Timeline for The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?
Current License: CC BY-SA 4.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jan 6, 2019 at 20:15 | review | Close votes | |||
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| S Jan 6, 2019 at 19:55 | history | suggested | CommunityBot | CC BY-SA 4.0 |
trying to beautify formula that's resisting very hard.
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| Jan 6, 2019 at 16:57 | review | Suggested edits | |||
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| Apr 3, 2017 at 17:22 | history | edited | Martin Sleziak |
removed (tag-removed) tag (The question has been bumped anyway.)
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| Apr 3, 2017 at 9:33 | history | edited | Vincent | CC BY-SA 3.0 |
link was broken, replaced it with new link to the same article on the same mother site
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| Sep 6, 2014 at 19:01 | history | edited | Sebastien Palcoux |
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| Aug 11, 2013 at 8:13 | vote | accept | Sebastien Palcoux | ||
| Jul 25, 2013 at 17:24 | review | Close votes | |||
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| Jul 25, 2013 at 17:06 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
I rewrite the big formula properly.
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| Jun 13, 2013 at 19:21 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 11, 2013 at 16:08 | comment | added | stankewicz | The analysis is unlikely to be simpler. There is apparently such a polynomial with 12 variables but whose degree is well over 100 thousand. Alternately, there's a prime producing polynomial whose degree is 5, but is in 42 variables. mathdl.maa.org/images/upload_library/22/Ford/… | |
| Jun 9, 2013 at 10:11 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 9, 2013 at 10:06 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 9, 2013 at 1:17 | comment | added | Sebastien Palcoux | The reason is that this polynomial is explicit, so that we can work with, and maybe use differential calculus on to investigate some results on the prime numbers. If anyone published another explicit polynomial with less than 26 variables, maybe its analysis would be more simple... Do you know one ? | |
| Jun 9, 2013 at 0:50 | comment | added | The User | Actually, Matiyasevich has proven that you can encode any algorithm into a polynomial using only 9 variables (not 26!). Is there a particular reason why you think that this polynomial is especially nice? | |
| Jun 8, 2013 at 22:47 | comment | added | Andrés E. Caicedo | The first of your bonus questions seems to me to be a bit lazy, surely unintentionally. The proof of the Jones-Sata-Wada-Wiens theorem tells you how to find the variables. | |
| Jun 8, 2013 at 22:34 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 7, 2013 at 18:56 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 7, 2013 at 18:52 | comment | added | Sebastien Palcoux | Thank you @François, it's an interesting remark, I upvoted your comment. Does anyone know a solution for $(a,b,c,..,z)$ to obtain the first primes numbers $2$, $3$ and $5$ ? It's a little off topic, but it's an interesting question. Does anyone can evaluate the minimal size of a solution for a fixed prime ? | |
| Jun 7, 2013 at 14:49 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 7, 2013 at 11:29 | comment | added | François Brunault | It seems to me that before tying to generate explicit sequences of large primes, a basic question is to generate an explicit single prime. | |
| Jun 7, 2013 at 10:51 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 7, 2013 at 10:01 | comment | added | Sebastien Palcoux | I don't know, we certainly can't generate large primes with a direct use of this polynomial in a reasonable time. But if we analyse this polynomial with differential calculus, maybe we can find some generic extremal points admitting neighborhood areas of positive range, and then generate easily computable (as large as we want) prime numbers. I don't know if it's possible, but it's worth a try... | |
| Jun 7, 2013 at 8:59 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 7, 2013 at 8:51 | comment | added | François Brunault | What is the largest prime that this polynomial has produced explicitly? | |
| Jun 7, 2013 at 8:15 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 7, 2013 at 7:42 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 7, 2013 at 0:42 | answer | added | user9072 | timeline score: 17 | |
| Jun 6, 2013 at 20:39 | history | edited | Sebastien Palcoux |
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| Jun 6, 2013 at 20:29 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 6, 2013 at 19:52 | comment | added | Sebastien Palcoux | Thank you Nathan for this citation of Paulo Ribenboim. I agree, this polynomial encodes an algorithm, but it's also a concrete polynomial, so it can be analyse by the tools of differential calculus to identify the extremal points and the area with positive range... I don't know if it's easy to compute, but now there exists very powerfull computers and supercomputers: $10^{6}$ times more powerfull than in 1989 when Paulo write its book. So does anyone can explain to me why all this investigation I'm talking about would be useless ? | |
| Jun 6, 2013 at 17:53 | history | edited | Sebastien Palcoux |
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| Jun 6, 2013 at 17:22 | comment | added | Nathan | As Paulo Ribenboim pointed out in The Book of Prime Number Records, "It should be noted that this polynomial also takes on negative values, and that a prime number may appear repeatedly as a value of the polynomial." It's my understanding that what this sort of polynomial does is encode an algorithm for listing the set. So there's no reason to expect it to give any particular insight not already implicit in the algorithm. | |
| Jun 6, 2013 at 17:10 | comment | added | Sebastien Palcoux | Thank you, you are probably true. But my question is about the use of calculus, so I put the modest tags "calculus". I hope it's ok for you. | |
| Jun 6, 2013 at 17:07 | history | edited | Sebastien Palcoux |
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| Jun 6, 2013 at 17:07 | comment | added | alvarezpaiva | @Sébastien: of course, everything is potentially useful, but maybe your question is not sufficiently focused to understand what you would like to do with this polynomial. By the way, I took the liberty of editing out the tags "differential geometry" and "calculus of variations", which do not seem to be related to the question. | |
| Jun 6, 2013 at 16:57 | history | edited | alvarezpaiva |
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| Jun 6, 2013 at 16:35 | comment | added | Sebastien Palcoux | Ok ok, there exists such a polynomial for every recursively enumerable set of numbers, but this polynomial here is concrete, we can analyse it, compute its variations, its extremal points..., by using modern supercomputer if necessary (they could not do it in 1976 ...). So what is the argument to prove that doing so will not be usefull to better understand the prime numbers? | |
| Jun 6, 2013 at 15:54 | comment | added | Andrés E. Caicedo | The term quid is looking for is r.e. (recursively enumerable), also called c.e. (computably enumerable). These are the sets of numbers that can be the output of a computer program running forever. | |
| Jun 6, 2013 at 15:46 | comment | added | user9072 | I think rather not. There is even a story related to this, in that some mathematician when being told about these types of results for primes in an informal way said something like this is great we will learn a lot about the prime numbers from it. Then, however, they were informaed that essatially same thing can be done with many other types of sets than the primes (the precise technical term for these sets escapes me at the moment, but they are not overly special sets) so not much hope to learn much on the primes specifically was the conclusion. | |
| Jun 6, 2013 at 15:29 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
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| Jun 6, 2013 at 14:21 | history | asked | Sebastien Palcoux | CC BY-SA 3.0 |