PATTERNS IN NUMBER THEORYTheophanes E. Raptis

Computational Applications Group

Division of Applied Technologies

NCSR Demokritos 2015

WHY NUMBER THEORY? Computers are based on integers even for

simulations of “continuous” systems – and yet they can build ‘virtual’ worlds!

Nuclear Spectra have a distinctively similar structure with prime distributions (Random Matrix Theory – Many Body Chaos).

SIC-POVM (Q. Cryptography): Samplings from the Bloch’s unit sphere lead to unexpected connections with t-Design combinatorics.

DISCRETE MATHEMATICS

Significance of the Axiomatic Basis (Think of geometry without Euclid!)

No “Axiomatization Algorithm” possible.

What is the simplest set of required axioms? How can we know that we use the best one?

How many alternatives possible?

FRACTALITY IN ARITHMETICS: WHENCE FROM?

Various number-theoretic functions exhibit self-similarity.

Simplest case: Digit-Sums in all alphabets.

• (Image Source: Wolfram Site)

Simple recursion schemes reproduce the above.

PRIMITIVE SELF-SIMILARITY

Remiscit Sets: Invariance under removal of a subset.

n-ary Tree Unfoldings: Lexicographically Ordered Combinatoric Powersets (“Lexicons”) of n symbol alphabets.

More primitive than Cantor Sets.

Rows: Periodicity, Columns: Quasi-periodicity

GENERAL QUASI-PERIODIC SYSTEM

Let { y1,…,yn } a set of harmonic functions of increasing periods (“Laplacian eigenfunctions”).

Let h( y,L ) a threshold sampler:

For any quantized interval { δτi,…, δτΝ } we get a set of binary “pulse” sequences, eg.

0011001100… 0001111110… Number theoretic equivalent:

2/1)/2()),(( LtySIGNLtyh i

2101210 ,mod),,( ppxxpppxxh

TRAILING ZEROS AND DIVISIBILITY 0 2 4 6 8 10 12 14 16

0 0 0 0 0 0 0 0 0

0 1 0 1 0 1 0 1 0

0 0 1 1 0 0 1 1 0

0 0 0 0 1 1 1 1 1

0 3 6 9 12 15 18 21 24

0 0 0 0 0 0 0 0 0

0 1 2 0 1 2 0 1 2

0 0 0 1 1 1 2 2 2

0 0 0 0 0 0 0 0 0

INTERPOLANTS FOR TZ(N,DIVISOR)

Tz(0:28 ,2):

Tz(0:35 ,3):

1)]([log

1

22

)1,2,(N

i

iih

)(log

1

22

)1,2,(N

k

kikh

)(log

1

33

)1,3,(N

k

kikh

EXAMPLE: COLLATZ DYNAMICS Branched mapping:

“Folded” forms:

Equivalent dynamics:

1

1)](2[log

1

,..)()2,13(

)](2[log

1

,..)()](2[log

1

,..)()2,(

2132)13(

2231)2(3

x

i

ihxTz

x

i

ihx

i

ihxTz

xx

xx

PRIMALITY VIA TZ

Let a list of primes < n.

Equivalently

IsPrime(n): (No trailing zeros in any base < n.)

),(),(2

),(1 ...: 21 kzzz pnT

kpnTpnT pppnNn

kpp ,...,1

)]([log

...,,

)1,,(

210

0:

n

kkk

pnhkk

ip ipjkijpnNn

kpp

ppZ pnT

10),(

DIVISOR FUNCTION Def.: Examples:

Properties: σ0( n ) ~ number of factors of

From the last one

nn ba

...... 212121210tttt pppp

1)(0 itn

‘HIDDEN’ DIMENSIONALITY From last formula and Tz:

Expand:

Transform to Sum-of-Products:

Products on rhs represent a Multi-Dim. Binary Grid!

1),()(0 iZ pnTn

1,),()1()( 0

)]([log

00

hpnhknn

k

kpk

ip

k

kipip

kpk

n

k

k pnhkn ,)1()(

)]([log

0

10

DIMENSIONAL ORDERING OF INTEGERS Sample [0,…,1024]

THE DIVISOR FUNCTION AS A DIRECT SUM Simpler Expression: Superposition of 1st line

of all ‘Lexicons’ for all alphabet bases.

Apply logical mask in the above array ~(A=0).

Equivalent:

1 2 3 4 5 6 7 8 9 10

1 0 1 0 1 0 1 0 1 0

1 2 0 1 2 0 1 2 0 1

1 2 3 0 1 2 3 0 1 2

1 2 3 4 0 1 2 3 4 0

1 2 3 4 5 6 7 8 9 10

0 1 0 1 0 1 0 1 0 1

0 0 1 0 0 1 0 0 1 0

0 0 0 1 0 0 0 1 0 0

0 0 0 0 1 0 0 0 0 1

0 0 0 0 0 1 0 0 0 0

n

k

knhn1

0 )1,,(1)(

Resulting Matrix (32 x 32) Particular case of

Quasi-periodicity. Open Problem: Do

other cases of positive sequencesof periods exist that exhibit global extrema? Are they associated with certain number theoretic properties?

Check

n

kjkj

n

kk ppnhpnh

1,1

),,(),1,,(

FUNDAMENTAL PROPERTY OF D-MATRIX Association of Divisors with Integer Partitions. ‘Geometry’ of the Divisor Set

Let D={ d1, …, dk } a list of divisors of n.

RSA 2-factor primes simultaneously satisfy:

Objective Function:

1)(

111

0 n

ii

iii dddn

Nzyxyxx

zyxn

,,,)(

innjnni

njnni

3/,3/,...,1

3/

zyx

Simple Example (n = 15, z = 7):

For all other values of z the 2 factors never meet along the same column.

The simple structure of 2-restricted partitions allows an easy and fast factorization algorithm.

7 12 15 16 15 12 7

7 6 5 4 3 2 1

1 2 3 4 5 6 7

OPTIMIZED VERSION WITH PRECOMPUTED ELEMENTS

The Matrix of all 2-Partitions Products is recursive!

We can avoid all multiplications.

Numeric structure: 2 2

3 4 3

4 6 6 4

5 8 9 8 5

6 10 12 12 10 6

7 12 15 16 15 12 7

8 14 18 20 20 18 14 8

Differences:0

1 -1

2 0 -2

3 1 -1 -3

4 2 0 -2 -4

5 3 1 -1 -3 -5

6 4 2 0 -2 -4 -6

7 5 3 1 -1 -3 -5 -7

kikii 2},0{,2/)12(},2{

NEXUS PRIMUS: A PROJECTIVE GEOMETRY? The Set of discrete diagonals of the Divisor

Matrix form a Bundle under continuation.

1

1))1(/*/()()( ii xinininy

Magnified Area near the center shows two envelopes (n = 512)

h(pi, pi): No roots

h(pi, 1): Roots structured.

DEVIL STAIRCASES

Let y(x) a strictly increasing positive envelope and q over y a regular or irregular ‘quantizer’.

Every such “staircase” admits a unique invertible decomposition as

Identity: Original signal encoded in sequence Cumulant Curve for Divisor Matrix:

))(,),(()),)(,()( 1, kqxkqxxhkkqxxhqyQ iiiii

1))(,,(),)(,( kqxkxhkkqxxh ii

1, iiq )1,1)(,( iqxxh

2/)1(1

nnnn

i

COMBINATORIC DESIGNS

Analysis of the structure of Powersets shows all designs to be related with Partition Functions

Simple example: Let L(b=6, N=10) the sextiary Lexicon of all 10 symbol strings. We ask for all the sequences that contain only groups of 3 and 5.

Symbol Counter functions:

Characteristic over a Partition Function:

N

i

ii bbibnnhnh1

0 1,...,0),,)(),(()(

}{1)(**

)(})}{({ ihji

ieMin