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

## Comments

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TopNewestThe 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. – Jonathon Eeftens · 2 years, 11 months ago

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– Tushar Gautam · 2 years, 11 months ago

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

Really? I thought I actually had something – Jonathon Eeftens · 2 years, 11 months ago

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– Tushar Gautam · 2 years, 11 months ago

It was just some different manner I means he showed using setsLog in to reply