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2015

Ulam Spiral Hidden

Patterns & Waves

Author: Mohammad Hefny

Ulam Spiral Hidden Patterns

Page 1 of 15

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

Page 2 of 15

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

Ulam Spiral Hidden Patterns

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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 {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 {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 {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 {1,2,3,.}

Figure 2: Classifying numbers by last digit

Ulam Spiral Hidden Patterns

Page 4 of 15

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 {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 {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 {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.

Ulam Spiral Hidden Patterns

Page 5 of 15

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

Figure 3: Bright dots are NON-PRIME OEN numbers

Ulam Spiral Hidden Patterns

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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 {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 {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 {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 {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 {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 {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 {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.

Ulam Spiral Hidden Patterns

Page 7 of 15

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

Ulam Spiral Hidden Patterns

Page 8 of 15

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

replicated every 11th 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.

Ula