**Prime Numbers in Binary**

We usually see numbers represented in base 10 "decimal" notation and almost all prime number lists use base 10. But we can represent numbers in any base we please. In base 10 we use 10 symbols 0,1,2,3,...,9 and in base n we use n symbols 0,1,2,3,...,(n-1)

The simplest base is 2, because in that base we have only 2 symbols 0,1

Base 2 is also call "binary" and writing numbers in binary makes them look like computer data.

In binary the positions represent 1,2,4,8,16,32,.. so the general representation of a positive integer n is..

n=sum{ai*2^i} where the coefficients {ai} are all 0 or 1 and the sum is over i from 0 to some upper limit.

Notice that, despite the left to right standard mathematical summation above, many people use the right to left notation when writing binary numbers. I don't like it, but I'll use it anyway!

So here's the first 50 prime numbers in binary..

2 10

3 11

5 101

7 111

11 1011

13 1101

17 10001

19 10011

23 10111

29 11101

31 11111

37 100101

41 101001

43 101011

47 101111

53 110101

59 111011

61 111101

67 1000011

71 1000111

73 1001001

79 1001111

83 1010011

89 1011001

97 1100001

101 1100101

103 1100111

107 1101011

109 1101101

113 1110001

127 1111111

131 10000011

137 10001001

139 10001011

149 10010101

151 10010111

157 10011101

163 10100011

167 10100111

173 10101101

179 10110011

181 10110101

191 10111111

193 11000001

197 11000101

199 11000111

211 11010011

223 11011111

227 11100011

229 11100101

Writing numbers in binary can help spot patterns we might not notice in other bases. For example..

In binary all prime numbers except 2 begin and end with 1.

The first 2 digits of the prime 71 is the prime 3 and the last 5 digits is the prime 17. So we could define a "+" operation and say that 3+17=71. Notice that the + operation depends on order, so 17+3=113 is different, but it's still a prime!

The prime 13 is just the prime 11 written backwards. The same is true for 23 and 29 and lots more. Many primes are just an earlier prime written backwards!

Some primes have all digits set to 1, so these primes are of the form (2^n)-1 where n is just the number of binary digits. Primes of this form are called Mersenne primes, named after Marin Mersenne, a French monk who studied them in the 17th century.

Here's another interesting thing. The twin prime pair (1049,1051)=(10000011001,10000011011) in binary. Now concatenate the binary bit streams to get 1000001100110000011011=2149403 and this is also a prime! Primes generating primes. Quite amazing.

I wonder what we would discover if we used sophisticated computer pattern recognition on prime numbers in binary format?

Content written and posted by Ken Abbott abbottsystems@gmail.com

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