# The formula for prime numbers

The hardest part about finding the relationship of primes is that the only relationship they have it having only factors of 1 and itself. Instead of finding a relationship with them, find a relationship of composites using the primes.

Most of you have seen this. I didn't realize this until playing around with prime numbers last week. They all fall into a chart of base 6.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

We know column 2, 4 and 6 get eliminated being a factor of 2 Column 3 is always a factor of 3. That leaves us with column 1 and 5

1 5 7 11 13 17 19 23 25 29 31 35

The process for finding the primes in those columns is by eliminating the composites.

We know the first series of primes are after 2 and 3( they eliminated themselves) are 5, 7, 11 and 13. We need two formulas. One for each column.

All primes squared fit into the first column(6n+1)

5^2 = 25 = 4(6)+1 7^2 = 49 = 8(6)+1 11^2 = 121 = 20(6)+1 13^2 = 169 = 28(6)+1

Since all the primes fit in the first column the formula would relate to the squares. This formula will predict all future numbers with a factor of that prime. Prime squared + 6(Prime)(N-1)

First column values 1
7prime
13prime
19prime
25eliminated
31prime
37prime
43prime
49eliminated
55eliminated
61prime
67prime
73prime
79prime
85eliminated
91eliminated
97prime
103prime
109prime
115eliminated
121eliminated
127prime
133eliminated
139prime
145eliminated
151prime
157prime
163prime
169eliminated
175eliminated(5,7)
181prime
187eliminated
193prime
199prime
205eliminated
211prime

5^2 + 6(5)(1-1)= 25

5^2 + 6(5)(2-1) = 55

5^2 + 6(5)(3-1) = 85

5^2 + 6(5)(4-1) = 115

5^2 + 6(5)(5-1) = 145

5^2 + 6(5)(6-1) = 175

5^2 + 6(5)(7-1) = 205

7^2 + 6(7)(1-1) = 49

7^2 + 6(7)(2-1) = 91

7^2 + 6(7)(3-1) = 133

7^2 + 6(7)(4-1) = 175

11^2 + 6(11)(1-1) = 121

11^2 + 6(11)(2-1) = 187

13^2 + 6(13)(1-1) = 169

13^2 + 6(13)(2-1) = 247

17^2 + 6(17)(1-1) = 289

The 5th column would be 6n-1. It's formula is 6(Prime)(N) - (Prime - closest factor to 6)Prime I'm not sure how to show closest factor to 6 in a formula. For Prime 7 you would subtract 6 remaining value of 1 For Prime 11 you would subtract 12 with a value of -1 For prime 29 you would subtract 30 with a value of -1 I'm sure you understand that now.

Fifth column values 5prime
11prime
17prime
23prime
29prime
35eliminated(5,7)
41prime
47prime
53prime
59prime
65eliminated(5,13)
71prime
77eliminated(7,11)
83prime
89prime
95eliminated
101prime
107prime
113prime
119eliminated(7,17)
125eliminated
131prime
137prime
143eliminated(11,13)
149prime
155eliminated
161eliminated
167prime
173prime
179prime
185eliminated
191prime
197prime
203eliminated
209eliminated
215eliminated

6(Prime)(N)-(Prime-6)Prime

6(5)(1) - (5-6)5 = 30-(-1)5 = 30+5 = 35

6(5)(2) - (5-6)5 = 60-(-1)5 = 60+5 = 65

6(5)(3) - (5-6)5 = 90-(-1)5 = 90+5 = 95

6(5)(4) - (5-6)5 = 120-(-1)5 = 120+5 = 125

6(5)(5) - (5-6)5 = 150-(-1)5 = 150+5 = 155

6(5)(6) - (5-6)5 = 180-(-1)5 = 180+5 = 185

6(5)(7) - (5-6)5 = 210-(-1)5 = 210+5 = 215

6(7)(1) - (7-6)5 = 42-(1)7 = 42-7 = 35

6(7)(2) - (7-6)5 = 84-(1)7 = 84-7 = 77

6(7)(3) - (7-6)5 = 126-(1)7 = 126-7 = 119

6(7)(4) - (7-6)5 = 168-(1)7 = 168-7 = 161

6(7)(5) - (7-6)5 = 210-(1)7 = 210-7 = 203

6(11)(1) - (11-12)11 = 66-(-1)11 = 66+11 = 77

6(11)(2) - (11-12)11 = 132-(-1)11 = 132+11 = 143

6(11)(3) - (11-12)11 = 198-(-1)11 = 198+11 = 209

6(13)(1) - (13-12)13 = 78-(1)13 = 78-13 = 65

6(13)(2) - (13-12)13 = 156-(1)13 = 156-13 = 143

6(13)(3) - (13-12)13 = 234-(1)13 = 234-13 = 221

6(17)(1) - (17-18)17 = 102-(-1)17 = 102+17 = 119 Note by Jonathon Eeftens
6 years, 3 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

The 1-36 up to was supposed to be in a 6 x 6 grid. Doing a copy and paste didn't seem to work out.

- 6 years, 3 months ago

The above work has already being presented last year in india in AMTI National maths conference.

- 6 years, 3 months ago

Really? I thought I actually had something

- 6 years, 3 months ago

It was just some different manner I means he showed using sets

- 6 years, 3 months ago