User talk:Moshe Levin

First 10^4 digits of Pi

First 10^6 digits of Pi

We start with A001223: 1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,... Local minima are shown in curly brackets: 1,2,2,{4,2,4},{4,2,4},{6,2,6},{4,2,4},6,{6,2,6},{4,2,6}{6,4,6},8,{4,2,4}{4,2,4},{14,4,6},{6,2,10}

Uploading graphics

If you're able, I suggest uploading your graphs as .png files rather than .jpg files. They will look sharper (and probably have smaller file sizes).

Charles R Greathouse IV 20:28, 27 October 2011 (UTC)

Also, you may consider moving this material to your user page, User:Moshe Levin; this page, User talk:Moshe Levin, is really designed for people to talk to you (like I am here). Just a thought. Charles R Greathouse IV 16:12, 1 December 2011 (UTC)

Forth side of the quadrilateral with maximal area.

Decimal expansion of the forth side, d ,of the quadrilateral with given sides (1,2,3), such that area K of the quadrilateral (1,2,3,d) is maximal.

4,1,1,3,0,9,0,5,8,4,3,2,4,9,5,1,2,5,3,6,2,6,6,5,4,9,9,9,3,3,8,2,6,0, 4,3,6,4,6,9,8,2,1,0,8,7,4,6,2,6,1,6,3,3,9,6,4,2,2,3,5,4,7,4,0,1,7,7, 7,8,7,1,3,4,8,8,8,3,0,7,6,2,2,1,3,7

Let three sides (a,b,c) of quadrilateral are given. What is the forth side, d, such that area of quadrilateral (a,b,c,d) is maximal?

Area K of quadrilateral (a,b,c,d) is maximal when the quadrilateral is cyclic and then

K=sqrt((p-a)(p-b)(p-c)(p-d)), with p=(1/2)(a+b+c+d), semiperimeter.

For given (a,b,c), possible values of d are between 0 (when quadrilateral reduces to triangle (a,b,c)) and (a+b+c) (when quadrilateral reduces to "line segment" (of length a+b+c) with zero area):

For d giving maximal area,we have cubic

d^3-(a^2+b^2+c^2)*d-2*a*b*c=0

with only one real positive solution 0<d<a+b+c.

(Note that this side is also diameter of circumcircle.)

For particular case a=1, b=2, c=3, we have

d=sqrt(56/3)*cos((1/3)*arccos(sqrt(243/686)))= 4.113090584324951253626654999...

Primes in square

Arrange nine positive (not necessarily distinct!) digits into 3x3 square such that

there are eight distinct 3-digit primes in each row, column and main diagonal.

There are 2792 (! not sure that counting is well-defined :=( ) different solutions for 3x3 square.

Lexicographically first solution is

1,1,3;2,3,9;7,9,7

with 8 distinct primes: 113,239,797,127,139,397,137,337 (total sum is 2286).

Lexicographically last solution is

9,8,3;7,3,9;7,9,7

with 8 distinct primes: 983,739,797,977,839,397,937,337 (total sum is 6006).

What about larger squares:

4x4 (10 distinct 4-d primes), 5x5 (12 distinct 5-d primes), etc.