Request for "Decimal Period of 1/X in Base Y" program

02272015, 06:13 AM
(This post was last modified: 03022015 07:06 AM by Joe Horn.)
Post: #1




Request for "Decimal Period of 1/X in Base Y" program
EDIT: This discussion's original title was:
Request for "Multiplicative Order of Y (mod X)" program ... but my actual goal is the "decimal period of 1/X in base Y", which I wrongly thought was the same thing. Hence the confusion as the discussion progresses below... Has anybody already written a Prime program to calculate the Number Theory value of "the multiplicative order of Y (mod X)"? One application of this concept is found in repeating decimal numbers. For example, 1/13 = 0.076923076923... where "076923" repeats forever. Notice that the repeating section of 1/13 is 6 digits long. In Number Theory, they express this as "the multiplicative order of 10 (mod 13) = 6" with the 10 coming from the fact that decimal numbers (as the name implies) use base 10. "Multiplicative order" is often shortened to just "order". Another example: In hexadecimal, 1/Dh (that's 1/13 in decimal) = 0.13B13B13B...h, where the three hex digits "13B" repeat forever. So the order of 16 (mod 13) = 3. In Mathematica, it's implemented as MultiplicativeOrder[y,x]. Thanks in advance to anybody who has already programmed their Prime to find the order of Y (mod X). It shouldn't be too hard, since Prime has EULER, IDIVIS, and MODPOW built in. <0ɸ0> Joe 

02272015, 08:49 AM
Post: #2




RE: Request for "Multiplicative Order of Y (mod X)" program
Here is a simple nonCAS variant:
Code: EXPORT MultiplicativeOrder(a,base) 

02272015, 09:35 AM
Post: #3




RE: Request for "Multiplicative Order of Y (mod X)" program
(02272015 06:13 AM)Joe Horn Wrote: Has anybody already written a Prime program to calculate the Number Theory value of "the multiplicative order of Y (mod X)"? Are you only interested in integer values of X & Y? 

02272015, 09:57 AM
(This post was last modified: 02272015 10:21 AM by Gerald H.)
Post: #4




RE: Request for "Multiplicative Order of Y (mod X)" program
IF you're only interested in integer values for X & Y, this 50G prog does the job:
Input: 2: Modulus 1: Number whose orderis to be calculated Returns the order you requested. :: CK2&Dispatch # FFFF :: 2DUP FPTR2 ^ZGCDext FPTR2 ^ZIsOne? NOTcase2drop Z0_ OVER FPTR2 ^QMod DUPUNROT OVER FPTR2 ^EULER FPTR2 ^ZFACTO ROTDROP DUPLENCOMP #2/ ZERO_DO DUPUNROT INDEX@ #2* #1+DUP #1+ SUBCOMP INCOMPDROP OVERUNROT COERCE FPTR2 ^PPow# ROTSWAP FPTR2 ^ZQUOText 4ROLLOVER 6PICK FPTR2 ^ModPow 5PICK FPTR2 ^QMod BEGIN FPTR2 ^DupZIsOne? NOT_WHILE :: 3PICK 6PICK FPTR2 ^ModPow 5PICK FPTR2 ^QMod SWAP3PICK FPTR2 ^QMul SWAP ; REPEAT DROPSWAPDROP 4PICKSWAP ROT LOOP DROP 4UNROLL3DROP ; ; I'm sure you Prime buffs won't have much trouble translating that into Prime. 

02272015, 01:55 PM
Post: #5




RE: Request for "Multiplicative Order of Y (mod X)" program
(02272015 09:57 AM)Gerald H Wrote: I'm sure you Prime buffs won't have much trouble translating that into Prime. LOL Bob Prosperi 

02272015, 04:01 PM
Post: #6




RE: Request for "Multiplicative Order of Y (mod X)" program
(02272015 08:49 AM)Thomas Ritschel Wrote: Here is a simple nonCAS variant: Thanks, Thomas! Now I gotta turn your program into an exactinteger CAS program. (02272015 09:57 AM)Gerald H Wrote: IF you're only interested in integer values for X & Y, this 50G prog does the job: ... Here's the URPL program I currently use on the 50g: Code: << Input: Y, X (exact mode, of course) Output: order of Y (mod X) <0ɸ0> Joe 

02272015, 04:34 PM
(This post was last modified: 02272015 04:38 PM by Gerald H.)
Post: #7




RE: Request for "Multiplicative Order of Y (mod X)" program
(02272015 04:01 PM)Joe Horn Wrote:(02272015 08:49 AM)Thomas Ritschel Wrote: Here is a simple nonCAS variant: Nice, but the POWMOD is a very heavy calculation for, eg 777 mod 541*10007 takes about 5 sec with your prog. 

02282015, 07:30 AM
Post: #8




RE: Request for "Multiplicative Order of Y (mod X)" program
(02272015 04:01 PM)Joe Horn Wrote:(02272015 08:49 AM)Thomas Ritschel Wrote: Here is a simple nonCAS variant: & thanks for publishing your programme, it made me question the structure of mine. I've now incorporated removing the common factors at the start instead of the crazy return zero. Lord knows why I did that! 

02282015, 09:39 AM
Post: #9




RE: Request for "Multiplicative Order of Y (mod X)" program
(02282015 07:30 AM)Gerald H Wrote:(02272015 04:01 PM)Joe Horn Wrote: Thanks, Thomas! Now I gotta turn your program into an exactinteger CAS program. Dear Gerald, I write to you to encourage you in your endeavours & to correct you. Your programme is correct & Joe Horn's erroneous. Try the order of 777 modulo 3600, then raise 777 to that power mod 3600  is the answer one? You just have too much respect for that nogood Joe Horn & too little confidence in yourself. & remember Robert Record's words on Ptolemy. I know you will come to the right conclusion, For I am A Gerald From The Future. 

02282015, 01:14 PM
Post: #10




RE: Request for "Multiplicative Order of Y (mod X)" program
(02282015 09:39 AM)Gerald H Wrote: Your programme is correct & Joe Horn's erroneous. Mathematica's MultiplicativeOrder(777,3600) does not return an answer, because 777 and 3600 are not coprime. But MultiplicativeOrder(777/3,3600) returns 60... same as my program. I'm just guessing, but perhaps this means that the GCD should be removed from the first input rather than the second, like this: Code: << <0ɸ0> Joe 

02282015, 01:56 PM
Post: #11




RE: Request for "Multiplicative Order of Y (mod X)" program
(02282015 01:14 PM)Joe Horn Wrote:(02282015 09:39 AM)Gerald H Wrote: Your programme is correct & Joe Horn's erroneous. No, the meaning is that if GCD of the two numbers is greater than one then NO power of that number is equal to one modulo the 2nd number. 

03012015, 01:21 AM
Post: #12




RE: Request for "Multiplicative Order of Y (mod X)" program
(02282015 01:56 PM)Gerald H Wrote: No, the meaning is that if GCD of the two numbers is greater than one then NO power of that number is equal to one modulo the 2nd number. Understood. But what I'm looking for is "the period of 1/X in base Y". It's only slowly dawning on me that this is NOT the same as "the order of Y mod X". Wolfram Mathworld says, "The number of digits in the repeating portion of the decimal expansion of a rational number can also be found directly from the multiplicative order of its denominator." But that assumes that the multiplicative order of the denominator exists, which apparently is not true, even when it *is* a repeating decimal. Example: 1/14 is a repeating decimal with a period of 6, and that's why my program returns 6 for an input of 10 & 14. But Mathematica's MultiplicativeOrder(10,14) refuses to return an answer for the reason you cited (10 and 14 are not coprime), just like Thomas' Prime program (which runs forever until halted by the user). So, how would you suggest calculating the period of the repeating decimal of 1/x where x is not coprime with 10? Please accept my apology if it seems like I'm changing the subject, which I might be doing inadvertently! When I started this thread, I had thought that "period of 1/x base y" was the same as "order of y mod x" once the GCD of x and y was removed from x (which is why my program starts by removing the GCD). That's certainly true when y=10. Can it be false for y<>10? Thanks in advance for teaching this student who is always eager to learn. <0ɸ0> Joe 

03012015, 06:28 AM
(This post was last modified: 03012015 07:29 AM by Gerald H.)
Post: #13




RE: Request for "Multiplicative Order of Y (mod X)" program
(03012015 01:21 AM)Joe Horn Wrote:(02282015 01:56 PM)Gerald H Wrote: No, the meaning is that if GCD of the two numbers is greater than one then NO power of that number is equal to one modulo the 2nd number. For that you should read this: http://mathworld.wolfram.com/DecimalExpansion.html I have edited out my misleading comment. 

03012015, 08:18 AM
Post: #14




RE: Request for "Multiplicative Order of Y (mod X)" program
ie Write fraction in lowest terms, a/b, remove factors of 2 & 5 from b, call this c, then calculate order of 10 modulo c  that, I hope, does the trick, ie returns number of repeating digits.


03012015, 09:04 AM
(This post was last modified: 03012015 09:17 AM by Gerald H.)
Post: #15




RE: Request for "Multiplicative Order of Y (mod X)" program
& here the prog to do it:
:: CK2&Dispatch # FFFF :: FPTR2 ^ZTrialDiv2 DROP BEGIN DUP Z5_ FPTR2 ^ZMod FPTR2 ^QIsZero? WHILE :: Z5_ FPTR2 ^ZQUOText ; REPEAT FPTR2 ^DupZIsOne? case2drop Z0_ DUPUNROT FPTR2 ^ZGCDext FPTR2 ^ZQUOText Z10_ xORD (name for my prog above) ; ; Again, I leave Prime translation to the experts. 

03012015, 09:48 AM
Post: #16




RE: Request for "Multiplicative Order of Y (mod X)" program
But this prog is better:
:: CK2&Dispatch # FFFF :: FPTR2 ^ZTrialDiv2 DROP BEGIN DUP Z5_ FPTR2 ^ZMod FPTR2 ^QIsZero? WHILE :: Z5_ FPTR2 ^ZQUOText ; REPEAT DUPUNROT FPTR2 ^ZGCDext FPTR2 ^ZQUOText FPTR2 ^DupZIsOne? casedrop Z0_ Z10_ xORD ; ; 

03012015, 05:40 PM
(This post was last modified: 03012015 05:43 PM by Joe Horn.)
Post: #17




RE: Request for "Multiplicative Order of Y (mod X)" program
(02272015 09:57 AM)Gerald H Wrote: IF you're only interested in integer values for X & Y, this 50G prog does the job: ... The 9th line of your program won't compile using the builtin tools. What is Z0_? EDIT: Never mind, I figured out it means ZINT 0. What assembler is that for? <0ɸ0> Joe 

03012015, 06:17 PM
Post: #18




RE: Request for "Multiplicative Order of Y (mod X)" program
(03012015 09:48 AM)Gerald H Wrote: But this prog is better: ... Uh oh. Running it after 12 ENTER 23 yields 22, but the correct answer for the period of 1/23 in base 12 (as returned by my URPL program above) is 11, as can be seen by actually writing 1/23 in base 12: 0.06316948421 06316948421 06316948421 06316948421 ... Am I misunderstanding something here? It happens. Thanks in advance for your patience. Disclaimer: My program is correct not based on any personal authority which I neither have nor have ever claimed to have, but it SEEMS TO ME to be correct based on empirical data. I only mention this due to some creepy and cryptic references recently made here and in the General Software Library about the danger of giving credence to anybody based on authority. <0ɸ0> Joe 

03012015, 07:02 PM
Post: #19




RE: Request for "Multiplicative Order of Y (mod X)" program
(03012015 06:17 PM)Joe Horn Wrote:(03012015 09:48 AM)Gerald H Wrote: But this prog is better: ... Don't understand why you're doing the calculation in base 12  my prog is only good for base 10 representation. Entering 12 & 23 returns length of period of 12/23 in base 10. Nothing "creepy or cryptic"  I just have too much respect for you. You have produced excellent progs for years & I'm very impressed & more readily impute error to myself than you. I'm not the only one affected by this respect for the greats, Robert Record's words are a lighthouse for us all. 

03012015, 09:55 PM
Post: #20




RE: Request for "Multiplicative Order of Y (mod X)" program
(03012015 07:02 PM)Gerald H Wrote: Don't understand why you're doing the calculation in base 12  my prog is only good for base 10 representation. Entering 12 & 23 returns length of period of 12/23 in base 10. As I mentioned in the original posting, I'm looking for the period of the repeating "decimal" expansion of 1/Y in base X. The example I gave there was 1/13 in hex, so Y=13, X=16 (1/Dh = 0.13B13B13B...h) which has a period of 3. I'm looking for a program that handles any base, not just decimal and hex. I think that my URPL program does this, but it's much slower than your ORD program, so I greatly appreciate all your insights here. (03012015 07:02 PM)Gerald H Wrote: Nothing "creepy or cryptic"  I just have too much respect for you. You have produced excellent progs for years & I'm very impressed & more readily impute error to myself than you. We're all in the same boat. Putting anybody on too high a pedestal can capsize the boat. <0ɸ0> Joe 

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