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| Euler Quote | Reply | | Goldbach's Conjecture posted on: 7/2/2006 2:40:47 AM It is said that Goldbach wrote in a letter to Euler saying that every integer greater than 2 can be written as the sum of three primes, to which Euler proposed what is now known as Goldbach's conjecture that every even number can be written as the sum of two primes. Euler said that the two statements are equivalent, as the second implies the first- can someone explain to me why exactly they are equivalent? |
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Euler
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Goldbach's Conjecture
replied on: 7/2/2006 2:41:46 AM I mean in mathematical terms I should mention.. |
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illoyd
Quote | Reply | This message was updated on 7/2/2006 9:50:04 AM by illoyd |
Goldbach's Conjecture
replied on: 7/2/2006 9:49:21 AM Clearly the case for the sum of three primes cannot be held as the sum of thee primes is odd except if one of them is 2. The case for using two is obvious E= 2+p1+p2 There is a case for four primes since: E= p1+p2+p3+p4 is even but E has to be 0 mod 4. |
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Euler
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Goldbach's Conjecture
replied on: 7/2/2006 3:31:32 PM Of course! I don't know how I missed that either! Thanks!! |
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contrapositive
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Goldbach's Conjecture
replied on: 8/25/2006 3:25:01 AM quote: Assume you have proved the second - every even number can be written as the sum of two primes. Call this (*) Let n denote the number we wish to write as the sum of three primes. If n is even, then by (*) there exists two prime numbers whose sum is n-2. Neither of this pair can be two - or n-2 would be odd (contradiction). Thus, the three primes which sum to n are 2 and the two which sum to n-2. If n is odd, you're spoilt for choice. Subtract an odd prime - the result is an even number. By (*) that even number can be written as the sum of two primes. Thus, your three primes are the subtracted prime and the pair which add to the even number. ~~~~~ Proving it in the reverse direction pretty much follows by reversing what was done above. As n is even consider n + 2, one of the three primes which sums to n+2 is 2 (since requires even + odd + odd as only one even available), the other two will sum to n. |
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HallsofIvy
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Goldbach's Conjecture
replied on: 4/6/2007 1:50:42 PM I don't think you need to be that deep. If n is even, then so is n-2. If n-2= p+ q then n= p+ q+ 2, a sum of three primes. If n is odd, then n-3 is even. If n- 3= p+ q then n= p+ q+ 3, again a sum of 3 primes. |
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cramwit
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Goldbach's Conjecture
replied on: 11/14/2007 12:09:43 PM Goldbach's conjecture hasn't been proven has it? I was thinking about it & what it requires/stipulates that for any even number > 2, 'n' there must be two primes that are equidistant from n/2. say x & y are the two summation primes lower to higher respectively & k is their difference from n/2. x + k = n/2 = y - k but has anyone ever proved that there must be a prime in the interval between n/2 & n for every number where n/2 is not prime? [if n/2 is prime then the conjecture is true, k=0] obviously there is a prime number from the interval from 0 to n/2 where n>2 because 2 & 3 are prime. if one can't prove there is a prime 'y' from the interval from n/2 to n then proving there is one with more specific properties is essentially impossible, yes? Statistically it seems apparent, but that would not be sufficient for proof. If someone could point me to a proof i would appreciate it. I think that proof might have ramifications. If that can't be proved then as a prerequisite for Goldbach's, Goldbach's becomes unprovable. |
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cramwit
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Goldbach's Conjecture
replied on: 11/14/2007 2:25:51 PM trying to get a handle on the local terrain around a random number, for any number n for any prime p must be a factor of a 'local' [radius of p/2] number or itself of (p-1)/2 or less? a number with a factor of 7 must be within 3-0 of any number n. 7 can only hit as a factor every 7 numbers. I wonder if some kind of maximum locality of a prime proof might exist? if it does the magnitude of n must have some bearing on it. Something to do with the maximum numbers that can be consumed by the regular intervals of primes given some random relationship. there must be some limit to the number of numbers 'local' to n consumed/covered by the primes<n before holes or gaps appear. Sort of like calculate the maximum spread coverage & it must be that or less. Since the primes less than n are 'given' it is trying to find the limits on the high side of n. |
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illoyd
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Goldbach's Conjecture
replied on: 11/15/2007 1:57:40 PM I think you are all trying to say that prime numbers must have some specific characteristics so that Goldbach’s conjecture can be proved, but such characteristics of primes seem to be elusive. No one has yet come up with a formula for example, of the interval between primes. |
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cramwit
Quote | Reply | This message was updated on 11/16/2007 12:14:32 PM by cramwit |
Goldbach's Conjecture
replied on: 11/16/2007 1:43:19 AM something that can be definitively said about non-prime spans, the densest coverage is by the smaller primes so combining them maximizes the coverage area/span define 'primefactorial's -> 2*3*5 . . . *pn its span of coverage is exactly p(n+1)-2 primes are by definition the uncovered numbers [factor gap numbers] since they start at two we have to subtract the one & since the 'gap'[unknown?] place is the next prime we have to subtract one from that value to exclude it from coverage; p(n+1)-2 ie. 2*3*5*7*11*13 is 13p! the next prime is 17 17-2=15 so the longest certain non-prime span is 15, one away [above or below] the primefactorial. about the prime factorial we have a maximized coverage area reflected one number away on either side of it the primefactorial is like the zero on the number line where the pattern is reverse/reflected on the negative side of the . . . . . . .-10123456789 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 .3. 3. 3. 3. 3. 3. 3. 3. 3. 3. 3 .5. . 5. . 5. . 5. . 5. . 5. . 5 xxx x xxx xxxxx x xxxxx xxx x xxx notice the gaps on either side are at the 7 (& -7), which is the next prime we subtract the 7 value & the 1 place as 'uncovered'[unknown] 7-2=5; which is our coverage spread either side of the zero or primefactorial point so one can say with certainty for any prime factorial one away [above & below] will have the next prime value beyond the highest prime in the prime factorial minus 2 as a certain covered [non-prime] spans. one can't know if the gaps beyond the certain spread areas are prime or not, but the spread areas can NOT be prime. I am pretty sure that it is true for implicit prime factorials too. ie 2^2*3*5^3*13=19500 where there is an implicit 2*3*5 prime factorial in the number. with a 7-2=5 covered span/spread above & below & one away from the implicit prime factorial. it's not much but it might be a start. primes seem not uncommon on one or both sides of a prime factorial & at the far ends of their spreads. It might be useful as a tool in eliminating looking at a lot of numbers that can be said with certainty not to be prime. |
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illoyd
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Goldbach's Conjecture
replied on: 11/16/2007 4:24:15 AM Yes I think this approach helps us to understand what primes are not and where they are likely to be. If it leads eventually to some definition of the characteristic of a prime then it will have been very useful. However I am reminded of the early so called astronomers who believed that the stars were produced by light coming though holes in the sky ceiling. Is this what you are doing mathematically? |
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cramwit
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Goldbach's Conjecture
replied on: 11/16/2007 12:19:09 PM i have lots of time & am compelled to joust & slay windmills across hillsides everywhere. Wherever you find a broken windmill, rest assured, i have been there. I vow the wind shall never be enslaved. Viva La Breeze. |
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illoyd
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Goldbach's Conjecture
replied on: 11/16/2007 2:08:30 PM Yes but we are looking for a Galileo not a Chaucer or who ever you are quoting. The beauty about number theory it that it is of little use to any one and every solution seems to lead to a more difficult problem! Keep going |
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cramwit
Quote | Reply | This message was updated on 11/16/2007 9:44:35 PM by cramwit |
Goldbach's Conjecture
replied on: 11/16/2007 9:38:08 PM the reference was Cervantes's Don Quixote who is seeking to fight dragons & [mis]interprets windmills as dragons. Topologically speaking inside & outside of a sphere is identical. They are both infinite sets of points in 3D space mutually bifurcated from one another by the sphere. All the 'stars' we see in the night sky is actually all the eyes of a light filled potato. So in a sense ancient philosophers may have been correct. [teasing, but only sort of] With entangled particles that are immediately adjacent or in fact the same particle separated by vast gulfs of apparent static 3D space the notion of inverting the universe [3D space] may not be a completely fantastic idea. That stars might be adjacent at their cores is not completely out of the question, or at least having minute interconnections/adjacencies. I think the idea of geometric space as opposed to topological space [topological space is oxymoron?] is static attachment to locality. I think it might be that in geometric space the particle or point has some description of its local environment. Perhaps an adjacency list, where you can only get new adjacencies from already adjacent points. Maybe to have regular n dimensional space it has a specific number of allowed adjacencies, & must trade/swap them with its adjacent partners. That would fit with the necessity of transiting essentially all intermediate points to get from point a to point b. Point/particles could have an absolute, complete description of the Universe, but an automata model seems more efficient/workable to my mere mortal mind. back on topic: in any event i was thinking that because the primefactorial by its determination necessarily excludes the p(n+1) prime it also forces the far [not the immediate, next-to] gaps to also not have as a factor p(n+1), precisely because they are in the p(n+1) position from the primefactorial. I know it is not much, but p(n+1) is the next densest prime so it the best possible improvement to the probability of those positions being prime. Interestingly because the far gaps necessarily eliminate the p(n+1) where the immediate next-to gaps don't, they must have a slightly higher probability of being prime than the immediately adjacent gaps. I have some vague, probably unworkable or false, notion that if one could sort of reach around infinity, inverting it that there would be some much more workable/efficient logic to view prime numbers from, as well as other things. Sort of coverage becomes primes & primes become coverage. Interestingly it is only locally about zero that the first positive case of a prime interval is in fact 'prime'. Also is the first negative case of a number considered 'prime'? is negative 7 prime or only positive 7 prime? Seems possibly arbitrary. A negative integer is still an integer quotient. I wonder if 'i' [imaginary] values of primes would have any interesting currency? I am pretty illiterate on the subject, but such is life. wandering off topic again: Abstractly perhaps like the sphere bifurcates 3D space at least be able to use the minimal(?) infinity of the sphere as a handle rather than wading through either the internal or external infinities of points. Bringing up the question, is the infinite set of points of the sphere in fact any less of an infinity than the 3D sets of points? topologically shouldn't they all be identical? Maybe the sphere points have a reduced adjacency lists? Not exactly the same length list as a strictly flat plane. Question, would the sphere points have potentially one less adjacency or one more than flat plane points? I'm not clear on that. Sort of both isn't it, depending on inside & outside? The curve of the sphere must be determined by having points more exposed to the exterior of the sphere than the points on the interior of the sphere. So perhaps much of our topology is some mix of quasi-geometric notions layered on the pure logic of absolute topology. In a sense it means the relationships of the immediate points to & of the sphere determine which set of infinite points is 'inside' & which is 'outside' the sphere. Either a few more than a flat plane contact or a few less than a flat plane contact determine 'inside' & 'outside' That is a sort of weird thought. makes it seem more arbitrary than one's actual experience would. But i suppose from higher dimensions any microscopic portion of 3D is accessible to infinite infinities. off topic, . . . again, many sci-fi superluminal transports include some kind of inversion of infinity for an instant, that also lasts eternity as you transit from one point in the universe to the other. I was thinking it might not be that you have to invert the entire infinity/universe, but just some portion of it. Of course that sounds like the classic 'space folding' model of space jumping. I am sort of wary of folding space too much, because it might damage space, sort of like metal fatigue. of course if it is already happening naturally, maybe it is not a big problem. Of course being turned inside-out through infinity for eternity, even for an instant does not sound at all pleasant. hope you don't mind my wild traversals of the mind. But at least their is one iota of hopefully solid, determinant on topic subject matter. |
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cramwit
Quote | Reply | This message was updated on 12/1/2007 10:30:22 AM by cramwit |
Goldbach's Conjecture
replied on: 12/1/2007 10:24:13 AM Apologies for my meandering mind. Back on topic. Goldbach's conjecture states that for any even number there are two primes that sum it. So there must be some k value between 2 & n such that n-k & n+k are both prime. Goldbach's Restated: for every number n>3 there must be 2 primes equi-difference/distance from it, above & below such that for some k, 0<=k<(n-2), n - k & n + k are both prime end further exploration I think the prime existence interval proof for n to 2n is relatively easy if it hasn't already been proved, so i will set that aside for now. there are two intervals to consider, 2 through n & n through 2n-2 The distribution of primes below a prime square are a relatively flat/even above the occurrence of that prime ('pr') as prime, ie. from pr+1 to pr^2-1 the relatively consistent & only considerations over that interval is all the modulo's of primes below pr (2,3,5...pq) we find the next prime squared above 2n, much like the integer-ceiling function except it is prime squares instead call it pr^2 the only modulos that can be relevant [prime blockers] for all the primes below pr are (2,3,5...pq) n itself may fall below some prime square between n & 2n so any proposed k must have certain specific deterministic modulo 'not prime' attributes. there must be modulo specifications on the low side of n and reflective modulo specifications on the high side with certain low end specs dropped & high end specs added as n-k goes below certain points & n+k goes above other points. note: by 'reflective' modulos i mean k(mod7)->2 being a prime factor of 7 below n then k(mod7)->5 above n must also be a prime factor of 7 [rhythmic prime block points] note: these modulo specifications are the stipulations of non-primes so any k that misses all the modulo specifications, must logically, by default, be prime. repeating that the distribution over the non-inclusive intervals between a prime & its square is relatively flat this may not be as difficult as the above stipulations seem to make it sound. so one method to prove Goldbach's is to find some unavoidable k that fits [misses] all the above & below modulo stipulations created by any n, where 0<=k<(n-2) i suppose one could first work with assuming a complete & infinitely flat prime filter & see if that was provable. [a few limited prime blockers] If so then one could go on to see/prove if any possible irregularities of that flat distribution filter would be impossible to avoid an evenly distanced pair. |
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cramwit
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Goldbach's Conjecture
replied on: 12/2/2007 5:24:48 PM basically a number n [1/2 of 2n, Goldbach's even number] floats among some pattern of non-prime [prime factor] blocking rhythms/intervals. [randomly along Eratosthenes's sieve] If n is prime all its prime factors [below the immediate prime square above it] are non-zero then n + n = 2n fits Goldbach's conjecture. If n has prime factors it's relative prime blocking intervals [to avoid] are going to have a zero modulus for those primes. for those primes which are not factors of n it will have an unequal paired modulos high & low [with the exception of 2 which is inherently symmetric] to avoid. so the possible modulos from 2 that can not be prime will be either 1 or 0 the possible modulos of 3 will be (the pair) 1,2 or 0 so we seek some k value between 0 & n-2 [ 0<=k<n-2 ] (k)mod2-> 0 or 1 (k)mod3-> 0 or 1,2 (k)mod5-> 0 or 1,4 or 2,3 (k)mod7-> 0 or 1,6 or 2,5 or 3,4 (k)mod11-> 0 or 1,10 or 2,9 or 3,8 or 4,7 or 5,6 etc. basically 0 or an unmatched a & b pair where a+b=p [except for 2 where singularly a modulo of 1 could be the case] eg. (k)mod2->1 (k)mod3->1,2 (k)mod5->0 (k)mod7->2,5 up to [excluding] the prime square above n, in this example 11^2=121 off on side exploratory side tangent for this n it must be a factor of 5 & odd as well as having either (n)mod3-> 1 or 2 & (n)mod7-> 2 or 5 & n<121 so {5,15,25,...115} 75mod7->5 might work except 75mod3->0 65mod7->2 so that would work AND 65mod3->2 so n could be 65 end side tangent again if n has no zero modulos for any of the primes below the next immediate prime square above it then it is prime & fits the easiest case of n + n = 2n if n has at least one zero modulo then we seek a k value that misses/avoids all its modulo specifications. missing zero modulos for its prime factors and some specific set of mixed modulo pairs for each of its non-prime factors. as k goes out further & further from the center, n value, on the low end [n-k] as it crosses below some square of one of its considered modulos, the low correlated 1/2 of that pair may be eliminated from consideration. Conversely as we go out from n with increasing k values [n+k] & arrive at a prime square, one will have to add a single, unpaired, high end modulo for that new prime filter/blocker. So if Goldbach's is to be provable, at least along this vein of thought it must be provable that two equal distances 'k' from n are both prime. ie. this k value misses/avoids all the specified modulo blocks. Presumably including the messy inclusion of ancillary primes as they are intersected. If nothing else it does point to an alternate method of finding paired prime differences for 2n. (Goldbach prime pairs) |
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