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Mathematics Forum
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| illoyd Quote | Reply | This message was updated on 4/2/2006 4:55:36 AM by illoyd | Goldbach's conjecture posted on: 4/1/2006 2:38:20 PM If Goldbach's conjecture is true, then every integer has an associated pair of prime numbers one equi-distance above and below it. e.g 23 has 17 and 29, 50 has 47 and 53 etc. If we take the highest known prime number, add 1 and then find its associated prime number pairs, would not this be a quicker way of finding the highest prime number? Proof: Goldbach Conjecture states: For all 2*m there exists two primes such that 2*m= p1+p2 therefore for all m there exists two primes such that m= (p1+p2)/2 Thus m is midway between p1 and p2 except for the special case where p1 or p2 =2 where p1=2 and p2= m+2. QED |
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The Riemann Hypothesis
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Goldbach's conjecture
replied on: 4/1/2006 7:26:56 PM I don't think so. Why don't you put a worked example on here. |
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illoyd
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Goldbach's conjecture
replied on: 4/2/2006 5:25:24 AM Proof: Goldbach Conjecture states: For all 2*m there exists two primes such that 2*m= p1+p2 therefore for all m there exists two primes such that m= (p1+p2)/2 Thus m is midway between p1 and p2 except for the special case where p1 or p2 =2 where p1=2 and p2= m+2. QED Ivor Lloyd |
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3.14159
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Goldbach's conjecture
replied on: 4/2/2006 1:47:30 PM In your proof you can drop the special case part, since if p1 is 2, p2 will have to be 2, otherwise their sum will be odd, and Goldbach's conjecture doesn't say anything about odd numbers. And this wouldn't really be any faster than current methods, since you'd have to find a prime below it and match it up with a prime above it, so you're checking 2 numbers instead of 1, which is going to be slower. |
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illoyd
Quote | Reply | This message was updated on 4/2/2006 3:02:11 PM by illoyd |
Goldbach's conjecture
replied on: 4/2/2006 3:01:28 PM Whilst I agree with you about the special case, I included this for completeness. However, only one number would need to be checked as the lower prime number would be already on the list of known prime numbers. The method would enable the selection of possible higher prime numbers rather than checking every odd number above the last known prime number. Incidently, this interpretation of Goldbach's conjecture also shows that there is an infinite number of primes. |
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The Riemann Hypothesis
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Goldbach's conjecture
replied on: 4/20/2006 3:41:51 AM The converse of the theorem isn't proven though. If you take and even number it doesn't mean that any prime below it will have an equidistant prime counterpart. You have to go through all primes below the number, then find the difference of the known prime and the number, add that to the number, then test to see if the resulting number is prime. Very inefficient. |
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Euler
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Goldbach's conjecture
replied on: 6/4/2006 3:40:06 AM Should Goldbach's Conjecture be proven (though it hasn't been for nearly 300 years), your assumption is partially correct. If you are basing the higher prime you are to find on twice the given number then you will be successful, though it will simply be a higher prime- not necessarily the next highest prime. And, in addition to this, I would like to point out that whoever said Goldbach's Conjecture can be utilized to prove the existance of infintely many primes is correct- and not only that, but it could be used to show that as you approach infinity every even number is comprised of infinitely many prime pairs. |
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illoyd
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Goldbach's conjecture
replied on: 6/12/2006 2:03:06 PM Yes, I only claimed to find a larger prime number, not the next consecutive prime number. Also, we cannot as yet claim that every prime number is pairwise associated with another prime and an integer. I have been looking more closely at the number of prime pairs associated with each even number. Although Goldbach claimed one pair, I claim that above 6, there are a least two as the table below shows. Also, The number of pairs increases the higher the even number. Indeed 990 has 51 such prime pairs! Column 1: Even Number range Column 2: Minimum number of pairs in range Coumn 3: Maximum Number of pairs in range 6 - 50 2 6 50 - 98 3 10 100-148 4 13 150-198 4 14 200-248 5 18 250-298 7 20 300-348 6 23 350-398 7 27 400-448 8 30 450-498 9 30 500-548 10 32 550-598 11 32 600-648 11 40 650-698 11 41 700-748 13 39 750-798 13 44 800-848 14 52 850-898 14 48 900-948 14 46 950-998 14 51 |
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Euler
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Goldbach's conjecture
replied on: 6/13/2006 11:22:15 PM Oh! Sorry, I thought you said you could find the next highest prime, but I looked back at the post and, as you said, you did say "the highest prime."- which does not necesarily mean the next one. Sorry for the mix up. As for my statement about the infinitly many pairs of primes which add up to a single even number, as you approach infinty- I am well aware of the fact that it is not yet proven. I was simply saying that it is true should Goldbach's conjecture be proven, which I know because I have proven it myself with the assumption that Goldbach's conjecture is true. It may interest you to know though that I am currently refining an equation I have developed which very closely models the sequence of the prime numbers, and in turn provides a very accurate number for the pi function, much more accurate than the PNT. I developed this equation in the hopes of proving Goldbach's conjecutre, so as soon as I finish refining it I hope to continue on that path. |
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The Riemann Hypothesis
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Goldbach's conjecture
replied on: 6/14/2006 5:22:11 AM I still don't understand how the conjecture can be used to generate primes. Can someone give me an example algorithm that assumes the conjecture is correct? |
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illoyd
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Goldbach's conjecture
replied on: 6/14/2006 3:16:27 PM No. I only said that the method reduces the number of candidates for primes. Let us take an example. Supose the largest known prime was 31 then even number above is 32 and the first candidate for a larger prime is 32+32-1 (i.e. equidestant above from 1) which is 63 which is not prime. Try 32+32-3= 61, which is prime. We have tried two numbers rather than all the numbers between 31 and 61 which are 1,3,7 or 9 modulo 10 which is 33, 37, 39, 41, 43,47, 49, 51,53,57,59,61. We would have tried 12 numbers to get a prime this high, but we would have missed a few. going on from 61: 62+62-1=123 no 62+62-3=121 no 62+62-5=119 no 62+62-7=117 no 62+62-11=113 yes! ie 5 trials instead of about 24 tries. I hope this helps. |
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Euler
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Goldbach's conjecture
replied on: 6/14/2006 5:28:35 PM quote: Just a little, incredibly little in fact, technical detail. To be entirely accurate, you should say that it will very likely reduce the number of candidates, as it is possible- especially considering how no pattern to the primes has been discovered- that the only primes between a given even number and twice that even number are less than that even number plus half the range of the traditional case- in which case, your method produces a greater amount of candidates to go through than the other way. But, like I said, it is highly, highly, highly unlikely- just wanted to say something on the matter. |
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Euler
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Goldbach's conjecture
replied on: 6/14/2006 5:36:49 PM In order to clarify I should say, using one of your examples, take the given even number 62, then the total number of tries you would have to make the other way is, as you said, 24. So if all the primes between 62 and 2*62 happen to be less than 84 (which I know they don't, but speaking purely hypothetically), it would take less than 12 tries using the other way, and it would, conversely, take more than 12 tries using your method. Maybe I am misunderstanding something, but I believe what I said is true... |
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