2015

Ulam Spiral Hidden

Patterns & Waves

Author: Mohammad Hefny

Ulam Spiral Hidden Patterns

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Contents Abstract: .......................................................................................................................................... 2

Numbers & Divisors Dependency ................................................................................................ 3

An OEN number can have 4 states: ......................................................................................... 4

An SVEN number can have 3states: ........................................................................................ 6

An NEIN number can have 4 states: ........................................................................................ 6

Divisors Waves............................................................................................................................. 7

List of All Waves: ..................................................................................................................... 9

A Hint on Waves Formula ...................................................................................................... 10

Ulman Spiral Pattern Explained ................................................................................................. 11

Conclusion ..................................................................................................................................... 14

References ..................................................................................................................................... 15

Ulam Spiral Hidden Patterns

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Abstract:

The Ulam spiral visualization method of prime numbers reveals some patterns and relations that

can be spotted by eye, whoever no formula can link between all prime numbers locations in this

diagram.

In Figure 1, black dots are prime numbers while white area is non-prime dots that are clustered

together. We can see in above diagram some discontinued diagonal lines formed by black dots

i.e. prime numbers.

In this study, focus will be made on white area rather than back dots. As this article suggests, the

tendency of forming diagonal lines of prime numbers are only the negative image of hidden

patterns exist in the white area.

Figure 1: Ulam Spiral Diagram

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Numbers & Divisors Dependency

As per definition a prime number is a natural number greater than 1 that has no positive

divisors other than 1 and itself.

We will classify all numbers either prime or non-prime by its last digit to:

a. OEN: For numbers ended by 1 – will be called Ones-Ended Numbers

This set of numbers can be described as:

(10N +1) where N ε {1,2,3,….}

N represents the index –or order- of the number. For example index =0 is 1 and index =

1 is 11 and index = 2 is 21.

b. TREN: For numbers ended by 3 – will be called Three-Ended Numbers

This set of numbers can be described as:

(3N +1) where N ε {1,2,3,….}

N represents the index –or order- of the number. For example index =0 is 3 and index =

1 is 13 and index = 2 is 23.

c. SVEN: For numbers ended by 7 – will be called Seven-Ended Numbers

This set of numbers can be described as:

(10N +7) where N ε {1,2,3,….}

N represents the index –or order- of the number. For example index =0 is 7 and index =

1 is 17 and index = 2 is 27.

d. NIEN: For numbers ended by 9 – will be called Nine-Ended Numbers

This set of numbers can be described as:

(10N +9) where N ε {1,2,3,….}

Figure 2: Classifying numbers by last digit

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N represents the index –or order- of the number. For example index =0 is 9 and index =

1 is 19 and index = 2 is 29.

We will just ignore even numbers and numbers ended by five as they all non-prime except 2 & 5,

as all even numbers have 2 as a positive divisor and all five-ended numbers have 5 as a positive

divisor.

An OEN number can have 4 states:

a. To have a positive two divisors, both are OEN such as 121, 341 …etc.

b. To have a positive two divisors, one is TREN & the second should be SVEN such as

21, 51, 91 …etc.

c. To have a positive two divisors, both are NIEN such as 81, 171 …etc.

d. If not one of the above then the number is a prime number that is ended by 1. Such

as 1,11,31,41.

OEN of the first state can be describes as:

(10N + 1) * (10M + 1) = (10L + 1) where N,M,L ε {1,2,3,….} eq.1

again N, M & L are indices of numbers all ended by 1 i.e. OEN.

OEN of the second state can be describes as:

(10N + 3) * (10M + 7) = (10L + 1) where N,M,L ε {1,2,3,….} eq.2

N is an index of a TREN number , M is an index of a SVEN number while L is an index of

OEN number.

OEN of the third state can be describes as:

(10N + 9) * (10M +9) = (10L + 1) where N,M,L ε {1,2,3,….} eq.3

N, M are indices of NIEN numbers while L is an index of OEN number.

Now the forth state which is prime numbers that are ended by 1. Cannot be described using a

formula. They are the numbers that remains in OEN set after taking out the three other states.

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Same technique can be applied on TRED, SVEN & NEIN numbers.

Figure 3: Bright dots are NON-PRIME OEN numbers

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An TRED number can have 3 states:

a. To have a positive two divisors, one is TREN & the second should be OEN such as 33,

143 …etc.

(10N + 1) * (10M + 3) = (10L + 3) where N,M,L ε {1,2,3,….} eq.4

b. To have a positive two divisors, one is SVEN & the second should be NEIN such as

63, 133, 153 …etc.

(10N + 7) * (10M + 9) = (10L + 3) where N,M,L ε {1,2,3,….} eq.5

c. If not one of the above then the number is a prime number that is ended by 3. Such

as 3, 13, 23, 43 …etc.

An SVEN number can have 3states:

a. To have a positive two divisors, one is SVEN & the second should be OEN such as 77,

187, 217 …etc.

(10N + 7) * (10M + 1) = (10L + 7) where N,M,L ε {1,2,3,….} eq.6

b. To have a positive two divisors, one is TREN & the second should be NEIN such as

27, 57, 117 …etc.

(10N + 3) * (10M + 9) = (10L + 7) where N,M,L ε {1,2,3,….} eq.7

c. If not one of the above then the number is a prime number that is ended by 7. Such

as 7, 17, 37 …etc.

An NEIN number can have 4 states:

a. To have a positive two divisors, both are TREN such as 9, 39, 169 …etc.

(10N + 3) * (10M + 3) = (10L + 9) where N,M,L ε {1,2,3,….} eq.8

b. To have a positive two divisors, one is OEN & the second should be NIEN such as

209, 319, 589 …etc.

(10N + 1) * (10M + 9) = (10L + 9) where N,M,L ε {1,2,3,….} eq.9

c. To have a positive two divisors, both are SVEN such as 49, 119, 629 …etc.

(10N + 7) * (10M + 7) = (10L + 9) where N,M,L ε {1,2,3,….} eq.10

d. If not one of the above then the number is a prime number that is ended by 9. Such

as 19, 29, 59, 79 …etc.

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Divisors Waves

There are two reasons behind why prime numbers are not distributed uniformly:

1- Prime numbers ended with 1 and 3 and 7 & 9 are all handled together as if they were

related, while there are not. i.e. there is no relation between 13 & 19.

2- Taking numbers ends with 1 OEN:

a. Non prime numbers as defined above “Stated of OEN Numbers” are controlled

by three equations eq. 1, & eq.2 & eq.3 and these equations are not dependent

on each others.

If we study these multiples and there intersections with numbers ends with 1 axis, we will find it

as waves each wave has a wave length and a phase.

Let us start by numbers ended by 1.

The pattern is described as a start –phase- and a length –wave length-. The start is given as

number index i.e. for number (10L +1) L is the number index.

The patterns are

a. (1,11) , (2,21), (3,31) …… ( n , (10n + 1)) pattern 1

b. (2,3) , (9,13), (16,33) ……. ( 7n + 2 , (10n + 3)) pattern 2

c. (8,9), (17,19), (26,29) …… ( 9n + 8 , (10n + 9)) pattern 3

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What we see here is read as follows: There is a wave start at index 1 which is number “11” and

replicated every 11’th index which means at index 1 + 11 = index 12 = number 121, and index 1

+ 11 + 11 = index 23 = number 231 ….etc. this is only generated from (1,11).

Same technique for (2,21) the first number is index 2 = number 21 and the second is index 23 =

231 and the third number is index 44 = number 441 and they are all can be divided by 21 the

original wave phase “the index 2”.

It is logic to say that these

Only in pattern 1 the start point in the pattern is a Prime Number, unless it exists in pattern 2 or

pattern 3. For example 21 is prime for pattern 1 but could be reached by pattern 2 in (2,3)

where 2 is the index number of 21.

In another way, any prime number is a start of a wave with a length of is value, for example 11

starts a wave from prime_index 1 with wave length = 11. In case of number 21 it is the same,

but it is hit by a wave comes from 3.

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Another interesting thing that is logic to expect is the symmetric in above table for values 1 x 1

& 9 x 9 where for numbers ended by 3 & 7 it is not symmetric.

List of All Waves:

OEN Waves:

Start Point Wave Length or Step Examples

N 10 * N + 1 (121,231,…) OEN x OEN

7 * N + 2 10 * N + 3 (21,51,…) TRED x SVEN

9 * N + 8 10 * N + 9 (81,171,361,…) NIEN x NIEN

N is index of a OEN number

TREN Waves:

Start Point Wave Length or Step Examples

N 10 * N + 3 (33,143,…) TRED x OEN

9 * N + 6 10 * N + 7 (63,133,153, …) SVEN x NIEN

N is index of a TRED number

SVEN Waves:

Start Point Wave Length or Step Examples

N 10 * N + 7 (77, 187, …) SVEN x OEN

9 * N + 2 10 * N + 3 (27, 57,…) TRED x NIEN

N is index of a SVEN number

NIEN Waves:

Start Point Wave Length or Step Examples

N 10 * N + 9 (99,189, 209,…) NEIN x OEN

3 * N 10 * N + 3 (39,169, …) TREN x TREN

7 * N + 4 10 * N + 7 (119,287, …) SVEN x SVEN

N is index of a NIEN number

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A Hint on Waves Formula

Take wave result in multiplying SVEN number by TRED number to generate OEN non prime.

Wave is (7 * N + 2 , 10 * N + 3).

When substituting in this wave :

Use N =0

7 * N + 2 = 2 This is the first index which equals to number (10 * N + 1) = 21.

Then increment with step (10 * N + 3) => 2 , 2 + 3, 2 + 3 + 3 …etc. => 21, 51, 81 …..etc.

Then increment N by one to go to the second wave.

7 * N + 2 = 9 which is equal to number 91

Then increment with step (10 * N + 3) => 9 , 9 + 13, 9 + 13 + 13 …etc. => 21, 221, 351 …..etc.

NOTE: for OEN x OEN we skip the very first result as it could be prime, unless it is intersected

with a wave comes from TRED or SVEN.

Same done with TRED , SVEN & NIEN numbers.

These waves enable the generation of non-prime number in one step by substituting in N of the

desired formla. We will use these wave formulas to generate Ulma Spiral Diagram directly

without the need to use classical approach of using isPrime function.

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Ulman Spiral Pattern Explained

As we can see in figure 5, even

numbers in Ulman spiral define a

well-formed pattern.

Again in figure 6 we can see a well-

formed pattern of numbers that are

multiples of 5. This by the way

including numbers ended by 0 which

are even numbers as well, so parts

of these patterns are overlapped.

Actually even if we remove 10’s

from figure 6, we will still get a

pattern, it is only a matter of

changing part of the patter into

black.

If we combine even numbers and

numbers ended by 5 in one

diagram as in figure 6 we can see

diagonal lines appear.

Now coming to less well-formed

patterns, they are patterns that can

be spotted in OEN, TRED, SVEN &

NEIN. There reason behind this is

that these set of numbers which

contains prime numbers in between,

and due to waves equations

mentioned above, waves

intersections that comes from other

sets helps to reduce the

homogeneity and continuity of the

pattern.

Figure 5: Red are even numbers.

Figure 4: Highlighted dots are numbers ended by 5 or 0. i.e. has 5 as a positive divisor

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In figure 13,14,15,16 we started

added OEN, NIEN, TREN & SVEN in

order to data in figure 6.

Figure 8: Green dots are non-prime numbers ended by 3 Figure 7: Yellow dots are Non-Prime numbers ended by 7

Figure 6: Even & numbers ended by 5 are plotted in white.

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Figure 10: A merge between Non-Prim TRED & SVEN. Figure 9: Purple dots are non-prime numbers ended by 9.

Figure 13: Adding OEN to Even & 5s Figure 14: OEN & NIEN added to even & 5's

Figure 12: Ulam Spiral Figure 11: OEN, TREN, NIEN added to even & 5's

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Conclusion

Ulam Spiral can be generated without the need to have a list of prime numbers or test if a

number is a prime or not.

Diagonal patterns are due to non-prime even numbers, and non-prime numbers ended by 5.

OEN, TREN & NIEN sets have waves of multiplies with a defined formula , that can be used to

generate non-prime numbers without directly without the need to test if the generated number

is prime or not, or the need of generate these numbers in sequence. We can choose N as any

arbitrary positive integer and substitute in Wave formulas to get a non-prime number.

Adding non-prime (OEN, TREN, NIEN) numbers just make distortion to this pattern and that

gives the impression that prime numbers in Ulman Spiral are related together in lines. While it is

vice versa, even numbers and numbers ended by 5 create the pattern and prime number just

distorted it.

Python Script to Generate Ulam is here

https://www.dropbox.com/s/kl5fzccigl0132g/UlamSpiralGenerator.v1.py?dl=0

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References

Prime Number Validator – Author Mohammad Hefny

http://www.slideshare.net/MHefny/prime-number-validator

Wikipedia (Ulam Spiral): http://en.wikipedia.org/wiki/Ulam_spiral

The Distribution of Prime Numbers on the Square Root Spiral - Authors Harry K. Hahn

http://arxiv.org/ftp/arxiv/papers/0801/0801.1441.pdf