Dear Math Fun, I would normally send this to the sequence-fans mailing list but that list appears to be defunct. But hopefully there is enough fun here to make it appropriate for this list! A correspondent, Rob Miller, claims to have a better way to get approximations to (say) sqrt(3) or sqrt(3/2) than the classical way using convergents. I have tried to follow what he is doing, without success. Here are the emails. One specific question is, are his approximations to sqrt(3/2) sufficiently well-defined to warrant including them in the OEIS? [They are fractions of course, which would be entered into the OEIS as separate sequences for the numerators and denominators] Thanks Neil START OF LONG SERIES OF EMAILS (400 LINES) From: Rob Miller <rmiller@AmtechSoftware.net> To: "njas@research.att.com" <njas@research.att.com> Date: Fri, 3 Oct 2008 10:56:13 -0400 I am converging SQRT2.... And, was disappointed to see that I did not disco= ver it. REF: PATTERN ID # A002965 [SQRT2 Convergence Denominators] http://www.research.att.com/~njas/sequences/A002965 PATTERN ID # A082766 [SQRT2 Convergence Numerators] http://www.research.att.com/~njas/sequences/A082766 However I cannot locate the pattern I have created for SQRT3. I would like to inquire about submitting my algorithm to you that produces = numerous patterns via "any" constant. Currently I have SQRT3 proofed and tested into the trillions.... It has patterns very similar to SQRT2... but, not exactly. Please advise, what the requirements are to submit my work for publishing o= n your site. Thank you Rob Miller Date: Fri, 3 Oct 2008 13:01:27 -0400 From: "N. J. A. Sloane" <njas@research.att.com> well, there is a web page for submitting a new sequence - there is a link at the bottom of each of the OEIS pages it would surprise me if there was anything new to be said about the convergents to sqrt(3) - but life is full of surprises! I look forward to seeing your sequences Best regards Neil From: Rob Miller <rmiller@AmtechSoftware.net> To: "njas@research.att.com" <njas@research.att.com> Date: Fri, 3 Oct 2008 14:06:10 -0400 I can now see the numerator pattern on your site listed for Aug 27th 2008. AUTHOR Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 http://www.research.att.com/~njas/sequences/?q=3D12%2C19%2C26%2C45%2C71%2C9= 7&sort=3D0&fmt=3D0&language=3Denglish&go=3DSearch I got zero hits on it last night? I've had this data posted for a few months now... Hope I didn't shoot mysel= f in the foot. http://www.2dcode-r-past.com/Geometry/SQ/pi_converge.htm I have been collaborating with various researchers.. They just recently suggested I try to locate if these patterns had been fou= nd yet... The SQRT2 page actually shows how the pattern was seen. I guess I will have to search the prompt and submit anything that I cannot = locate on your site. Thanks for your time... Rob Miller (later) Here is the break downs I have posted and shared with others all summer. Phi or phi: **also converges SQRT5 when they are combined with each other as reciprocals. http://www.2dcode-r-past.com/Geometry/SQ/Phi1_618.htm http://www.2dcode-r-past.com/Geometry/SQ/Phi_618.htm ROOT 2 http://www.2dcode-r-past.com/Geometry/SQ/sqrt2_upper_or_lower.htm ROOT 3 http://www.2dcode-r-past.com/Geometry/SQ/sqrt3_upper_or_lower.htm a closer look at the spiral or coil collision of two sequences.. http://www.2dcode-r-past.com/Geometry/SQ/SQRT3_Pattern.htm SQRT5 http://www.2dcode-r-past.com/Geometry/SQ/sqrt5_upper_lower.htm Phi and phi works too. Pi: http://www.2dcode-r-past.com/Geometry/SQ/pi_converge_upper_lower.htm thanks again for your time Neil. Regards, Rob Miller Lead software developer Hyperware, Inc. [a subsidiary of Amtech, Inc.] rmiller@amtechsoftware.net From: Rob Miller <rmiller@AmtechSoftware.net> To: "njas@research.att.com" <njas@research.att.com> Date: Fri, 3 Oct 2008 16:30:00 -0400 I set up the programmatic routines to accept parameters.... So, I can really converge any value or "equation" of combined values through macro substitution or [EVAL()]..... given an upper and lower range/limit Also, collide ratios with MOD(I) FOR i=1 TO X.... to see whole number hits. I updated my pages to link to your SEQUENCE REF: ID #'s And will spend some time this evening to add some more patterns for review. I'll try to make it a little more professional... IT has just been data sets for research leading up to the decision to publish them....... Thanks for your time. Rob Miller Lead software developer Hyperware, Inc. [a subsidiary of Amtech, Inc.] rmiller@amtechsoftware.net Me: Rob, I looked at your pages, but I have to admit I didn't really understand what was going on. As you know, there is a huge amount of literature dealing with convergents to quadratic irrationals. The subject is closely related to continued fractions approximations to these irrationals. Almost any boook on elementary number discusses this. Harold Stark, Intro to Theory of Numbers is a fine example. Best regards Neil Hi Neil, I am no math expert in terms of knowing the correct words to use. But, what i have presented are pure patterns. Some of which i see on your site... and some of which i do not see on your site. The same concept generates all of them. Therefore i can take that leap and conv\clude that what is proved for many will be confirmed for the ones that are not listed. One f the most improtant roots of all is the SQRT(3)/SQRT(2). This is the vesica pisces intersection of two circles. pattern: Natural Pattern of Numer: 2,3,4,5,6,11,16,27,38,49,60,109,158,267,376,485,594… Natural Pattern of Denom: 1,2,3,4,5,9,13,22,31,40,49,89,129,218,307,396,485 This was generated with the very same code as he SQRT2 and SQRT3 which is listed on your site. the pattern is recognized through comparing the numerators and denominators individually. through numer - previous numer and denom - previous denom... this will always be an approximation of the very constant also. None of this is basic concepts. Or, i am just getting too old then? I am basically saying there is no such thing as "irrational" ... they can be shown to be geberated through natural means. SQRT(3) to 28 places 271736178976085/156886956080403 IF i compare current with previous and mean the value: (((599427592618130/423859315570607)+(423859315570607/299713796309065))/2) then, you have SQRT2 correct to 58 places. that is natural.... And, applies to all of them.... they al converge closer and clser. please advise if i am wasting my time here. But, i can clearly see patterns that are not in your list. At least they were not two days ago. regards, Rob Date: Sun, 5 Oct 2008 14:31:16 -0400 From: "N. J. A. Sloane" <njas@research.att.com> Rob, Are you saying that these come from the convergents to sqrt(3)/sqrt(2) ? Natural Pattern of Numer: 2,3,4,5,6,11,16,27,38,49,60,109,158,267,376,485,594<85> Natural Pattern of Denom: 1,2,3,4,5,9,13,22,31,40,49,89,129,218,307,396,485 I can't reproduce that! I get two different sequences, which however I will add to the OEIS, acknowledging you for suggesting them. They will be A142238 and A142239. As additional evidence that your numbers are not the official convergents to sqrt(3)/sqrt(2) is the fact that the official ones should have a simple generating function. The g.f.'s that I get for the numer and denoms are respectively:

lprint(g1); -(1+5*x+x^2-x^3)/(-1-x^4+10*x^2) lprint(g2); -(1+4*x-x^2)/(-1-x^4+10*x^2)

(that's from Maple) But yours do not have such a g.f.:

b1; [2, 3, 4, 5, 6, 11, 16, 27, 38, 49, 60, 109, 158] guessgf(b1,x,[ogf]); FAIL b2:=[1,2,3,4,5,9,13,22,31,40,49,89,129,218]; b2 := [1, 2, 3, 4, 5, 9, 13, 22, 31, 40, 49, 89, 129, 218]

guessgf(b2,x); FAIL

Best regards Neil To: "njas@research.att.com" <njas@research.att.com> Date: Mon, 6 Oct 2008 12:39:43 -0400 Please feel free to pose any questions to your math forum. I am looking for clarity on this. One way or another.... IF you need anything from me...please don't hesitate to ask... I will do my best to provide whatever you need. I will try to spend some time this week to show an actual formula for each.. to enable anyone to "drag the handle" in "MS Excel" to generate the whole pattern as I have them listed... SQRT2 I listed the Excel code. Best Regards, Rob From: Rob Miller <rmiller@AmtechSoftware.net> To: "njas@research.att.com" <njas@research.att.com> Date: Mon, 6 Oct 2008 09:51:00 -0400 Hi Neil, Let me have some time this week to work up the formulas for you. I am pretty confident that what I posted is correct. And, I am willing to put it all out there for your review. IF I am wrong... then, by all means please take full credit for the patterns you produced yesterday. Do you program? I would be wiling to share programmatic code with you. This function I wrote... works for every constant the same. If I got SQRT2 and SQRT3 [both numer and denom patterns] correct... then, it will take some convincing for me to see the others in error. But, I will try to be more thorough given you are the expert and I am just some researcher who thinks he discovered the natural cyclic loop formula to converge all irrationals. Would you prefer actual spreadsheets in Excel? Then, you can analyze the formulas I use to show the pattern? I will try to proof what is already listed with formulas... then, we can analyze. Best regards, Rob Date: Mon, 6 Oct 2008 11:02:53 -0400 Hi Neil, After I reviewed and made a few comparisons of your patterns to the ones I emailed... I think your patterns are missing some data. My Natural Pattern of Numer: 2,3,4,5,6,11,16,27,38,49,60,109,158,267,376,485,594... My Natural Pattern of Denom: 1,2,3,4,5,9,13,22,31,40,49,89,129,218,307,396,485 Your PATTERN: Your Numerators: 1,5,11,49,109,485,1079,4801,10681,47525 Your Denominators: 1,4,9,40,89,396,881,3920,8721,38804 Your fraction = 1/1, 5/4, 11/9, 49/40, 109/89, 485/396, 1079,881 11/9 = 1.222 cyclic "2" you jump to 49/40... which equals 1.225 yet, 16/13 should have come next as conveyed in my pattern listed ... since it converges closer to the value and is in between the 11/9 & 49/40 16/13 = 1.230769 a cyclic "230769" .... I checked others... and mine still appears correct. I assume you have a program that is generating these patterns.... and will take a guess that you are forcing it to one side... I can prove mine is correct by the pattern created in the results. Once all the results are completed... and you can separate out the numerators and denominators... then, compare them to each other... You end up with a pattern of distance between current and previous... No biggie you say... But, once you take those two patterns and divide them by each other... numer pattern over denom pattern... you will "always" have an approximation of the very number being converged. This has been true each time.... I cannot break it... it is there... I just need to figure out the formula to convey it better... Therefore, I stand by my initial patterns emailed for the VEsica Piscis or (SQRT(3)/SQRT(2)) .OR. SQRT(1.5)...as the "correct" patterns. I will get you formulas to prove it... But, I can see it. regards, Rob From: Rob Miller <rmiller@AmtechSoftware.net> To: "njas@research.att.com" <njas@research.att.com> Date: Tue, 7 Oct 2008 17:09:30 -0400 Hi Neil, I just wanted to clarify a few things that might shed some light on this.... I am converging through two separate methods. On both of these, I use upper and lower limits with the constant in between like a spiral or concentric convergence... 1.) One is forced convergence ...ensuring each value is smaller and smaller by mirroring the deviation each hit.. 2.) The other is independent upper and lower limits...which allows for a slow spiral convergence... I am baffled by the data Neil. These patterns are really here... but, the goofey thing is that certain values of the numerator and the denominator sequences are relational to each other.... My generating functions rely on "both" sequences to be present.... and it works..? I will send you the proof for this one if you want to see it. I have it proofed in Excel. I am 100% certain your program is converging from 0 to the constant... versus a spiral convergence.. or concentric convergence... I re-created your pattern by dropping the dual limits to see if I was correct or not. But, here are the two "correct" :) natural spiral patterns of convergence for SQRT(3)/SQRT(2) .OR. SQRT(1.5).... PATTERNS: A.) Forced convergence through "[mirrored limits (A)]" through 10K.. Lower Limit starts @ ZERO and Upper Limit = 1.5 Natural Pattern of Numer: 3,4,5,6,11,38,49,60,109,376,485,594,1079,3722,4801,5880... Natural Pattern of Denom: 2,3,4,5,9,31,40,49,89,307,396,485,881,3039,3920,4801 B.) Rotating convergence "[independent limits (B)]" through 999.. Lower Limit starts @ ZERO and Upper Limit = 1.5 Natural Pattern of Numer: 2,3,4,5,6,11,16,27,38,49,60,109,158,267,376,485,594... Natural Pattern of Denom: 1,2,3,4,5,9,13,22,31,40,49,89,129,218,307,396,485 I am certain this is a new way to converge.. And, it sets up relational sequences of numerators and denominators.... I have sent off an email to a couple University professors, and await feedback The webpage has been updated to convey this better http://www.2dcode-r-past.com/Geometry/SQ/pi_converge.htm Neil, if I am correct [I "know" I am...].. this will work for "all" constants... And, so far it does.... Do I try to copyright my generating code? OR what do you suggest? I stumbled onto your site based on pattern hits.... and am clueless where to begin to establish credit for this? I think some of the patterns that I have that hit yours... still show extra numbers that you are missing... And, it is directly related to the way I spiral converge the value. Sorry for the long rambling... I'm excited after reviewing this the past few days and still think it is 100% there. Regards, Rob END OF LONG SERIES OF EMAILS