An iteration produces all the prime numbers

02132021, 06:05 PM
Post: #1




An iteration produces all the prime numbers
This very much appeals to me: initially it's rather a surprise.
Take the number 2.920050977316134712092562917112019... and perform the following iteration: x := IP(x)*FP(x)+IP(x) (Where IP is the integer part and FP the fractional part.) Note the integer parts of the resulting sequence! Ref: this paper or this video or this wikipedia section or this OEIS page. 

02132021, 09:38 PM
(This post was last modified: 02132021 09:39 PM by Valentin Albillo.)
Post: #2




RE: An iteration produces all the prime numbers
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Hi, EdS2: (02132021 06:05 PM)EdS2 Wrote: This very much appeals to me: initially it's rather a surprise. There are infinitely many such constants that produce primes (this one, Mills' constant, etc) but though they're somewhat interesting from a 'recreational maths' point of view, they're useless as primeproducing constants because you need to know in advance the prime sequence to compute them, which makes the subject rather circular: You can use the constant to produce primes but you need the primes to produce the constant. It would be quite another matter if any such constant could be computed to arbitrary precision some other way without involving the primes, say as a limit, an infinite summatory, an integral ... Alas, that's never the case so far and it's highly dubious it will ever be. Regards. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

02142021, 12:38 PM
Post: #3




RE: An iteration produces all the prime numbers
Indeed, so, all true. I still liked what I saw!
Thanks for Mills' constant (1.3063778838630806904686144926...)  that's perhaps a little more accessible, as we only need an expression, not an iteration. (Then again, it produces only primes, less impressive to me than producing all primes, although that is of course a matter of taste.) 

02142021, 12:49 PM
Post: #4




RE: An iteration produces all the prime numbers
Hello all,
is there a relation between the precision of that constant and which greatest prime you can produce with it. That would be fancy: Let's say you know the constant with n decimal precision and the highest prime you can produce is say p(n). And with that prime you can improve this constant to a precision n+1 (or more) and with that you can produce the greatest prime let it call P(n+1) which is the successor of p(n), or the next two (or more) successors of p(n)... Why I have to think about Münchhausen's lift oneself up by his own bootstraps? 

02142021, 03:05 PM
Post: #5




RE: An iteration produces all the prime numbers
For fairly simple methods of quickly producing primes (faster than the sieve of Eratosthenes), check out "wheels." The idea is to start with 2 (or seed with 2,3,5,7...) and as one goes, one dynamically changes the sieving process to pickup more primes. First one gets 6*K+(1,5) then 30*K+(1,7,11,13,17,19,23,29) and so forth. I like the 30*k method as I can store 30 blocks of 30 numbers in a single byte. Later, one needs fancier methods. It becomes more efficient to store the distances between primes. Then even store that distance using a Universal Code.
https://en.wikipedia.org/wiki/Wheel_factorization https://en.wikipedia.org/wiki/Universal_...mpression) 

02142021, 03:38 PM
Post: #6




RE: An iteration produces all the prime numbers
(02142021 03:05 PM)ttw Wrote: First one gets 6*K+(1,5) then 30*K+(1,7,11,13,17,19,23,29) and so forth. I like the 30*k method as I can store 30 blocks of 30 numbers in a single byte. Later, one needs fancier methods. It becomes more efficient to store the distances between primes. Then even store that distance using a Universal Code. Storing primes using wheels is nice, but data compression ratio is still a constant wheel(2,3), compression ratio = 1 / (1  1/2  1/3 + 1/(2*3)) = 3 wheel(2,3,5), compression ratio = 1 / (1  1/2  1/3  1/5 + 1/(2*3) + 1/(2*5) + 1/(3*5)  1/(2*3*5)) = 3.75 ... Assuming we do not need random access, but simply O(1) way to get next prime (like OP formula). Is there a way to do better ? Does storing prime gaps give better compression ratio ? 

02152021, 01:27 AM
(This post was last modified: 02152021 01:29 AM by ttw.)
Post: #7




RE: An iteration produces all the prime numbers
One doesn't (for really big stuff) use the same wheel. When possible, one switches to a bigger wheel. Most of those I've seen use 30030 to store 5760 for an efficiency of 192/1001. Of course, the last time I did this seriously, I had a 100 megaword Cray YMP for computation (I didn't use the SSD) for 100,000,000 words *64 bits or 6,400,000,000 bits. Using 8 bits for 30 numbers (easy coding) gave me N=24,000,000,000.
The density of primes is about ln(N)/N which means the spacing is ln(n) on the average. This clearly increases as N gets large. For N=6,400,000,000 the average spacing is /Ln( 6,400,000,000 or about 22. The 30wheel has uses 30 numbers for 8 prime so the spacing is 15/4 or 3.75. A list of spacings is better. To make things more complicated, Yitang Zhang proved that there are infinitely many prime pairs with gap less than 70,000,000, the first result related to prime pairs. Of course,70,000,000 is bigger than 2. (The latest number gap which occurs infinitely often is 246.) https://online.kitp.ucsb.edu/online/coll...m_KITP.pdf 

02152021, 01:29 AM
(This post was last modified: 02152021 01:41 AM by Valentin Albillo.)
Post: #8




RE: An iteration produces all the prime numbers
(02142021 12:38 PM)EdS2 Wrote: Indeed, so, all true. I still liked what I saw! Mills' constant (1.306377883863...) produces primes which grow extremely fast as the article you linked states (2, 11, 1361, 2521008887, 16022236204009818131831320183, ~O(1e84)), each term has about 3x the digits of the previous term, so using it you can only produce and certify the primality of a very few before you are forced to check their primality using a probabilistic method for a little while and afterwards even that won't be feasible. However, there's an infinity of primegenerating Millslike procedures but with much reduced growing rates so they can be used to generate proven primes by the hundreds. On the other hand, there are some iterative procedures (many trivial, but not all) that do not depend on the accuracy of an irrational (transcendental ?) constant to generate an indefinite number of fullycertified primes. Primes are always a source of awe. One of the many many things that awed me is that you can generate the sequence of primes using the sequence of zeros of the Riemann's Zeta function, and you can in turn generate the zeros of the function using the primes. Perfectly symmetrical, one sequence encodes the other. By the way, the constant mentioned in your linked article can be generated to the full 13 digits given there by running this trivial 2line HP71B program, which produces the value in no time: 1 DESTROY ALL @ P=2 @ M=P @ S=1 2 FOR I=1 TO 12 @ P=FPRIM(P+1) @ S=S+(P1)/M @ M=M*P @ NEXT I @ DISP " 2";STR$(S) >RUN 2.920050977316 Regards. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

02152021, 05:02 PM
Post: #9




RE: An iteration produces all the prime numbers
What is FPRIM?


02152021, 05:34 PM
(This post was last modified: 02152021 05:37 PM by Valentin Albillo.)
Post: #10




RE: An iteration produces all the prime numbers
(02152021 05:02 PM)KeithB Wrote: What is FPRIM? FPRIM is a function which returns the next prime (either forward of backwards) from the given argument. For instance: FPRIM(6)= 7, FPRIM(4,1)=3, FPRIM(5)=5 It can be found in the JPC ROM. As a matter of course, i always keep the HP71B fitted with at least 150 Kb RAM, the MATH ROM, the JPC ROM, the HPIL ROM and the STRINGLX LEX file. Anything less is a maimed HP71B as far as i'm concerned. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

02162021, 03:34 PM
Post: #11




RE: An iteration produces all the prime numbers
13 digits  that's odd  I see the HP71B described as having 12 digits. Can you explain please, Valentin?
(As a related question, I wonder if there's any extended precision library for the 71B or any other HP calculator, to allow us to work with more digits?) 

02172021, 01:00 AM
Post: #12




RE: An iteration produces all the prime numbers
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Hi, EdS2: (02162021 03:34 PM)EdS2 Wrote: 13 digits  that's odd Correct. 13 is indeed an odd integer. Quote:I see the HP71B described as having 12 digits. Can you explain please, Valentin? Yes. The value featured in your linked article is 2 followed by 12 decimals, i.e., 13 digits in all. To get the full 12 decimals in a 12digit calc the integer part (2) gets in the way so my program simply gets rid of it by initializing the sum to 1, thus no integer part is ever involved and the 12digit HP71B can merrily compute the 12digit decimal part in its full glory (.920050977316), then it's a matter of just adding the 2 in front when printing and there you are, 2.920050977316, 13 digits. Quote:I wonder if there's any extended precision library for the 71B or any other HP calculator, to allow us to work with more digits? Well, computing multiprecision results with the HP71B is not that difficult at all once you've done it a few times, just for instance have a look at these PDf articles I wrote: HP71B Producing Digits of Pi one at a time HP71B Multiprecision E and its roots which can compute thousands of digits of Pi and e (and its roots, square, etc), respectively. As for a multiprecision "library", have a look at the one I wrote for this challenge o'mine: HP Challenge VA021  Short Sweet Math Challenge 20 April 1st Spring Special and in page 74 you'll find the following (abridged here): Appendix A: The multiprecision library I'll briefly discuss here the small(just 32 lines of code) "Adhoc multiprecision library" I've implemented specifically for this particular subchallenge (the following listing is numbered as if to be included in the same file as the main program without line numbers conflicting, but could reside in a separate program file in RAM as well, if intended to be used by other programs). SUB MMUL(A(),B(),C()) { Multiprecision MULtiplication } SUB MPOW(A(),N,C()) { Multiprecision POWer } SUB MSUM(A(),B(),C()) { Multiprecision SUM } SUB MSBI(A(),B()) { Multiprecision SuBtraction Inplace } SUB MMODK(A(),K,M) { Multiprecision MODulus singleprecision } SUB MDIVK(A(),K) { Multiprecision DIV (and modulus) singleprecision } SUB MNOR(A()) { Multiprecision NORmalization } SUB MTOP(A(),U) { Multiprecision TOP } SUB M2N(A(),N) { Multiprecision to realprecision Numeric variable } SUB N2M(N,A()) { Numeric Realprecision variable to Multiprecision } SUB MMOD(A(),B(),C()) { Multiprecision MODulus } SUB MFAC(N,A()) { Multiprecision FACtorial } Sample use: 69! MOD 13^{68} = 161707119156747337147933149666645422128978599226831537632782504385470771962 However, do not have any high hopes, it was written adhoc for the challenge so it's not general purpose by any means, though it might be useful to you as an example on how to proceed (read the Caveats). Regards. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

02172021, 02:14 AM
Post: #13




RE: An iteration produces all the prime numbers
(02152021 05:34 PM)Valentin Albillo Wrote: <snip> The HPIL ROM in your list, is that the ROM in the HPIL peripheral, or some other (development?) ROM? I've seen the BIN files HPILROMA.BIN, HPILROMB.BIN, and HPILROMH4.BIN with the first two being ROMs in the HPIL peripheral. The H4 bin file seems to be a version of the first two? I know STRINGLX functions can be found in JF's ULIB52 ROM. Just asking. I'm trying to put together a final list of available ROMs for the 71. Remember kids, "In a democracy, you get the government you deserve." 

02172021, 04:16 AM
Post: #14




RE: An iteration produces all the prime numbers
(02172021 02:14 AM)mfleming Wrote: The HPIL ROM in your list, is that the ROM in the HPIL peripheral, or some other (development?) ROM? This one: HPIL adaptor 82401A, i.e.: the plainvanilla one. Quote: I know STRINGLX functions can be found in JF's ULIB52 ROM. Don't know, I'm using the ~838byte individual LEX file, don't need anything else. You know, having many LEX files and ROMs with many keywords tends to slow down the HP71B, so having the MATH, JPC and HPIL ROMs is enough for me, I don't need more ROMs or LEX files. Quote:Just asking. I'm trying to put together a final list of available ROMs for the 71. Good luck with that. I suppose you've consulted Sylvain's HP71B Compendium and Joe Horn's and JF's sites. Regards. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

02172021, 07:10 AM
Post: #15




RE: An iteration produces all the prime numbers
(02172021 04:16 AM)Valentin Albillo Wrote: Good luck with that. I suppose you've consulted Sylvain's HP71B Compendium and Joe Horn's and JF's sites. And MoHPC flash and HHC flash and PPC flash and ... anything else? Thanks Valentin, ~Mark Remember kids, "In a democracy, you get the government you deserve." 

02172021, 08:09 AM
Post: #16




RE: An iteration produces all the prime numbers
(02172021 07:10 AM)mfleming Wrote:(02172021 04:16 AM)Valentin Albillo Wrote: Good luck with that. I suppose you've consulted Sylvain's HP71B Compendium and Joe Horn's and JF's sites.And MoHPC flash and HHC flash and PPC flash and ... anything else? and Matthias' module database. JF 

02172021, 10:00 AM
Post: #17




RE: An iteration produces all the prime numbers
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(02172021 01:00 AM)Valentin Albillo Wrote: oddOne of my favourite jokes... Quote:Thanks for explaining!Quote:I see the HP71B described as having 12 digits. Can you explain please, Valentin? Quote:Excellent  thanks again.Quote:I wonder if there's any extended precision library for the 71B or any other HP calculator, to allow us to work with more digits? 

02172021, 09:57 PM
Post: #18




RE: An iteration produces all the prime numbers
Hi, EdS2: (02172021 10:00 AM)EdS2 Wrote: Thanks for explaining! You're welcome. Quote:Excellent  thanks again. You're welcome again. Re the HP71B 2liner I posted above, which instantly computes the constant as given in your linked article (13 decimal places, 2.920050977316), it's quite possible to convert it to a multiprecision program which would compute the constant to the digits shown in this Wikipedia article: Formula for primes where the constant is given as 2.920050977316134712092562917112019. However, for variety's sake let's use instead SwissMicros awesome DM42 calc, which can run HP42S RPN programs and supports 34digit precision instead of just 12digit as both the HP42S and the HP71B do. Thus, I wrote the following 50step, 86byte RPN program for the DM42 to perform the feat: LBL "CCN" 1 5 X>Y? SIZE 30  STO 00 GTO 01 1 RCL/ 02 LBL 01 RCL 04 STO 03 RCL+ 03 RCL IND 00 STO IND 01 6 ENTER RCL 04 ISG 01 STO 01 X<> 03 RCL/ ST Y LBL 03 2 X=Y? FP GTO 02 STO 02 GTO 04 X=0? LBL 04 3 RCL ST Z GTO 00 2 STO 04 STOx 02 Rv + STO 05 LBL 00 LASTX END LBL 02 2 X>Y? RCL ST X STO+ 04 ISG 00 Let's execute it: XEQ "CCN" > 2.92005097732 [SHOW] > 2.920050977316134712092562917112019 which is computed instantly and exactly matches the Wikipedia's value. As the DM42 doesn't have any numbertheoretic functions in its instruction set (like PRIM, FPRIM, GCD, etc.), my program above generates and uses the prime numbers on the fly. 101 is the last prime needed. BTW, the program is a quick job, also written on the fly, so it's not optimized to any extent as it already runs instantly and takes little program memory. It can be optimized by improving stack use but I see little need, be my guest if you want to try. Regards. V. All My Articles & other Materials here: Valentin Albillo's HP Collection 

02182021, 08:42 AM
Post: #19




RE: An iteration produces all the prime numbers
very nice! I don't have a DM42, but I do have Free42 on my phone...


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