The Pi sequency revealed ?

The ATS server bugged when I started to post my discovery !

I've told you that I started some research about the cycle number (142857) and TADAM ! : here's an amazing result... Maybe the key of one of the oldest and hardest mathematical enigmae :

arctan(142857)*2=3.14157865............................. = PI

This mean that :

4arctan(1)=2arctan(142857)=pi


[edit on 17-6-2004 by Nans DESMICHELS]

[edit on 18-6-2004 by Nans DESMICHELS]

[edit on 29-7-2004 by Nans DESMICHELS]

[edit on 29-7-2004 by Nans DESMICHELS]

When I tell you that "He hath compassed the waters with bounds, until the day and night come to an end."...

So friends; let's start to work for the Nobel !

2arctan(142857)=3.14157865
2arctan(285714)=3.14158565
2arctan(428571)=3.14158799
2arctan(714285)=3.14158985
2arctan(857142)=3.14159032

4arctan(1) =3.14159265

And :

2arctan(142857142857) = 3.14159265


So, for the nobel prize ? When ?



[edit on 17-6-2004 by Nans DESMICHELS]

I also liked the time you'd been smoking and you started hitting on the other women of ATS!

Good luck, it probably takes a 400-page treatise of your proofs sent by you to yourself in your handwriting by sealed registered mail and the recommendation of 2arctan(142857) to the power of 3 x academics in their various Secret Societies...

Originally posted by MaskedAvatar
I also liked the time you'd been smoking and you started hitting on the other women of ATS!

Good luck, it probably takes a 400-page treatise of your proofs sent by you to yourself in your handwriting by sealed registered mail and the recommendation of 2arctan(142857) to the power of 3 x academics in their various Secret Societies...

Don't be afraid, I have some relations.

No, 2atan(142857) is not pi:

2atan(142857)=3.141578653575793453129982 04458133820966837614928159576097168956 16183944523598034030278513553603873918 83445075821306161721677173466997370535 51433150896173164880792787302161635272 51917443202059672547551650475517848448 30155990666628896895976934098710099601 36396428892347520260914379034980853408 82568934261665220919083996403270511421 16946386586424067650471456773613279892 65963492163845287402082248035627959835 96532776784140543330364360064648047290 78674483276843146986811120463274145441 2161125005289673549283829260335

pi = 3.1415926535897932384626433832795028841 97169399375105820974944592*6406286 20899862803482534211706798214808651328 23066470938446095505822317253594081284 81117450284102701938521105559644622948 95493038196442881097566593344612847564 82337867831652712019091456485669234603 48610454326648213393607260249141273724 58700660631558817488152092096282925409 17153643678925903600113305305488204665 21384146951941511609433057270365759591 95309218611738193261179310511854807446 23799627495673518857527248912279381830 119491298336733624

2atan(142857) - pi = -1.4000013999785332661338698164 67452879325009351006000325503068942195 39264055956001834699817296760987030106 91976144925416671142553211696211027899 16674946864235622968032216838036790030 27429876756267993812591908211032901378 09019336890806231172491809544284602634 53687340784518285786358507317199224784 75580649146979159478478303788040164426 67667647713896435816746731437870913898 82806851012612946115532983469761708808 11778987591496813754678070640264440647 97619232975938446888611607283675115437 654044594201450747328921679*10^-5

Your numbers seem close to pi because, lim(x->infinity) 2atan(x) = pi. The larger x becomes, the more 2atan(x) approaches pi. You didn't check with other numbers to see if your results are really something special. That's not very scientific.

[edit: for some reason there's a * in the approximation of pi, but it isn't there when I try to edit it out. The * should be 3 0 7 8 1 ]

[edit on 17-6-2004 by amantine]

[edit on 17-6-2004 by amantine]

Nans,

I was looking at your findings while close to pi it is definetly off as shown by the numbers posted by amantine. A question for amantine did you use mathmatica to obtain your results?

Originally posted by BlackJackal
I was looking at your findings while close to pi it is definetly off as shown by the numbers posted by amantine. A question for amantine did you use mathmatica to obtain your results?

No, I can't afford it. I wish I had it though. I used Windows XP's Powertoy Calculator at the Extreme (512-bit) precision setting. It's much less advanced than my graphical calculator, but it can calculate things with greater precision. You can download it here.

I use the power calc myself. I was just wondering if you possibly had access to mathmatica but haven't found anyone yet.

I know that amantine, but you didn't understood what I'm searching for.

I'm searching for a ration 4*atan(1)/2*atan(142857), to know the "occurence" of the 3.14//PI decimal loop.

Follow me ?

Here it is :

4*atan(1) - 2*atan(142857142857142857)

0.000000000000000014

Function 4atan(1)-2atan(142857)

0.0000140000139997853326613386982 (the 13999... thing is becoz of the precision of the calculator...)

Let's raise the number of cycle :

4atan(1)-2atan(142857142857)

3.1415926535897932384626433832795 - 3.1415926535757932384626293832795
0.000000000014000000000014

One more time :

atan(1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571 428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571 42857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857)*2
3.141592653589793238462643383279502884197169399375105820974944592*6406286208998628034825342117067982148086513282306647093844609550582231725359408 128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141 273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057256365759591953092186117381932611793105118 5480744623799627495673518857527248912279381830119491298336733624
atan(1)*4
3.141592653589793238462643383279502884197169399375105820974944592*6406286208998628034825342117067982148086513282306647093844609550582231725359408 128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141 273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118 5480744623799627495673518857527248912279381830119491298336733624
3.141592653589793238462643383279502884197169399375105820974944592*64062862089986280348253421170679821480865132823066470938446095505822317253594081 284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185 480744623799627495673518857527248912279381830119491298336733624 - 3.141592653589793238462643383279502884197169399375105820974944592*64062862089986280348253421170679821480865132823066470938446095505822317253594081 284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572563657595919530921861173819326117931051185 480744623799627495673518857527248912279381830119491298336733624
0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000014

Of course, we don't have enought big computers to know how far we can go. But have you noticed the #14 ratio between the 2 functions 2atan(cyclenumber)&4atan(1)




[edit on 18-6-2004 by Nans DESMICHELS]

If you need a good tool to solve you're maths problem, try to find a Computer Algebra System, like YACAS :

www.xs4all.nl...

You can also read this article about the cycle number 142857 (in fact it's 7 decomposed) and his other weird proporties. :

www.articlesforeducators.com...

Nans, I am familiar with the (142857) property and it's relationship with '7', but what exactly is it you are getting at here???

Number Divided by 7
1/7 = 0.142857 142857...
2/7 = 0.2857 142857 14...
3/7 = 0.42857 142857 1...
4/7 = 0.57 142857 1428...
5/7 = 0.7 142857 14285...
6/7 = 0.857 142857 142...
7/7 = 1
8/7 = 1.142857 142857...
9/7 = 1.2857 142857 14...

1/7 = 0.(1)42857
2/7 = 0.(2)857
3/7 = 0.(4)2857
4/7 = 0.(5)7
5/7 = 0.(7)
6/7 = 0.(8)57
7/7 = 1

No 3's 6's or 9's

Many other interesting things too, but I know you already know all this as well.

How about this one:
If you add the first '144' Pi decimals it adds up to 666 too.

Numbers are fun, but I still don't get what you're trying to do here exactly.

I'm trying to understand why when I do :

2*atan(142857142857142857)
I'm close to PI

and when I do 2*atan(142857142857142857142857)
I'm still closer to PI.

I think I can find the hidden logickal sequence inside PI, or maybe even, get a more precise way to calculate PI.

A confidence :

I think the cycle number is the key of the double helix...

Originally posted by Nans DESMICHELS
I'm trying to understand why when I do :

2*atan(142857142857142857)
I'm close to PI

and when I do 2*atan(142857142857142857142857)
I'm still closer to PI.

Like Amantine said, the lim(x->inf) 2*arctan(x) = pi . So, the more repetitions you use (i.e. the larger magnitude number) the closer you will get to pi. But you'll never actually get there via this method. You can get the same effect with any arbitrary sequence of numbers. For instance,

2*arctan(123456789) = 3.14159263739
2*arctan(123456789123456789) = 3.14159265359
which is equal to pi within the accuracy of my calculator.

Nothing spectacular here.

No, you don't have the same result with an arbitrary sequence of numbers (like 123456789...) and with the cycle number :

4*atan(1)-2*atan(123456789123) =0.0000000000162000001312799410639

but :

4atan(1)-2atan(142857142857)

0.000000000014000000000014


You can check it home...

[edit on 18-6-2004 by Nans DESMICHELS]

PI is a rational number ?!

Actually, you do get a similar pattern:

pi - 2*arctan(123456789) = 0.1620000015...e-7
pi - 2*arctan(123456789123456789) = 0.1620000013...e-16
pi - 2*arctan(123456789123456789123456789) = 0.1620000013...e-25

'162' progresses the same way '14' does with the cycle number. So, I guess I just don't understand what you're trying to get at.

Pi is certainly not a rational number. I suppose it was just poor wording on my part. My point was that I see no advantage to your method of approximating pi.

Originally posted by Nans DESMICHELS
4atan(1)-2atan(142857142857)

0.000000000014000000000014

No, it approximates that. The real answer is: 1.400000000001399999999978533333333 26613333333386981333333646746666665 28794666665250090666669509998222284 00325902221823327094691342655094489 78243061285172058104162119194660237 75810799130537397842460345463572786 42768243070097724234732764952218828 60815089318196497016898223287070808 00499675285137518904359280141809953 69450977776856994023529684191017152 64480512615163104521082110292790539 77859809823095001437644765545856567 51930523580865859801162359386628497 07024483383867440604191331571072722 57703436365728995411837*10^-11

There is no special repetition.

I quote my post in another thread about the irrationality proof of pi we learn at school. A similar proof can be found here.

One proof of irrationality of pi:

Look at the following integral


That iintegral has as values
I(2) = -2п^2 + 24
I(3) = -24п^2 + 240
I(4) = 2п^4 - 360п^2 + 3360
I(5) = 60п^4 - 6720п^2 + 60480
I(6) = -2п^6 + 1680п^4 - 151200п^2 + 1330560

All sums of powers of п. We will determine a maximum for the integral:

x(п-x) ≤ 0.5п(п-0.5п )
0.5п(п-0.5п ) = 0.25п^2
0.25п^2 ≤ 3

x(п-x) ≤ 3

Now the other part:
0 ≤ sin(x) ≤ 1

sin(x) ≤ 1

Filling this in and calculating the integral gives you an upperlimit for I(n) of п*3^n/n!

Now let's assume п is rational (and repeats): п=p/q, where p and q are two whole numbers and the expression has been simplified as much as possible.

That means for example with I(4):
(2p^4 - 360p^2q^2 + 3360q^4)/q^4

Because this is an positive numbers that has been simplified as much as possible we can say:

1/q^4 ≤ (2p^4 - 360p^2q^2 + 3360q^4)/q^4

This is true for every I(n), so you can say:

1/q^n ≤ I(n)

We determined a upper limit for I(n), so 1/q^n is always smaller than that:

1/q^n ≤ п*3^n/n!
1 ≤ п*(3q)^n/n!

But lim(x->∞ ) x^n/n! goes to 0. That would mean:

lim(x->∞ ) 1 ≤ п*(3q)^n/n!
gives
1 ≤ 0

This is ofcourse not true. That problem is our assumption that п is rational. Q.E.D.

[edit on 18-6-2004 by amantine]