mathematicians have been solving related problems that are "weaker" than the weak conjecture, and working towards "stronger" ones. For example, in 1937 the weak conjecture was proven for odd numbers greater than 314348907. In 1995 a version was proven based on the sum of no more than seven prime numbers, and in 2012 the ceiling was lowered to five primes. In 2013 the weak conjecture was claimed proven for numbers greater than 1030, while all numbers below 1030 have been verified by supercomputer to satisfy the conjecture; these together imply that the weak conjecture is true (although there is no general proof of it for all numbers). Goldbach's strong conjecture remains unsolved.

This comic plays on the "strong" and "weak" naming of Goldbach's conjectures by extending it to further degrees of strength or weakness. The "very weak" and "extremely weak" conjectures are indeed implied by Goldbach's weak conjecture, just as the weak conjecture is implied by the strong one. The "very strong" and "extremely strong" conjectures are extensions of Goldbach's strong conjecture, even as it is an extension of the weak conjecture. However, the "very weak" and "extremely weak" conjectures are so obviously true that they are hardly worth stating, while the "very strong" and "extremely strong" conjectures make such bold claims that they are obviously false.

Moreover, the "extremely weak" and "extremely strong" conjectures contradict each other, even though they're both derived (albeit in opposite directions) from the same initial conjectures.



http://www.explainxkcd.com/wiki/index.php/1310:_Goldbach_Conjectures
A falling knife has no handle.
98Clubs
98Clubs
  • Threads: 52
  • Posts: 1728
Joined: Jun 3, 2010
January 3rd, 2014 at 3:34:55 PM permalink
ONE is NOT a prime, its "UNIQUE" in that its divisible by one and itself only, but one IS itself. TWO is the ONLY even prime discovered. All primes greater than 6 have a terminal digit of either 1, 3, 7, or 9. (2 and 5 are the only exceptions known otherwise it would be all primes)
Some people need to reimagine their thinking.
AxiomOfChoice
AxiomOfChoice
  • Threads: 32
  • Posts: 5761
Joined: Sep 12, 2012
January 3rd, 2014 at 3:39:59 PM permalink
Quote: 98Clubs

TWO is the ONLY even prime discovered.



Do you suspect that there are undiscovered ones?
98Clubs
98Clubs
  • Threads: 52
  • Posts: 1728
Joined: Jun 3, 2010
January 3rd, 2014 at 3:56:03 PM permalink
No. Like my statement about "one" thats how I learned... so far its holding up. ;o)
Some people need to reimagine their thinking.
AxiomOfChoice
AxiomOfChoice
  • Threads: 32
  • Posts: 5761
Joined: Sep 12, 2012
January 3rd, 2014 at 4:09:35 PM permalink
Quote: 98Clubs

No. Like my statement about "one" thats how I learned... so far its holding up. ;o)



Am I missing some humor here?
  • Jump to: