The Pi sequency revealed ?, page 1
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reply posted on 17-6-2004 @ 10:06 AM by amantine
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]


reply posted on 18-6-2004 @ 02:58 AM by Nans DESMICHELS
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]


reply posted on 18-6-2004 @ 07:14 AM by amantine
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]
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