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A variability measure of sorts for prime distributions?
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Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
I suspect that this sort of thing has already been well analysed, but am curious to know of any results. 0 Attachment
Nice to catch up with you in these most unlikely surroundings after all these years separated by a moat 2000km wide. Evert few years I vaunt my ignorance profound on this forum, and spend the rest licking my wounds.
But, the lure of the call of the Jumble which is the Prime World brings me back.
My answer to tour query will, as usual, be wrong in one or more aspects, but that fact will like a magnet arouse the finer feelings which lay deep and hidden in the heart of every Mathematician, with their beautiful minds, (and Russel is one of us) even if he emerges from trapdoors with the aim of disembowelling you on the spot, sword of Gauss in hand, and the shield of Euler, (in the immortal woids of Mr James Durante in an oilier time). Euwe always fascinated me, as besides only holding the Title very briefly, had a name which had to be heard torather than arrived ay by analysis.
But, I digress.
In all my visits here, I cannot recall the words Ave and SD being mentioned. What that means I suspect is that the Lod reigns supreme here as a measure of probability, for reasons which are faily obvious. Basically, primes are the log of themselves apart from the next door neighbour as a general rule, separated my the natural moat of at the preceding and following even number.
So, you are searching for a function in which boring log plays a crucial part both as the area under observation, and the gaps which play in its region, with somww delightful interplay which occurs when there is in the vicinity a number which has a plenitude of small factors.
So, dust off your Ribenboim at once, to quote Mr Durante again. or get onto Mr E Bay, who probably has a dozen going cheap from members dismissed from thid Forum!
 In primenumbers@yahoogroups.com, "woodhodgson@..." <rupert.weather@...> wrote:
>
> Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
>
> I suspect that this sort of thing has already been well analysed, but am curious to know of any results.
> 0 Attachment
Pardon my typs, but mine eyes eyes are dimming, my ears are failing, and my hands are none too steady, and I am still too impetuous, a disadvantage in one's Dotty Age even more than in Uncouth Youth.
 In primenumbers@yahoogroups.com, "John" <mistermac39@...> wrote:
>
> Nice to catch up with you in these most unlikely surroundings after all these years separated by a moat 2000km wide. Evert few years I vaunt my ignorance profound on this forum, and spend the rest licking my wounds.
> But, the lure of the call of the Jumble which is the Prime World brings me back.
>
> My answer to tour query will, as usual, be wrong in one or more aspects, but that fact will like a magnet arouse the finer feelings which lay deep and hidden in the heart of every Mathematician, with their beautiful minds, (and Russel is one of us) even if he emerges from trapdoors with the aim of disembowelling you on the spot, sword of Gauss in hand, and the shield of Euler, (in the immortal woids of Mr James Durante in an oilier time). Euwe always fascinated me, as besides only holding the Title very briefly, had a name which had to be heard torather than arrived ay by analysis.
>
> But, I digress.
>
> In all my visits here, I cannot recall the words Ave and SD being mentioned. What that means I suspect is that the Lod reigns supreme here as a measure of probability, for reasons which are faily obvious. Basically, primes are the log of themselves apart from the next door neighbour as a general rule, separated my the natural moat of at the preceding and following even number.
>
> So, you are searching for a function in which boring log plays a crucial part both as the area under observation, and the gaps which play in its region, with somww delightful interplay which occurs when there is in the vicinity a number which has a plenitude of small factors.
>
> So, dust off your Ribenboim at once, to quote Mr Durante again. or get onto Mr E Bay, who probably has a dozen going cheap from members dismissed from thid Forum!
>
>  In primenumbers@yahoogroups.com, "woodhodgson@" <rupert.weather@> wrote:
> >
> > Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
> >
> > I suspect that this sort of thing has already been well analysed, but am curious to know of any results.
> >
> 0 Attachment
Even my typos are nested, and letters elided. You have been known to remark on my odious propensity to pun, but the aforesaid Russell, who never bothered a Prime in his life, held the sword of Euler fermatly on his hands all the whiles.
 In primenumbers@yahoogroups.com, "John" <mistermac39@...> wrote:
>
> Pardon my typs, but mine eyes eyes are dimming, my ears are failing, and my hands are none too steady, and I am still too impetuous, a disadvantage in one's Dotty Age even more than in Uncouth Youth.
>
>  In primenumbers@yahoogroups.com, "John" <mistermac39@> wrote:
> >
> > Nice to catch up with you in these most unlikely surroundings after all these years separated by a moat 2000km wide. Evert few years I vaunt my ignorance profound on this forum, and spend the rest licking my wounds.
> > But, the lure of the call of the Jumble which is the Prime World brings me back.
> >
> > My answer to tour query will, as usual, be wrong in one or more aspects, but that fact will like a magnet arouse the finer feelings which lay deep and hidden in the heart of every Mathematician, with their beautiful minds, (and Russel is one of us) even if he emerges from trapdoors with the aim of disembowelling you on the spot, sword of Gauss in hand, and the shield of Euler, (in the immortal woids of Mr James Durante in an oilier time). Euwe always fascinated me, as besides only holding the Title very briefly, had a name which had to be heard torather than arrived ay by analysis.
> >
> > But, I digress.
> >
> > In all my visits here, I cannot recall the words Ave and SD being mentioned. What that means I suspect is that the Lod reigns supreme here as a measure of probability, for reasons which are faily obvious. Basically, primes are the log of themselves apart from the next door neighbour as a general rule, separated my the natural moat of at the preceding and following even number.
> >
> > So, you are searching for a function in which boring log plays a crucial part both as the area under observation, and the gaps which play in its region, with somww delightful interplay which occurs when there is in the vicinity a number which has a plenitude of small factors.
> >
> > So, dust off your Ribenboim at once, to quote Mr Durante again. or get onto Mr E Bay, who probably has a dozen going cheap from members dismissed from thid Forum!
> >
> >  In primenumbers@yahoogroups.com, "woodhodgson@" <rupert.weather@> wrote:
> > >
> > > Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
> > >
> > > I suspect that this sort of thing has already been well analysed, but am curious to know of any results.
> > >
> >
> 0 Attachment
Where I say, (typos fixed):
*In all my visits here, I cannot recall the words Average and Standard Deviation being mentioned. What that means I suspect is that the Log reigns supreme here as a measure of probability, for reasons which are fairly obvious. Primes are the log of themselves apart from the next door neighbour as a general rule, separated by the natural moat of at the preceding and following even number.
So, you are searching for a function in which boring log plays a crucial part, *both* as to the area under observation, *and* the gaps which play in its region, with some delightful interplay which occurs when there is, in the vicinity, a number which has a plenitude of small factors.*
That last phrase is probably wrong, but what I getting at is that after 30, you have only 31 as a prime until 37.
After 210, you have only 211 until 223.
I think they call 6,30,210,2310 etc primorials, after the style of factorials, which means you get after the prime which follows in the all cases I have looked at, you get a bonus run free of primes, as they are going to have factors contained in the primorial.
Statistics do not really play a part here, as the sparcity of primes has to do with arithmetical reasons rather than the vagaries which often plague primes. There is another trap for young players in all this too, If I can spell it, Aurefellian Factors, tricky beasts indeed, which are not algebraic in the "normal" interpretation of the term. They could be described as prime factors which arrive before "due date" in series like the Fibonacci. F19 is an example. F19 is not prime, and has factors of 37, and from memory, 113. Those two numbers are 19*21 and 19*61
There is I am sure are ways of contriving situations where you find sparse prime pickings.
There is a lot of material on the site here to swat up before you can speak the lingo. You being a math graduate have a marked advantage there. Years ago, we had a house fire, and all my Chess, Music, and Maths Textbooks were destroyed. Lucky if you work at Uni with access to them, and learned papers galore, and a mentor to lead you. We auto didacts are at a disadvantage, as our teachers know nothing. LOL
 In primenumbers@yahoogroups.com, "John" <mistermac39@...> wrote:
>
> Nice to catch up with you in these most unlikely surroundings after all these years separated by a moat 2000km wide. Evert few years I vaunt my ignorance profound on this forum, and spend the rest licking my wounds.
> But, the lure of the call of the Jumble which is the Prime World brings me back.
>
> My answer to tour query will, as usual, be wrong in one or more aspects, but that fact will like a magnet arouse the finer feelings which lay deep and hidden in the heart of every Mathematician, with their beautiful minds, (and Russel is one of us) even if he emerges from trapdoors with the aim of disembowelling you on the spot, sword of Gauss in hand, and the shield of Euler, (in the immortal woids of Mr James Durante in an oilier time). Euwe always fascinated me, as besides only holding the Title very briefly, had a name which had to be heard torather than arrived ay by analysis.
>
> But, I digress.
>
> In all my visits here, I cannot recall the words Ave and SD being mentioned. What that means I suspect is that the Lod reigns supreme here as a measure of probability, for reasons which are faily obvious. Basically, primes are the log of themselves apart from the next door neighbour as a general rule, separated my the natural moat of at the preceding and following even number.
>
> So, you are searching for a function in which boring log plays a crucial part both as the area under observation, and the gaps which play in its region, with somww delightful interplay which occurs when there is in the vicinity a number which has a plenitude of small factors.
>
> So, dust off your Ribenboim at once, to quote Mr Durante again. or get onto Mr E Bay, who probably has a dozen going cheap from members dismissed from thid Forum!
>
>  In primenumbers@yahoogroups.com, "woodhodgson@" <rupert.weather@> wrote:
> >
> > Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
> >
> > I suspect that this sort of thing has already been well analysed, but am curious to know of any results.
> >
> 0 Attachment
If you want to get up to speed auickly, here are a few names you here around the traps as the leading lights of maths. Gauss, Newton, and the "old" masters, I am sure you know, but more modern names are Riemann, Hardy, (the mentor of Ramanujan), and still highly respected, Carmicheal, Pomerance, and all the prime hunters here. You will have no trouble with your background of writing in Pari, and some other "languages", and I can foersee you being numbered among the big time Prime Crackers and Factorisers.
Some of the tests are Fermat's Little theorem, Lucas Type series, the use of n^21, and the properties that flow from have a cubic function equal a quadratic, I sem to remember ECM has something to do with this.
Weistein's World of Numbers is a valuable internet, and other resource.
It is a pity that text books cost the earth. I think Hardy is essential, and the site mention Riebenboim, whose name i hope I get right.
Best.
 In primenumbers@yahoogroups.com, "John" <mistermac39@...> wrote:
>
> Where I say, (typos fixed):
>
> *In all my visits here, I cannot recall the words Average and Standard Deviation being mentioned. What that means I suspect is that the Log reigns supreme here as a measure of probability, for reasons which are fairly obvious. Primes are the log of themselves apart from the next door neighbour as a general rule, separated by the natural moat of at the preceding and following even number.
>
> So, you are searching for a function in which boring log plays a crucial part, *both* as to the area under observation, *and* the gaps which play in its region, with some delightful interplay which occurs when there is, in the vicinity, a number which has a plenitude of small factors.*
>
> That last phrase is probably wrong, but what I getting at is that after 30, you have only 31 as a prime until 37.
>
> After 210, you have only 211 until 223.
>
> I think they call 6,30,210,2310 etc primorials, after the style of factorials, which means you get after the prime which follows in the all cases I have looked at, you get a bonus run free of primes, as they are going to have factors contained in the primorial.
>
> Statistics do not really play a part here, as the sparcity of primes has to do with arithmetical reasons rather than the vagaries which often plague primes. There is another trap for young players in all this too, If I can spell it, Aurefellian Factors, tricky beasts indeed, which are not algebraic in the "normal" interpretation of the term. They could be described as prime factors which arrive before "due date" in series like the Fibonacci. F19 is an example. F19 is not prime, and has factors of 37, and from memory, 113. Those two numbers are 19*21 and 19*61
>
> There is I am sure are ways of contriving situations where you find sparse prime pickings.
>
> There is a lot of material on the site here to swat up before you can speak the lingo. You being a math graduate have a marked advantage there. Years ago, we had a house fire, and all my Chess, Music, and Maths Textbooks were destroyed. Lucky if you work at Uni with access to them, and learned papers galore, and a mentor to lead you. We auto didacts are at a disadvantage, as our teachers know nothing. LOL
>
>
>  In primenumbers@yahoogroups.com, "John" <mistermac39@> wrote:
> >
> > Nice to catch up with you in these most unlikely surroundings after all these years separated by a moat 2000km wide. Evert few years I vaunt my ignorance profound on this forum, and spend the rest licking my wounds.
> > But, the lure of the call of the Jumble which is the Prime World brings me back.
> >
> > My answer to tour query will, as usual, be wrong in one or more aspects, but that fact will like a magnet arouse the finer feelings which lay deep and hidden in the heart of every Mathematician, with their beautiful minds, (and Russel is one of us) even if he emerges from trapdoors with the aim of disembowelling you on the spot, sword of Gauss in hand, and the shield of Euler, (in the immortal woids of Mr James Durante in an oilier time). Euwe always fascinated me, as besides only holding the Title very briefly, had a name which had to be heard torather than arrived ay by analysis.
> >
> > But, I digress.
> >
> > In all my visits here, I cannot recall the words Ave and SD being mentioned. What that means I suspect is that the Lod reigns supreme here as a measure of probability, for reasons which are faily obvious. Basically, primes are the log of themselves apart from the next door neighbour as a general rule, separated my the natural moat of at the preceding and following even number.
> >
> > So, you are searching for a function in which boring log plays a crucial part both as the area under observation, and the gaps which play in its region, with somww delightful interplay which occurs when there is in the vicinity a number which has a plenitude of small factors.
> >
> > So, dust off your Ribenboim at once, to quote Mr Durante again. or get onto Mr E Bay, who probably has a dozen going cheap from members dismissed from thid Forum!
> >
> >  In primenumbers@yahoogroups.com, "woodhodgson@" <rupert.weather@> wrote:
> > >
> > > Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
> > >
> > > I suspect that this sort of thing has already been well analysed, but am curious to know of any results.
> > >
> >
> 0 Attachment
It is indeed good to catch up. My question is serious, even though I have never seen these statistical terms in use in number theory  but then I have only seen a minute portion of anything ever written or posted on the subject of primes in general.
I can recall first being interested in prime distributions in a general way as a schoolkid  one of the "crammer" types of textbook had a onepage table with factorisations to 1000, and the prime numbers in bold. Apart from the "quads", the thing that struck me most was the lumpiness of the 800s, contrasting with the smoothness of the 700s and 900s.
 In primenumbers@yahoogroups.com, "John" <mistermac39@...> wrote:
>
> Nice to catch up with you in these most unlikely surroundings after all these years separated by a moat 2000km wide. Evert few years I vaunt my ignorance profound on this forum, and spend the rest licking my wounds.
> But, the lure of the call of the Jumble which is the Prime World brings me back.
>
> My answer to tour query will, as usual, be wrong in one or more aspects, but that fact will like a magnet arouse the finer feelings which lay deep and hidden in the heart of every Mathematician, with their beautiful minds, (and Russel is one of us) even if he emerges from trapdoors with the aim of disembowelling you on the spot, sword of Gauss in hand, and the shield of Euler, (in the immortal woids of Mr James Durante in an oilier time). Euwe always fascinated me, as besides only holding the Title very briefly, had a name which had to be heard torather than arrived ay by analysis.
>
> But, I digress.
>
> In all my visits here, I cannot recall the words Ave and SD being mentioned. What that means I suspect is that the Lod reigns supreme here as a measure of probability, for reasons which are faily obvious. Basically, primes are the log of themselves apart from the next door neighbour as a general rule, separated my the natural moat of at the preceding and following even number.
>
> So, you are searching for a function in which boring log plays a crucial part both as the area under observation, and the gaps which play in its region, with somww delightful interplay which occurs when there is in the vicinity a number which has a plenitude of small factors.
>
> So, dust off your Ribenboim at once, to quote Mr Durante again. or get onto Mr E Bay, who probably has a dozen going cheap from members dismissed from thid Forum!
>
>  In primenumbers@yahoogroups.com, "woodhodgson@" <rupert.weather@> wrote:
> >
> > Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
> >
> > I suspect that this sort of thing has already been well analysed, but am curious to know of any results.
> >
> 0 Attachment
Because the heavyweight contributors to this forum encounter a lot of cranks on a site like this, and I admit to being a minor one, they are understandably reluctant to be diverted from their *Prime* aim.
It will not take you long to realise who the real experts are here, and since they know all the theory they only discuss as much as pertains to the subject in hand as a general rule.
I always read these tidbits, as they help to focus on the relationships, and hierarchies of importance of the elements of theory.
So, your query has not aroused a reply, as they clearly so not regard normal Statistics as very germane to prime numbers.
Pardon me then if some of what I say is trivially obvious, but I think I know the path that everyone who is interested in Prime Numbers as a fascinating subject in itself, like a keen Chess Amateur, like to sit at the feet of Masters and hear them expound.
So, I suggest that you start thinking the elements through, in layman terms first, and then learn the *Lingo* required to intelligently converse here, something I *know* you will be able to do once you set your mind to it.
Prime Numbers can be categorised in basic terms using the numbers 4 and 6.
Apart from 2, all Primes are expressible as 4n+1 or 4n1, two only categories. The theory says that over the long haul this makes two sets only which are equally populated.
An important fact is that the 4n+1 prime class can be expressed as the sum of two squares, whereas 4n1 cannot.
Apart from 2 and 3, all Primes are expressible as 6n+1 or 6n1, two only categories. The theory says that over the long haul this makes two sets only which are equally populated also. All twin primes apart from (3,5) are of form (6n1,6n+1).
Bone up on some very important ideas and symbols. Modular arithmetic, quadratic residues/nonresidues, quadratic reciprocity, Jacobi symbol, Legendre Symbol, smoothness, Euler's totient function (phi) and the tables of same, his theorem, their properties and manipulation. Also invedtigate all you can about a numerous number of primality tests, and the many methods of factorisation.
There is also the famous work done by Riemann that will prove a mine of fascination.
That should start you off. Go to the Home Page of the site for explanations of a all this elementary material.
 In primenumbers@yahoogroups.com, "woodhodgson@..." <rupert.weather@...> wrote:
>
> It is indeed good to catch up. My question is serious, even though I have never seen these statistical terms in use in number theory  but then I have only seen a minute portion of anything ever written or posted on the subject of primes in general.
>
> I can recall first being interested in prime distributions in a general way as a schoolkid  one of the "crammer" types of textbook had a onepage table with factorisations to 1000, and the prime numbers in bold. Apart from the "quads", the thing that struck me most was the lumpiness of the 800s, contrasting with the smoothness of the 700s and 900s.
>
>  In primenumbers@yahoogroups.com, "John" <mistermac39@> wrote:
> >
> > Nice to catch up with you in these most unlikely surroundings after all these years separated by a moat 2000km wide. Evert few years I vaunt my ignorance profound on this forum, and spend the rest licking my wounds.
> > But, the lure of the call of the Jumble which is the Prime World brings me back.
> >
> > My answer to tour query will, as usual, be wrong in one or more aspects, but that fact will like a magnet arouse the finer feelings which lay deep and hidden in the heart of every Mathematician, with their beautiful minds, (and Russel is one of us) even if he emerges from trapdoors with the aim of disembowelling you on the spot, sword of Gauss in hand, and the shield of Euler, (in the immortal woids of Mr James Durante in an oilier time). Euwe always fascinated me, as besides only holding the Title very briefly, had a name which had to be heard torather than arrived ay by analysis.
> >
> > But, I digress.
> >
> > In all my visits here, I cannot recall the words Ave and SD being mentioned. What that means I suspect is that the Lod reigns supreme here as a measure of probability, for reasons which are faily obvious. Basically, primes are the log of themselves apart from the next door neighbour as a general rule, separated my the natural moat of at the preceding and following even number.
> >
> > So, you are searching for a function in which boring log plays a crucial part both as the area under observation, and the gaps which play in its region, with somww delightful interplay which occurs when there is in the vicinity a number which has a plenitude of small factors.
> >
> > So, dust off your Ribenboim at once, to quote Mr Durante again. or get onto Mr E Bay, who probably has a dozen going cheap from members dismissed from thid Forum!
> >
> >  In primenumbers@yahoogroups.com, "woodhodgson@" <rupert.weather@> wrote:
> > >
> > > Suppose one considers a positive integer N, and looks at the integers in a "local" interval centred on N, say [0.9N, 1.1N] (meaning the ones bounded by these values)  the "width" of 0.2N is just my arbitrary selection and may not be very suitable. Obtain the set of all differences between consecutive primes in the interval, and compute the mean and standard deviation of the selection. The "coefficient of variation" thus obtained (standard deviation/mean) could be conisdered as a measure of the "local" variability of the prime distribution, for the given "width". How might this behave for varying N?
> > >
> > > I suspect that this sort of thing has already been well analysed, but am curious to know of any results.
> > >
> >
> 0 Attachment
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