[math-fun] Goldbach conjecture update
Since this topic has subsided from public view, a priesthood incorporating James Maynard, Andrew Granville, Terry Tao, has been assiduousy shaving Zhang's original 70 million, and refining and simplifying the proof to the point where a normal mathematically-inclined human (if such an entity exists) might possibly be able to follow it with the assistance of a powerful computer and a sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement. The current result asserts that there are infinitely many consecutive prime pairs with difference h for some h <= 600 . The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) . For those with time to spare and robust self-esteem, there is (considerably) more detail available at http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m... Fred Lunnon
Fred, Thanks for this. However, their work is for approximations to the twin prime conjecture. It doesn't say anything about Goldbach. Victor On Wed, Nov 20, 2013 at 9:46 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Since this topic has subsided from public view, a priesthood incorporating James Maynard, Andrew Granville, Terry Tao, has been assiduousy shaving Zhang's original 70 million, and refining and simplifying the proof to the point where a normal mathematically-inclined human (if such an entity exists) might possibly be able to follow it with the assistance of a powerful computer and a sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement.
The current result asserts that there are infinitely many consecutive prime pairs with difference h for some h <= 600 . The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) .
For those with time to spare and robust self-esteem, there is (considerably) more detail available at
http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m...
Fred Lunnon
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On Wed, Nov 20, 2013 at 9:46 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement.
The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) .
There must be some further restriction this . In particular, h must be a multiple of p, for any p <=m, or one of the numbers in the progression will be a multiple of p. Are there any additional restrictions?
For those with time to spare and robust self-esteem, there is (considerably) more detail available at
http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m...
On this page, a comment asks if any of this generalizes to arithmetic progressions, but there is no reply. Andy
The "progressions" was bungled shorthand for (infinitely many repetitions of) sets of m consecutive primes with specified differences; and of course the conjecture is the "twin prime" (don't listen to what I say; listen to what I mean). Thanks for corrections, and apologies for my typographical incompetence. WFL On 11/20/13, Andy Latto <andy.latto@pobox.com> wrote:
On Wed, Nov 20, 2013 at 9:46 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement.
The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) .
There must be some further restriction this . In particular, h must be a multiple of p, for any p <=m, or one of the numbers in the progression will be a multiple of p. Are there any additional restrictions?
For those with time to spare and robust self-esteem, there is (considerably) more detail available at
http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m...
On this page, a comment asks if any of this generalizes to arithmetic progressions, but there is no reply.
Andy _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Fred, anyone, Do you know if they are finding two (or N) consecutive primes, or if there might be more primes mixed in within the specified pattern? This doesn't happen for twin primes, or small-gap near-twins (sibs? cousins?), but is possible for larger gaps like 600. The infinite count has a curious consequence: since there are only a finite number of possible prime-or-composite patterns of a specific width, there is some specific pattern like Pcccpcc...P of width<600 that occurs infinitely often. (Capital P at each end, or with periodic spacing in the more-terms-arithmetic-progression case, are the theorem's required primes, while the interior lower case p/c's are the required-by-infinite-happenstance pattern.) Of course the most probable pattern is the one where there are no lower-case p's, i.e. where every non-forced number is composite. Rich -------- Quoting Fred Lunnon <fred.lunnon@gmail.com>:
The "progressions" was bungled shorthand for (infinitely many repetitions of) sets of m consecutive primes with specified differences; and of course the conjecture is the "twin prime" (don't listen to what I say; listen to what I mean).
Thanks for corrections, and apologies for my typographical incompetence.
WFL
On 11/20/13, Andy Latto <andy.latto@pobox.com> wrote:
On Wed, Nov 20, 2013 at 9:46 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement.
The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) .
There must be some further restriction this . In particular, h must be a multiple of p, for any p <=m, or one of the numbers in the progression will be a multiple of p. Are there any additional restrictions?
For those with time to spare and robust self-esteem, there is (considerably) more detail available at
http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m...
On this page, a comment asks if any of this generalizes to arithmetic progressions, but there is no reply.
Andy _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Ooof --- I did say "consecutive", and on closer examination managed to get that wrong as well: there may indeed be extra primes in between. I'll insure against further embarrassment by cribbing the opening paragraph from http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m... --- the author of which, among his other intimidating characteristics, has an irritating habit of getting multiple interlocking details right first time, maintaining all the while a stupendous rate of output ... << For each natural number {m}, let {H_m} denote the quantity \displaystyle H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n), where {p_n} denotes the {n\textsuperscript{th}} prime. In other words, {H_m} is the least quantity such that there are infinitely many intervals of length {H_m} that contain {m+1} or more primes. Thus, for instance, the twin prime conjecture is equivalent to the assertion that {H_1 = 2}, and the prime tuples conjecture would imply that {H_m} is equal to the diameter of the narrowest admissible tuple of cardinality {m+1} (thus we conjecturally have {H_1 = 2}, {H_2 = 6}, {H_3 = 8}, {H_4 = 12}, {H_5 = 16}, and so forth; see http://math.mit.edu/~primegaps/ for further continuation of this sequence).
WFL On 11/22/13, rcs@xmission.com <rcs@xmission.com> wrote:
Fred, anyone,
Do you know if they are finding two (or N) consecutive primes, or if there might be more primes mixed in within the specified pattern? This doesn't happen for twin primes, or small-gap near-twins (sibs? cousins?), but is possible for larger gaps like 600.
The infinite count has a curious consequence: since there are only a finite number of possible prime-or-composite patterns of a specific width, there is some specific pattern like Pcccpcc...P of width<600 that occurs infinitely often. (Capital P at each end, or with periodic spacing in the more-terms-arithmetic-progression case, are the theorem's required primes, while the interior lower case p/c's are the required-by-infinite-happenstance pattern.)
Of course the most probable pattern is the one where there are no lower-case p's, i.e. where every non-forced number is composite.
Rich
-------- Quoting Fred Lunnon <fred.lunnon@gmail.com>:
The "progressions" was bungled shorthand for (infinitely many repetitions of) sets of m consecutive primes with specified differences; and of course the conjecture is the "twin prime" (don't listen to what I say; listen to what I mean).
Thanks for corrections, and apologies for my typographical incompetence.
WFL
On 11/20/13, Andy Latto <andy.latto@pobox.com> wrote:
On Wed, Nov 20, 2013 at 9:46 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement.
The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) .
There must be some further restriction this . In particular, h must be a multiple of p, for any p <=m, or one of the numbers in the progression will be a multiple of p. Are there any additional restrictions?
For those with time to spare and robust self-esteem, there is (considerably) more detail available at
http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m...
On this page, a comment asks if any of this generalizes to arithmetic progressions, but there is no reply.
Andy _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Just wondering if anyone knows: Do all of these results, from the original Zhang constant down to Maynard's 600, imply only that there exists *at least one* number N no more than the constant such that there are infinitely many primes separated by exactly N ??? Or is there some reasoning within the proofs of these theorems that would imply more than one such N ??? --Dan On 2013-11-20, at 6:46 AM, Fred Lunnon wrote:
Since this topic has subsided from public view, a priesthood incorporating James Maynard, Andrew Granville, Terry Tao, has been assiduousy shaving Zhang's original 70 million, and refining and simplifying the proof to the point where a normal mathematically-inclined human (if such an entity exists) might possibly be able to follow it with the assistance of a powerful computer and a sabbatical year or two --- by which time it would no doubt have undergone further drastic improvement.
The current result asserts that there are infinitely many consecutive prime pairs with difference h for some h <= 600 . The result generalises to finite arithmetic progressions of m consecutive primes with span h , for which instead h <= exp(8m + 5) .
For those with time to spare and robust self-esteem, there is (considerably) more detail available at http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-m...
Fred Lunnon
participants (5)
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Andy Latto -
Dan Asimov -
Fred Lunnon -
rcs＠xmission.com -
Victor Miller