# Jean-Francois Colonna - Number Theory

(Pictures of Number Theory)

## The 126.646 first digits -base 6- of the square root of 2 displayed as a tridimensional random walk.

Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

digit=0 ==> move(+D,0,0) digit=1 ==> move(-D,0,0) digit=2 ==> move(0,+D,0) digit=3 ==> move(0,-D,0) digit=4 ==> move(0,0,+D) digit=5 ==> move(0,0,-D)

with D=1.

## The 126.646 first digits -base 6- of the golden ratio displayed as a tridimensional random walk.

Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

digit=0 ==> move(+D,0,0)

digit=1 ==> move(-D,0,0)

digit=2 ==> move(0,+D,0)

digit=3 ==> move(0,-D,0)

digit=4 ==> move(0,0,+D)

digit=5 ==> move(0,0,-D)

with D=1.

## The 126.922 first digits -base 6- of 'e' displayed as a tridimensional random walk.

Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

digit=0 ==> move(+D,0,0)

digit=1 ==> move(-D,0,0)

digit=2 ==> move(0,+D,0)

digit=3 ==> move(0,-D,0)

digit=4 ==> move(0,0,+D)

digit=5 ==> move(0,0,-D)

with D=1.

## The 126.646 first digits -base 6- of 'pi' - displayed as a tridimensional random walk.

Each digit N -base 6- defines the current step of an “absolute” tridimensional random walk:

digit=1 ==> move(-D,0,0)

digit=2 ==> move(0,+D,0)

digit=3 ==> move(0,-D,0)

digit=4 ==> move(0,0,+D)

digit=5 ==> move(0,0,-D)

with D=1.

## Generation of the 63x63 first Conway's surreal complex numbers

## Tridimensional Hilbert Curve -iteration 4-.

## A pseudo-periodical Penrose tiling of the Golden Decagon.

## Tridimensional display of the Riemann Zeta function inside (-10.0,+20.0)x(-15.0,+15.0).

## Tridimensional display of the Riemann Zeta function inside (+0.1,+0.9)x(0,+50).

Here is the meaning of the three {X,Y,Z} display coordinates:

## The Goldbach conjecture -the Goldbach comet or the Goldbach rainbow- from 6 to 411678.

**The Goldbach conjecture**states that each even integer number N greater or equal to 4 can be written as the sum of two prime numbers. For example:

4 = 2+2 6 = 3+3 8 = 3+5 [= 5+3] 10 = 3+7 = 5+5 [= 7+3] (...) 70 = 3+67 = 11+59 = 17+53 = 23+47 = 29+41 [= 41+29 = 47+23 = 53+17 = 59+11 = 67+3] (...) 990 = 7+983 = 13+977 = 19+971 = 23+967 = 37+953 = 43+947 = 53+937 = 61+929 = 71+919 = 79+911 = 83+907 = 103+887 = 107+883 = 109+881 = 113+877 = 127+863 = 131+859 = 137+853 = 151+839 = 163+827 = 167+823 = 179+811 = 181+809 = 193+797 = 229+761 = 233+757 = 239+751 = 251+739 = 257+733 = 263+727 = 271+719 = 281+709 = 307+683 = 313+677 = 317+673 = 331+659 = 337+653 = 347+643 = 349+641 = 359+631 = 373+617 = 383+607 = 389+601 = 397+593 = 419+571 = 421+569 = 433+557 = 443+547 = 449+541 = 467+523 = 487+503 = 491+499 [= 499+491 = 503+487 = 523+467 = 541+449 = 547+443 = 557+433 = 569+421 = 571+419 = 593+397 = 601+389 = 607+383 = 617+373 = 631+359 = 641+349 = 643+347 = 653+337 = 659+331 = 673+317 = 677+313 = 683+307 = 709+281 = 719+271 = 727+263 = 733+257 = 739+251 = 751+239 = 757+233 = 761+229 = 797+193 = 809+181 = 811+179 = 823+167 = 827+163 = 839+151 = 853+137 = 859+131 = 863+127 = 877+113 = 881+109 = 883+107 = 887+103 = 907+83 = 911+79 = 919+71 = 929+61 = 937+53 = 947+43 = 953+37 = 967+23 = 971+19 = 977+13 = 983+7] (...)

The horizontal axis represents the even numbers N={6, 8, 10, 12,…} starting at 6 (for compatibility with the other related visualizations). The “altitude” of each point exhibits the number of decompositions of N as sums of two prime numbers.

## The ABC conjecture.

The horizontal and vertical axes display respectively two whole numbers A and B. Each disk display a couple of coprime numbers A and B:

GCD(A,B)=1

The number C is the sum of A and B:

C = A+B

The function Radical(N) gives the product of the prime factors (with an exponent equals to 1) of N. For example:

N = 1960 = 2^3 .5^1 .7^2

Radical(1960) = 2^1 .5^1 .7^1 = 2.5.7 = 70

Then the following function is computed:

k(A,B,C) = log(C) / log(Radical(A.B.C))

**The ABC conjecture** states that k(A,B,C) is less than a certain constant (unknown, but greater than 1 and hopefully lesser than 2…) whatever the values of A and B.

The surface and the luminance of each disk are proportional to k(A,B,C).

For this picture, the numbers A and B belong to [1,100] giving birth to the following values:

min(k(A,B,C))=0.37117806024788 max(k(A,B,C))=1.22629438553090

## The Syracuse conjecture -polar coordinates display-

The Syracuse sequence is defined as follows:

U(0) = N (an integer number)

if U(n) is even :

U(n+1) = U(n)/2

else :

U(n+1) = 3.U(n) + 1

**The Syracuse conjecture**states that sooner or later the {[[4,] 2,] 1} sequence will appear whatever the starting number N (and then repeats itself obviously*ad vitam eternam*). For example with N=7:U(0) = 7 U(1) = 22 U(2) = 11 U(3) = 34 U(4) = 17 U(5) = 52 U(6) = 26 U(7) = 13 U(8) = 40 U(9) = 20 U(10) = 10 U(11) = 5 U(12) = 16 U(13) = 8 U(14) =

**4**U(15) =**2**U(16) =**1**

- This picture is a circular display of sixteen different sequences from U(0)=5 (lower left) to U(0)=20 (upper right). For each sequence U(n) the following “star” is generated:
Rho(n) = U(n) (with a renormalization inside [0,1])

Teta(n) = 2.pi.n/(nm+1)

X(n) = Rho(n).cos(Teta(n)) Y(n) = Rho(n).sin(Teta(n))

where ‘nm’ denotes the maximal value of ‘n’:

U(nm) = 1

The colors used are a function of ‘n’ (from Dark Blue [n=0] to White with an increasing luminance).

## The additive persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right-.

Starting from the origin of the coordinates (at the center of the picture), one follows a square spiral-like path and numbers each integer point encountered (1, 2, 3,…).

Then one displays the N-th point with the false color f(PA(N,B)) where:

PA(N,B) = the additive persistence of N for the base B, f(...) = an arbitrary ascending function

Let’s define PA(N,B) with an obvious example:

B = 10 N = 856 (= 8xB^2 + 5xB^1 + 6xB^0)

Then the following sequence is computed:

856 ---> (8+5+6) = 19 ---> (1+9) = 10 ---> (1+0) = 1

It takes three (3) steps to reach a one digit number. Then:

PA(856,10) = 3

## The multiplicative persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right-.

Starting from the origin of the coordinates (at the center of the picture), one follows a square spiral-like path and numbers each integer point encountered (1, 2, 3,…).

Then one displays the N-th point with the false color f(PM(N,B)) where:

PM(N,B) = the multiplicative persistence of N for the base B, f(...) = an arbitrary ascending function.

Let’s define PM(N,B) with an obvious example:

B = 10 N = 77 (= 7xB^1 + 7xB^0)

Then the following sequence is computed:

77 ---> (7x7) = 49 ---> (4x9) = 36 ---> (3x6) = 18 ---> (1x8) = 8 It takes four (4) steps to reach a one digit number (by the way it is the longest sequence with a two digit number). Then:

PM(77,10) = 4

A conjecture states that PM(N,10) cannot exceed 11...

Here is an example of a longer sequence:

48699984 ---> 4478976 ---> 338688 ---> 27648 ---> 2688 ---> 768 ---> 336 ---> 54 ---> 20 ---> 0

## 12 evenly distributed points on a sphere -an Icosahedron- by means of simulated annealing.

## 3-foil torus knot on its torus.

## Three successive elementary monodimensional binary cellular automata -106,90,86- with 1 yellow starting point -bottom middle-.

An elementary monodimensional binary automaton is a monodimensional set of cells. At time ‘t’, each cell (with coordinate ‘x’) has a value ‘CELL(x,t)’ that equals either 0 (**B**lack) or 1 (**W**hite) and has two neighbours (one at its left ‘CELL(x-1,t)’ and one at its right ‘CELL(x+1,t)’). The points outside the picture (at left and at right) are assumed to be **W**hite. The time evolution of this set of cells is defined by means of rules.

This picture was computed using successively the three following elementary monodimensional binary cellular automata:

**automaton 86** (for t E [401,574]) -white background-

**automaton 90** (for t E [101,400]) -light orange background-

**automaton 106** (for t E [0,100]) -dark orange background-

The vertical axis is the time axis and the initial conditions are displayed on the bottom line.

## Tridimensional display of the dynamics of the bidimensional John Conway's life game

The third coordinate ‘Z’ is the time ‘T’, whereas the two space coordinates X’ and ‘Y’ are periodical

## The tridimensional John Conway's life game with random initial conditions -25% of occupied cells-.

The bidimensional life game was initially defined by Conway. It uses an empty square mesh (all vertices are turned **off**). At time t=0 some vertices are occupied (they are turned **on**): this is the initial state. To go from the time t to the time t+1, it suffices to count for each vertex -or “Cell”- C(x,y) the number N of its neighbours (it is less than or equal to 3^2-1=8) and then to possibly change the state of M according to the following bidimensional automata rules:

[R1 = Birth] ((C(t).IS.off).AND.(N == 3)) ==> C(t+1)on[R2 = Death] ((C(t).IS.on).AND.((N < 2).OR.(N > 3))) ==> C(t+1)off[R3] other cases ==> C(t+1)=C(t)

The boundary conditions can be periodical or not.

This process can extended in a tridimensional space. The number N of neighbours of the vertex -or “Cell”- C(x,y,z) is computed (it is less than or equal to 3^3-1=26) and the preceding rules can be extended as follows:

[R1 = Birth] ((C(t).IS.off).AND.((N >= NB1).AND.(N <= NB2))) ==> C(t+1)on[R2 = Death] ((C(t).IS.on).AND.((N < ND1).OR.(N > ND2))) ==> C(t+1)off[R3] other cases ==> C(t+1)=C(t)

The bidimensional and tridimensional processes can be extended one step further using two binary lists ‘LD’ and ‘LA’ (“Dead” -**off**- and “Alive” -**on**- respectively):

[R1 = Birth] ((C(t).IS.off).AND.(LD[N] == 1)) ==>C(t+1)=on[R2 = Death] ((C(t).IS.on).AND.(LA[N] == 1)) ==>C(t+1)=off[R3] other cases ==> C(t+1)=C(t)

(“1” means “to change the state” and “0” means “the state is unchanged”).

For this picture, the parameters have the following values:

NB1=10 NB2=14 ND1=8 ND2=16

## Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.

## Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.

## The 64 first lines of the Pascal's Triangle.

The white peaks display the coefficients that are divisible by 2, 3, 5 and 7 (when the dark blue ones displays the coefficients that are not divisible by 2, 3, 5 and 7). The other colors display the other cases… The height of each point is proportional to the luminance of its color.

## Artistic display of a Sudoku grid.

Here is the used grid with each digit (from 1 to 9) displayed with its own color:

1 2 3 | 4 5 6 | 7 8 9 4 5 6 | 7 8 9 | 1 2 3 7 8 9 | 1 2 3 | 4 5 6 ------+-------+------ 2 3 4 | 5 6 7 | 8 9 1 5 6 7 | 8 9 1 | 2 3 4 8 9 1 | 2 3 4 | 5 6 7 ------+-------+------ 3 4 5 | 6 7 8 | 9 1 2 6 7 8 | 9 1 2 | 3 4 5 9 1 2 | 3 4 5 | 6 7 8