JSH: Depressing reality, prime reality
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From: JSH on 8 Mar 2010 22:38 On Mar 8, 7:24 pm, Rotwang <sg...(a)hotmail.co.uk> wrote: > JSH wrote: > > On Mar 8, 7:04 pm, Rotwang <sg...(a)hotmail.co.uk> wrote: > >> JSH wrote: > >>> On Mar 8, 6:56 pm, Rotwang <sg...(a)hotmail.co.uk> wrote: > >>>> Jesse F. Hughes wrote: > >>>>> JSH <jst...(a)gmail.com> writes: > >>>>>>>http://img96.imageshack.us/img96/8200/axiom19.png > >>>>>> I don't chase links. > >>>>>>> The first hit is Jim's post on Google Groups. > >>>>>> Not for me. Did you use Google or Google Groups? > >>>>> If you'd simply look at the link, you'd know. > >>>> Indeed, but since James is concerned that looking at a .png file will > >>>> give him a virus: I used Google. > >>>>>> And try Yahoo! or Bing as well. > >>>>> Searching for "Musatov's Axiom 19" brings up Jim's post on every site > >>>>> but Bing. So much the worse for Bing, since Jim's post is the most > >>>>> relevant mention of "Musatov's Axiom 19". > >>>> Actually, Jim's post was the second hit on Bing when I tried. It was the > >>>> first hit on Yahoo!. > >>> Oh, search is: prime residue axiom > >>> Or you can just try: residue axiom > >>> Or you can try: prime residue > >> Huh? You wrote > > >> You're deluded. Google your try, and then Google: prime residue > >> axiom > > >> Or go to any web search engine. Here it does not have to be > >> Google. > > >> The world listens to me. It does not listen to you. > > >> His try was "Musatov's Axiom 19". What do you imagine the results of a > >> completely different search are supposed to prove? > > > First, take off the quotes. > > > And second, quit playing stupid. Now if I don't have quotes, why do > > you? > > > Also, do that search on: prime residue > > > Or search: residue axiom > > > Yes, in MY THREAD, you may have a situation where putting quotes on > > that particular phrase is pulling into search engines now. But try > > the search again across search engines WITHOUT PUTTING IT IN QUOTES. > > > Can you do that? > > Already did, as you would know if you were to look at the image I posted > (guess reality testing isn't much use if you're too scared to look at > reality). All the searches whose results I reported were without quotes. Oh. I don't chase links. I was just looking at replies which had it in quotes. Check tomorrow. It'll drop. My threads don't last in search strings, usually, and he's getting a boost from this one. The real test is the test of time. In the meantime, I dominate in this area across several search strings related to primes: prime residue axiom prime residue residue axiom Try clipping your "Musatov" search in similar ways and see what you get. James Harris
From: Transfer Principle on 8 Mar 2010 23:50 On Mar 8, 5:12 am, "Ostap S. B. M. Bender Jr." <ostap_bender_1...(a)hotmail.com> wrote: > On Mar 7, 7:50 am, JSH <jst...(a)gmail.com> wrote: > > Reality of course is that I first wondered if > > primes might have no preference with each other > Wait. Didn't you earlier discover that primes prefer composites? I usually don't bother to post in JSH threads, but there's something I'd like to point out here: > you derive that some primes "might have preference with each other". In another JSH thread last month, a standard theorist found a way to express (not _prove_, of course, but just express) JSH's claim using more rigorous mathematical terminology. Instead of referring to "preferences," the standard theorist mentioned the notion of asymptotic density, as follows. Let P(k) denote the kth prime. That is to say, P(k) is defined recursively as: P(1) = 2 P(k+1) = min({peN | p>P(k) & p prime}) Then JSH's claim can be stated as follows: if m and n are coprime, then the set {keN | (P(k) == n) mod m} has asymptotic density 1/phi(m) (phi is Euler's totient). One standard theorist claimed that the result is true and follows from Dirichlet's theorem, but Dirichlet's theorem only shows that these sets are _infinite_, not what their density is. Another standard theorist claimed that the result is false, and used quadratic residues in his argument. Indeed, he pointed out that if r is a quadratic residue mod m, and n is a quadratic nonresidue mod m, then for many natural numbers M, card({keN | k<M & (P(k) == n) mod m}) > card({keN | k<M & (P(k) == r) mod m}) One can see why this may be the case -- since r is a quadratic residue mod m, by definition, the set on the right contains squares, and by definition squares are not primes. However, the poster failed to mention whether this affects the asymptotic density at all, so that: density({keN | (P(k) == n) mod m}) > 1/phi(m), n a nonresidue mod m density({keN | (P(k) == r) mod m}) < 1/phi(m), r a residue mod m > > The best never stop at one. > Andrew Wiles has zero. Unlike you, he can prove theorems, so he has no > need to "discover" axioms. Until a standard theorist gives a proof, or at least a reference to a proof, of either JSH's claim or its negation, the possibility remains that his claim is undecidable in ZFC. In that case, JSH would be justified in calling his claim an "axiom." Indeed, since Euclid's Fifth Postulate and AC have been shown to be undecidable in neutral geometry and ZF, respectively, one can add either these or their negations as _axioms_ and produce a new theory. > > There are no one-hit wonders in major mathematical discovery. > Most so-called "mathematicians" have not discovered a single new axiom > in their entire miserable lives. "Discovering" axioms isn't meaningful, but discovering nontrivial statements that are undecidable in ZFC is -- especially if one can derive interesting results from them. Consider Riemann's Hypothesis. So far neither RH nor its negation have been proved in ZFC, and the possibility remains that RH might be undecidable in ZFC. But there are some results in number theory that require RH. Therefore, if RH is shown to be undecidable in ZFC, some number theorists might prefer to work in ZFC+RH and not ZFC. It's conceivable that JSH's claim might also lead to interesting results in number theory. Although I don't really believe that JSH has a proof of the Twin Primes Conjecture, if someday in the future someone proves the Twin Primes Conjecture, but the proof requires JSH's claim, and that claim has been proved undecidable in ZFC, then number theorists might choose to work in ZFC+JSH rather than ZFC, thus vindicating JSH's axiom once and for all.
From: Ostap S. B. M. Bender Jr. on 9 Mar 2010 03:30 On Mar 8, 8:50 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Mar 8, 5:12 am, "Ostap S. B. M. Bender Jr." > > <ostap_bender_1...(a)hotmail.com> wrote: > > On Mar 7, 7:50 am, JSH <jst...(a)gmail.com> wrote: > > > Reality of course is that I first wondered if > > > primes might have no preference with each other > > Wait. Didn't you earlier discover that primes prefer composites? > > I usually don't bother to post in JSH threads, but there's > something I'd like to point out here: > > > you derive that some primes "might have preference with each other". > > In another JSH thread last month, a standard theorist found a way > to express (not _prove_, of course, but just express) JSH's claim > using more rigorous mathematical terminology. Instead of referring > to "preferences," the standard theorist mentioned the notion of > asymptotic density, as follows. > > Let P(k) denote the kth prime. That is to say, P(k) is defined > recursively as: > > P(1) = 2 > P(k+1) = min({peN | p>P(k) & p prime}) > > Then JSH's claim can be stated as follows: if m and n are coprime, > then the set > > {keN | (P(k) == n) mod m} > > has asymptotic density 1/phi(m) (phi is Euler's totient). > I am brand new to these discussions, so please explain the background to me. What exactly does JSH's "axiom", as expressed by JSH himself, say? Also, has anybody BEFORE JSH posed (and solved) this problem before? For example, is it at all related to the Dirichlet's theorem? > > One standard theorist claimed that the result is true and follows > from Dirichlet's theorem, but Dirichlet's theorem only shows that > these sets are _infinite_, not what their density is. > Really? That's not what English and Russian Wikipedias say: http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_primes_in_arithmetic_progressions In number theory, Dirichlet's theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ⥠0. In other words: there are infinitely many primes which are congruent to a modulo d. Stronger forms of Dirichlet's theorem state that different arithmetic progressions with the same modulus have approximately the same proportions of primes. Further, the proportion of primes in each of those is: 1/phi(m), where phi is Euler's totient function. Russian Wiki: \lim_{s\to 1+}\frac{\sum_p\frac1{p^s}}{\ln\frac1{s-1}}= \frac1{\varphi(k)} > > Another standard theorist claimed that the result is false, and > used quadratic residues in his argument. Indeed, he pointed out > that if r is a quadratic residue mod m, and n is a quadratic > nonresidue mod m, then for many natural numbers M, > > card({keN | k<M & (P(k) == n) mod m}) > card({keN | k<M & (P(k) == r) > mod m}) > > One can see why this may be the case -- since r is a quadratic > residue mod m, by definition, the set on the right contains > squares, and by definition squares are not primes. However, the > poster failed to mention whether this affects the asymptotic > density at all, so that: > > density({keN | (P(k) == n) mod m}) > 1/phi(m), n a nonresidue mod m > density({keN | (P(k) == r) mod m}) < 1/phi(m), r a residue mod m > > > > The best never stop at one. > > Andrew Wiles has zero. Unlike you, he can prove theorems, so he has no > > need to "discover" axioms. > So, let me understand what you have told me: A couple of months or years ago, JSH (or "some standard theorist") has posed a hypothesis (let's call it hypothesis JSH) that: -------------------------- if m and n are coprime, then the set {keN | (P(k) == n) mod m} has asymptotic density 1/phi(m) -------------------------- Now, it sounds like an interesting problem to resolve this JSH, and maybe a very difficult problem unless it follows immediately from Dirichlet's Theorem (I am too lazy to check). But there are thousands of interesting and very difficult problems/hypotheses in number theory alone. At what point will you add these hypotheses to the set of axioms avout the natural numbers? For example, the Fermat Theorem remained unresolved for several centuries, despite the fact that thousands of brilliant mathematicians tried in vain to prove it. And yet, I had never heard of anybody advocating that the Fermat's Theorem be made into a new axiom. And low and behold - Wiles came and proved it. Not an axiom. Bummer. :-) Or let's take the Twin Prime conjecture itself. Has anybody declared it a new axiom yet? HardyâLittlewood conjecture? Polignac's conjecture? Has anybody made them into axioms yet? :-) Just because two guys on the Internet tried to resolve JSH and couldn't, doesn't mean it's unsolvable, does it? > > Until a standard theorist gives a proof, or at least a reference > to a proof, of either JSH's claim or its negation, the possibility > remains that his claim is undecidable in ZFC. In that case, JSH > would be justified in calling his claim an "axiom." Indeed, since > Euclid's Fifth Postulate and AC have been shown to be undecidable > in neutral geometry and ZF, respectively, one can add either these > or their negations as _axioms_ and produce a new theory. > What reason do you have to expect this "JSH hypothesis" to be more likely to be undecidable than the Twin Prime conjecture or any of the other thousands of open conjectures? > > > > There are no one-hit wonders in major mathematical discovery. > > Most so-called "mathematicians" have not discovered a single new axiom > > in their entire miserable lives. > > "Discovering" axioms isn't meaningful, but discovering nontrivial > statements that are undecidable in ZFC is -- especially if one can > derive interesting results from them. > > Consider Riemann's Hypothesis. So far neither RH nor its negation > have been proved in ZFC, and the possibility remains that RH might > be undecidable in ZFC. But there are some results in number theory > that require RH. Therefore, if RH is shown to be undecidable in ZFC, > some number theorists might prefer to work in ZFC+RH and not ZFC. > > It's conceivable that JSH's claim might also lead to interesting > results in number theory. Although I don't really believe that JSH > has a proof of the Twin Primes Conjecture, > So, you think that JSH doesn't have a proof of the Twin Primes Conjecture but aren't sure that he doesn't. Could you share with me what "proof" of the Twin Primes Conjecture JSH has given so far, so that I could see why you aren't sure? > > if someday in the future > someone proves the Twin Primes Conjecture, but the proof requires > JSH's claim, and that claim has been proved undecidable in ZFC, > then number theorists might choose to work in ZFC+JSH rather than > ZFC, thus vindicating JSH's axiom once and for all. > The same can be said not only about JSH but about ANY open hypothesis. What's different about JSH?
From: Ostap S. B. M. Bender Jr. on 9 Mar 2010 03:43 On Mar 9, 12:30 am, "Ostap S. B. M. Bender Jr." <ostap_bender_1...(a)hotmail.com> wrote: > On Mar 8, 8:50 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > > On Mar 8, 5:12 am, "Ostap S. B. M. Bender Jr." > > > <ostap_bender_1...(a)hotmail.com> wrote: > > > On Mar 7, 7:50 am, JSH <jst...(a)gmail.com> wrote: > > > > Reality of course is that I first wondered if > > > > primes might have no preference with each other > > > Wait. Didn't you earlier discover that primes prefer composites? > > > I usually don't bother to post in JSH threads, but there's > > something I'd like to point out here: > > > > you derive that some primes "might have preference with each other". > > > In another JSH thread last month, a standard theorist found a way > > to express (not _prove_, of course, but just express) JSH's claim > > using more rigorous mathematical terminology. Instead of referring > > to "preferences," the standard theorist mentioned the notion of > > asymptotic density, as follows. > > > Let P(k) denote the kth prime. That is to say, P(k) is defined > > recursively as: > > > P(1) = 2 > > P(k+1) = min({peN | p>P(k) & p prime}) > > > Then JSH's claim can be stated as follows: if m and n are coprime, > > then the set > > > {keN | (P(k) == n) mod m} > > > has asymptotic density 1/phi(m) (phi is Euler's totient). > > I am brand new to these discussions, so please explain the background > to me. What exactly does JSH's "axiom", as expressed by JSH himself, > say? > > Also, has anybody BEFORE JSH posed (and solved) this problem before? > For example, is it at all related to the Dirichlet's theorem? > > > > > One standard theorist claimed that the result is true and follows > > from Dirichlet's theorem, but Dirichlet's theorem only shows that > > these sets are _infinite_, not what their density is. > > Really? That's not what English and Russian Wikipedias say: > > http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_primes_in_arith... > > In number theory, Dirichlet's theorem, states that for any two > positive coprime integers a and d, there are infinitely many primes of > the form a + nd, where n ⥠0. In other words: there are infinitely > many primes which are congruent to a modulo d.  Stronger forms of > Dirichlet's theorem state that  different arithmetic progressions with > the same modulus have approximately the same proportions of primes. > Further, the proportion of primes in each of those is: 1/phi(m), where > phi is Euler's totient function. > > Russian Wiki: > >  \lim_{s\to 1+}\frac{\sum_p\frac1{p^s}}{\ln\frac1{s-1}}= > \frac1{\varphi(k)} > > > > > > > Another standard theorist claimed that the result is false, and > > used quadratic residues in his argument. Indeed, he pointed out > > that if r is a quadratic residue mod m, and n is a quadratic > > nonresidue mod m, then for many natural numbers M, > > > card({keN | k<M & (P(k) == n) mod m}) > card({keN | k<M & (P(k) == r) > > mod m}) > > > One can see why this may be the case -- since r is a quadratic > > residue mod m, by definition, the set on the right contains > > squares, and by definition squares are not primes. However, the > > poster failed to mention whether this affects the asymptotic > > density at all, so that: > > > density({keN | (P(k) == n) mod m}) > 1/phi(m), n a nonresidue mod m > > density({keN | (P(k) == r) mod m}) < 1/phi(m), r a residue mod m > > > > > The best never stop at one. > > > Andrew Wiles has zero. Unlike you, he can prove theorems, so he has no > > > need to "discover" axioms. > > So, let me understand what you have told me: > > A couple of months or years ago, JSH (or "some standard theorist") has > posed a hypothesis (let's call it hypothesis JSH) that: > > -------------------------- >  if m and n are coprime, then the set > >  {keN | (P(k) == n) mod m} > >  has asymptotic density 1/phi(m) > > -------------------------- > > Now, it sounds like an interesting problem to resolve this JSH, and > maybe a very difficult problem unless it follows immediately from > Dirichlet's Theorem (I am too lazy to check).  But there are thousands > of interesting and very difficult problems/hypotheses in number theory > alone. At what point will you add these hypotheses to the set of > axioms avout the natural numbers? > > For example, the Fermat Theorem remained unresolved for several > centuries, despite the fact that thousands of brilliant mathematicians > tried in vain to prove it. And yet, I had never heard of anybody > advocating that the Fermat's Theorem be made into a new axiom. And low > and behold - Wiles came and proved it. Not an axiom. Bummer. :-) > > Or let's take the Twin Prime conjecture itself. Has anybody declared > it a new axiom yet? > > HardyâLittlewood conjecture? Polignac's conjecture? Has anybody made > them into axioms yet? :-) > > Just because two guys on the Internet tried to resolve JSH and > couldn't, doesn't mean it's unsolvable, does it? > > > > > Until a standard theorist gives a proof, or at least a reference > > to a proof, of either JSH's claim or its negation, the possibility > > remains that his claim is undecidable in ZFC. In that case, JSH > > would be justified in calling his claim an "axiom." Indeed, since > > Euclid's Fifth Postulate and AC have been shown to be undecidable > > in neutral geometry and ZF, respectively, one can add either these > > or their negations as _axioms_ and produce a new theory. > > What reason do you have to expect this "JSH  hypothesis" to be more > likely to be undecidable than the Twin Prime conjecture or any of the > other thousands of open conjectures? > > > > > > > > > There are no one-hit wonders in major mathematical discovery. > > > Most so-called "mathematicians" have not discovered a single new axiom > > > in their entire miserable lives. > > > "Discovering" axioms isn't meaningful, but discovering nontrivial > > statements that are undecidable in ZFC is -- especially if one can > > derive interesting results from them. > > > Consider Riemann's Hypothesis. So far neither RH nor its negation > > have been proved in ZFC, and the possibility remains that RH might > > be undecidable in ZFC. But there are some results in number theory > > that require RH. Therefore, if RH is shown to be undecidable in ZFC, > > some number theorists might prefer to work in ZFC+RH and not ZFC. > > > It's conceivable that JSH's claim might also lead to interesting > > results in number theory. Although I don't really believe that JSH > > has a proof of the Twin Primes Conjecture, > > So, you think that JSH doesn't have a proof of the Twin Primes > Conjecture but aren't sure that he doesn't. Could you share with me > what "proof" of the Twin Primes Conjecture JSH  has given so far, so > that I could see why you aren't sure? > > > > >  if someday in the future > > someone proves the Twin Primes Conjecture, but the proof requires > > JSH's claim, and that claim has been proved undecidable in ZFC, > > then number theorists might choose to work in ZFC+JSH rather than > > ZFC, thus vindicating JSH's axiom once and for all. > I am too lazy to check, but wouldn't the proof (using only ZFC) that JSH doesn't contradict ZFC, be actually a proof that JSH is true in ZFC? > > The same can be said not only about JSH but about ANY open > hypothesis.  What's different about JSH?
From: Michael Stemper on 9 Mar 2010 09:00
In article <7fbe8630-0467-497d-9ee4-d169927afe8e(a)k6g2000prg.googlegroups.com>, Arthur <nabikov1900(a)yahoo.com> writes: >On Mar 5, 6:08=A0pm, JSH <jst...(a)gmail.com> wrote: >> There is a lot of satisfaction with having my own axiom, which I had >> the honor of naming as I'm the discoverer, which is of course, the >> prime residue axiom, and yes, posters can reply in the negative or >> derisively, but there you see the difference between finding something >> and talk. >You seem to say that you have discovered a new "axiom" in number >theory. For it to be a new axiom, it is necessary to: > >1. Prove that it cannot be derived from the already existing axioms as >a theorem > >2. Explain why you believe that this axiom/fact is correct, and there >exist no counterexamples to it. > >What seems to have happened is that you were trying to prove some >theorem but couldn't. So, you declared it a "new axiom". That's a very >lazy way of approaching science. What if every time a teacher gives a >homework to high school or colege students to prove something, the >students declare that no proofs are needed because these homework >problems are "new axioms"? Think of the improvement in productivity possible with this approach. If the mathematician's guild, with its medieval insistence that every theorem be worked out from axioms dating back fifty, one hundred, two hundred years old, can be overthrown, new methods of production could be put in place. Replace the individual mathematician, working alone with only a white board, with huge teams of axiom-discoverers. Each one could prove scores of theorems per day, resolving many long-standing conundra by the simple expedient of discovering new axioms. After clearing away all of this backlog, vast new vistas of mathematics can then be explored. I believe that your modest proposal would be a good first step in vocational training for the new breed of axiom-discoverers. -- Michael F. Stemper #include <Standard_Disclaimer> "Writing about jazz is like dancing about architecture" - Thelonious Monk |