[VA] SRC#002 Almost integers and other beasties

12132018, 10:30 PM
Post: #1




[VA] SRC#002 Almost integers and other beasties
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Hi all, welcome to my SRC#002  Almost integers and other beasties. Here I'll show an assortment of the results I can get using my HP71B IDENTIFY program version 2.0. The original version 1.0 was extensively discussed and demonstrated with examples galore in my article Boldly Going ... Identifying Constants, published more than 10 years ago. You can download it as a PDF document using this link: Boldly Going ... Identifying Constants Shortly after publishing it I expanded its already substantial capabilities with important additional features such as the ability to find Minimal Polynomials and other implicit expressions, which greatly increased the recognition of arbitrary constants, and further this version 2.0 can be used creatively to find interesting, uncanny expressions never before seen, like the following ones I found and which you might enjoy seeing and checking using your trusty HP calculator:
Go ahead, check them, and I'd love to see any and all comments you would have on the matter, as well as your own uncanny expressions of a similar nature (Gerson, I'm looking at you ), please post your very best, original ones discovered by you (no 3rdparty ones harvested on the Internet, please) as replies in this thread. Regards. V. . 

12142018, 04:41 AM
Post: #2




RE: [VA] SRC#002 Almost integers and other beasties
The PDF had fonts the I couldn't read, but that doesn't stop me from commenting. (I is the internet.)
If interpreted your comments correctly, you compare by comparing fractions in lowest terms. This suggests (if you are not doing this already, converting decimals to continued fractions and do comparisons by generating a single partial quotient at each step. This could lead to a nice speedup as grossly different numbers could be eliminated quickly. 

12142018, 03:14 PM
Post: #3




RE: [VA] SRC#002 Almost integers and other beasties
(12132018 10:30 PM)Valentin Albillo Wrote: Go ahead, check them, and I'd love to see any and all comments you would have on the matter, as well as your own uncanny expressions of a similar nature (Gerson, I'm looking at you ), please post your very best, original ones discovered by you (no 3rdparty ones harvested on the Internet, please) as replies in this thread. Hello, Valentin, More or less in the same vein,
Here are a few more original nearintegers and nearidentities: \[2\left ( \pi + e  \psi \right ) = 4.9999776\] \[2\left ( e\tan^{1}\left ( e \right ) \right )=2.9999978\] \[\ln \left ( \frac{16\ln 878}{\ln \left ( 16\ln 878 \right )}\right )=3.14159265377\] \[\frac{e^{\frac{23}{4}}}{100+\frac{1}{100+\frac{1}{\sqrt{100\sqrt{5}}}}}=3.14159265354\] \[3.141593\frac{\sqrt{3}}{5\times 10^{6}}=3.1415926535898\] \[\frac{\ln \left ( \sqrt{8} \cdot 10^{8}\right )}{\ln \pi }=16.999994\] \[\frac{\ln \left (2\cdot \varphi ^{39}\right )}{\ln \pi }=17.00000026\] Best regards, Gerson 

12142018, 08:29 PM
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RE: [VA] SRC#002 Almost integers and other beasties  
12142018, 08:58 PM
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RE: [VA] SRC#002 Almost integers and other beasties
(12142018 03:14 PM)Gerson W. Barbosa Wrote: \[\ln \left ( \frac{16\ln 878}{\ln \left ( 16\ln 878 \right )}\right )=3.14159265377\] There is something beautiful and compelling about this one, at least for me! Both you guys truly amaze me... in a good way, just to be clear.... Bob Prosperi 

12142018, 09:41 PM
(This post was last modified: 12142018 09:47 PM by Gerson W. Barbosa.)
Post: #6




RE: [VA] SRC#002 Almost integers and other beasties
(12142018 08:29 PM)Thomas Klemm Wrote:(12132018 10:30 PM)Valentin Albillo Wrote: no 3rdparty ones harvested on the Internet, please I would humbly suggest an even more comprehensive test: \(e^{\pi }\pi + \left ( \frac{9}{3\times 10^{2}\frac{2\sqrt{3}}{10^{3}\ln \left ( 2+\sqrt{2} \right )}} \right )^{2}=20.000000000000000\) 

12152018, 12:19 AM
Post: #7




RE: [VA] SRC#002 Almost integers and other beasties
Or this one, for 10digit calculators:
\(\pi^{2}+\frac{e^{2}}{33\left(\ln\left(\pi\right)\right)^{4}}=10.00000000\) 

12152018, 09:20 PM
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RE: [VA] SRC#002 Almost integers and other beasties
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Hi, ttw: (12142018 04:41 AM)ttw Wrote: This suggests (if you are not doing this already, converting decimals to continued fractions and do comparisons by generating a single partial quotient at each step. Thanks for your interest and comment. I do use a continued fraction algorithm to convert the arbitrary constant supplied by the user to a rational fraction, generating partial quotients one by one till the usersupplied accuracy is met. Regards. V. . 

12152018, 09:56 PM
(This post was last modified: 12162018 03:48 AM by Valentin Albillo.)
Post: #9




RE: [VA] SRC#002 Almost integers and other beasties
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Hi, Gerson: (12142018 03:14 PM)Gerson W. Barbosa Wrote: More or less in the same vein, Thanks for your excellent findings, I was sure you'd never fail to contribute some amazing nearidentities to this thread. As Bob Prosperi already pointed out, I too find this one particularly beautiful: Quote:\[ \ln \left ( \frac{16\ln 878}{\ln \left ( 16\ln 878 \right )}\right )=3.14159265377 \] Good finding indeed ! By the way, it's quite nice that the simple function x/Ln(x) sometimes gives almostinteger results for integer arguments (which means its graphic passes extremely close to integercoordinates grid points), such as the following, in increasing order of "closeness": x x/Ln(x)  17 6.0002541... 163 31.9999987... 53453 4910.0000012... 110673 9529.0000006... 715533 53078.0000004... so that we have, for instance, 53453/Ln(53453) = 4910.0000012... In your case the argument x=16*Ln(878) results in x/Ln(x) being 23,1406926369... which is almost the famous Gelfond's constant = e^Pi (the easiest transcendental number to compute to high precision) so its natural logarithm is very nearly Pi itself. Nice catch ! Have a fine weekend and best regards V. . 

12162018, 09:06 AM
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RE: [VA] SRC#002 Almost integers and other beasties
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(12152018 09:56 PM)Valentin Albillo Wrote: By the way, it's quite nice that the simple function x/Ln(x) sometimes gives almostinteger results for integer arguments...Thanks, that leads to a rabbit hole of interesting links (OEIS and Mathoverflow.) 

12162018, 03:51 PM
Post: #11




RE: [VA] SRC#002 Almost integers and other beasties
Exp(Pi*Sqrt(163)) is one of the classic examples. The explanation is rather complicated. Other expressions, for example:
({[Sqrt(5)+1]/2}^n)/Sqrt(5) is close to the Fibonacci numbers; in fact ({[Sqrt(5)+1]/2}^n{[1Sqrt(5)]/2}^n)/Sqrt(5) is the wellknown Binet formula for Fibonacci numbers. This sequence works by successive approximation to an integer the 163 sequence just seems to happen. 

12162018, 10:43 PM
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RE: [VA] SRC#002 Almost integers and other beasties
(12162018 09:06 AM)EdS2 Wrote: Thanks, that leads to a rabbit hole of interesting links (OEIS and Mathoverflow.) Interesting. I did not consult the OEIS for the results I gave above for x/Ln(x), I simply obtained them myself by running this trivial HP71B program I wrote in JF Garnier's Emu71 to quickly find them: 1 DESTROY ALL @ M=1 @ I=2 2 X=I/LN(I) @ N=ABS(XIROUND(X)) @ IF N<M THEN M=N @ DISP I;,X 3 I=I+1 @ GOTO 2 >RUN 2 2.88539008178 5 3.10667467281 9 4.09607651981 13 5.06832618827 17 6.00025410569 163 31.9999987385 53453 4910.00000122 110673 9529.00000068 715533 53078.0000004 1432276 101044 ... ... Substituting X=I/LN(I) at line 2 by some other function and rerunning the program will result in a new set of almostinteger values, for instance: 2 X=I/TAN(I) ... >RUN 2 .915315108721 3 21.0457576543 7 8.03260795684 37 44.0072133321 48 39.9957590124 128 123.004197859 170 460.0010337 1489 12899.9995967 2106 986.000155144 11923 15493.9999873 i.e.: 1489/Tan(1489) = 12899.9995967 ~ 12900 and so on and so forth. Trivial variations of this trivial program will produce an infinitude of almostinteger valued expressions of all kinds. Thanks for your interest and links. V. . 

12192018, 10:23 PM
(This post was last modified: 12202018 09:56 PM by Valentin Albillo.)
Post: #13




RE: [VA] SRC#002 Almost integers and other beasties
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Hi, all: (12162018 10:43 PM)I Wrote: Trivial variations of this trivial program will produce an infinitude of almostinteger valued expressions of all kinds. A few additional, nice almostinteger results obtained that way: 5e^Acos(178/181) = 6.0000000066 9e^Acos(538/541) = 10.00000000023 8e^Acos(430/433) = 9.00000000048 Ln 146 + Sin 614 = 4.00000800014 Ln 455 + Cos 188 = 7.00000034 Ln 231 + Tan 87 = 4.00000023 Gamma(314/709) = 2.00000047 All trig functions, in radians. V. . 

01022019, 11:45 PM
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RE: [VA] SRC#002 Almost integers and other beasties
640000*x^5768000*φ^2*x^4+3000+ln(2)=0


01032019, 06:29 AM
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RE: [VA] SRC#002 Almost integers and other beasties
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Hi, Gerson: (01022019 11:45 PM)Gerson W. Barbosa Wrote: 640000*x^5768000*φ^2*x^4+3000+ln(2)=0 I only have an iPad at hand right now so running this extremely quick'n'dirty Newton on it produces the intended root of your polynomial, namely: 10 def fnf(x)=640000*x^5768000*p^2*x^4+3000+log(2) 20 def fnd(x)=(fnf(x+0.0001)fnf(x0.0001))/0.0002 30 p=(1+sqr(5))/2:input x0:home 35 for i=1 to 15 40 x1=x0fnf(x0)/fnd(x0) 50 print x1;" ";fnf(x1) 60 x0=x1 70 next i Run ?10 8.167851990851785 14317006965.841368 6.715802401810718 4653139559.305097 5.573461555087618 1501801189.590903 4.687642795302511 477761112.5232644 4.021032838991736 147136983.11165047 3.551962197641171 41803117.36241603 3.2712995361031516 9506051.502593396 3.1593486482128053 1132110.8530623894 3.141983056899174 24349.060661761047 3.141592847539331 12.090443311217141 3.1415926535896097 0.0000030975368739971643 3.14159265358956 3.170697671084355e8 3.1415926535895604 1.904654323148236e9 3.1415926535895604 1.904654323148236e9 3.1415926535895604 1.904654323148236e9 which is a nice approximation to Pi, congrats and thanks for sharing. Perhaps it's even more accurate than what the iPad produces but right now I can't tell ... Regards. V . 

01032019, 06:48 AM
(This post was last modified: 01032019 06:49 AM by Paul Dale.)
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RE: [VA] SRC#002 Almost integers and other beasties  
01032019, 07:36 AM
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RE: [VA] SRC#002 Almost integers and other beasties  
01032019, 11:56 AM
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RE: [VA] SRC#002 Almost integers and other beasties
(01032019 06:48 AM)Paul Dale Wrote: 3.1415926535897932 Or for spotting mistakes like this one (YouTube video). I used to remember the first 16 digits when I used my HP200LX on a regular basis (its screen has gone dark and I have no spare part to replace it again). Anyway, I don't remember any "747" sequence occurring so early in \( \pi \). Gerson. 

01032019, 12:51 PM
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RE: [VA] SRC#002 Almost integers and other beasties
The first "747" occurs at position 740 (thanks to the \(\pi\) searcher).
In the fullness of \(\pi\), this is right near the start (of course). Pauli 

01032019, 10:33 PM
Post: #20




RE: [VA] SRC#002 Almost integers and other beasties
(12192018 10:23 PM)I Wrote: A few additional, nice almostinteger results obtained that way: For completeness, I forgot to include this remarkable one which I discovered and posted here last March: Sin(9*(Sin 1 + Cos 40)) = 0.999999999999999830826985368... which differs from the integer 1 by about 1e16. V. . 

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