# Tag Archives: octal

# Ace Base

# Clock around the Rock

If you like minimalism, you should like binary. There is unsurpassable simplicity and elegance in the idea that any number can be reduced to a series of 1’s and 0’s. It’s unsurpassable because you can’t get any simpler: unless you use finger-counting, two symbols are the minimum possible. But with those two – a stark 1 and 0, true and false, yin and yang, sun and moon, black and white – you can conquer any number you please. 2 = 10_{[2]}. 5 = 101. 100 = 1100100. 666 = 1010011010. 2013 = 11111011101. 9^9 = 387420489 = 10111000101111001000101001001. You can also perform any mathematics you please, from counting sheep to modelling the evolution of the universe.

But one disadvantage of binary, from the human point of view, is that numbers get long quickly: every doubling in size adds an extra digit. You can overcome that disadvantage using octal or hexadecimal, which compress blocks of binary into single digits, but those number systems need more symbols: eight and sixteen, as their names suggest. There’s an elegance there too, but binary goes masked, hiding its minimalist appeal beneath apparent complexity. It doesn’t need to wear a mask for computers, but human beings can appreciate bare binary too, even with our weak memories and easily tiring nervous systems. I especially like minimalist binary when it’s put to work on those most maximalist of numbers: the primes. You can compare integers, or whole numbers, to minerals. Some are like mica or shale, breaking readily into smaller parts, but primes are like granite or some other ultra-hard, resistant rock. In other words, some integers are easy to divide by other integers and some, like the primes, are not. Compare 256 with 257. 256 = 2^8, so it’s divisible by 128, 64, 32, 16, 8, 4, 2 and 1. 257 is a prime, so it’s divisible by nothing but itself and 1. Powers of two are easy to calculate and, in binary, very easy to represent:

2^0 = 1 = 1

2^1 = 2 = 10_{[2]}

2^2 = 4 = 100

2^3 = 8 = 1000

2^4 = 16 = 10000

2^5 = 32 = 100000

2^6 = 64 = 1000000

2^7 = 128 = 10000000

2^8 = 256 = 100000000

Primes are the opposite: hard to calculate and usually hard to represent, whatever the base:

02 = 000010_{[2]}

03 = 000011

05 = 000101

07 = 000111

11 = 001011

13 = 001101

17 = 010001

19 = 010011

23 = 010111

29 = 011101

31 = 011111

37 = 100101

41 = 101001

43 = 101011

Maximalist numbers, minimalist base: it’s a potent combination. But “brimes”, or binary primes, nearly all have one thing in common. Apart from 2, a special case, each brime must begin and end with 1. For the digits in-between, the God of Mathematics seems to be tossing a coin, putting 1 for heads, 0 for tails. But sometimes the coin will come up all heads or all tails: 127 = 1111111_{[2]} and 257 = 100000001, for example. Brimes like that have a stark simplicity amid the jumble of 83 = 1010011_{[2]}, 113 = 1110001, 239 = 11101111, 251 = 11111011, 277 = 100010101, and so on. Brimes like 127 and 257 are also palindromes, or the same reading in both directions. But less simple brimes can be palindromes too:

73 = 1001001

107 = 1101011

313 = 100111001

443 = 110111011

1193 = 10010101001

1453 = 10110101101

1571 = 11000100011

1619 = 11001010011

1787 = 11011111011

1831 = 11100100111

1879 = 11101010111

But, whether they’re palindromes or not, all brimes except 2 begin and end with 1, so they can be represented as rings, like this:

Those twelve bits, or binary digits, actually represent the thirteen bits of 5227 = 1,010,001,101,011. Start at twelve o’clock (digit 1 of the prime) and count clockwise, adding 1’s and 0’s till you reach 12 o’clock again and add the final 1. Then you’ve clocked around the rock and created the granite of 5227, which can’t be divided by any integers but itself and 1. Another way to see the brime-ring is as an Ouroboros (pronounced “or-ROB-or-us”), a serpent or dragon biting its own tail, like this:

But you don’t have to start clocking around the rock at midday or midnight. Take the Ouroboprime of 5227 and start at eleven o’clock (digit 12 of the prime), adding 1’s and 0’s as you move clockwise. When you’ve clocked around the rock, you’ll have created the granite of 6709, another prime:

Other Ouroboprimes produce brimes both clockwise and anti-clockwise, like 47 = 101,111.

Clockwise

101,111 = 47

111,011 = 59

111,101 = 61

Anti-Clockwise

111,101 = 61

111,011 = 59

101,111 = 47

If you demand the clock-rocked brime produce distinct primes, you sometimes get more in one direction than the other. Here is 151 = 10,010,111:

Clockwise

10,010,111 = 151

11,100,101 = 229

Anti-Clockwise

11,101,001 = 233

11,010,011 = 211

10,100,111 = 167

10,011,101 = 157

The most productive brime I’ve discovered so far is 2,326,439 = 1,000,110,111,111,110,100,111_{[2]}, which produces fifteen distinct primes:

Clockwise (7 brimes)

1,000,110,111,111,110,100,111 = 2326439

1,100,011,011,111,111,010,011 = 3260371

1,110,100,111,000,110,111,111 = 3830207

1,111,101,001,110,001,101,111 = 4103279

1,111,110,100,111,000,110,111 = 4148791

1,111,111,010,011,100,011,011 = 4171547

1,101,111,111,101,001,110,001 = 3668593

Anti-Clockwise (8 brimes)

1,110,010,111,111,110,110,001 = 3768241

1,100,101,111,111,101,100,011 = 3342179

1,111,111,011,000,111,001,011 = 4174283

1,111,110,110,001,110,010,111 = 4154263

1,111,101,100,011,100,101,111 = 4114223

1,111,011,000,111,001,011,111 = 4034143

1,110,110,001,110,010,111,111 = 3873983

1,000,111,001,011,111,111,011 = 2332667

**Appendix: Deciminimalist Primes**

Some primes in base ten use only the two most basic symbols too. That is, primes like 11_{[10]}, 101_{[10]}, 10111_{[10]} and 1011001_{[10]} are composed of only 1’s and 0’s. Furthermore, when these numbers are read as binary instead, they are still prime: 11_{[2]} = 3, 101_{[2]} = 5, 10111_{[2]} = 23 and 1011001_{[2]} = 89. Here is an incomplete list of these deciminimalist primes:

11_{[10]} = 1,011_{[2]}; 11_{[2]} = 3_{[10]} is also prime.

101_{[10]} = 1,100,101_{[2]}; 101_{[2]} = 5_{[10]} is also prime.

10,111_{[10]} = 10,011,101,111,111_{[2]}; 10,111_{[2]} = 23_{[10]} is also prime.

101,111_{[10]} = 11,000,101,011,110,111_{[2]}; 101,111_{[2]} = 47_{[10]} is also prime.

1,011,001_{[10]} = 11,110,110,110,100,111,001_{[2]}; 1,011,001_{[2]} = 89_{[10]} is also prime.

1,100,101_{[10]} = 100,001,100,100,101,000,101_{[2]}; 1,100,101_{[2]} = 101_{[10]} is also prime.

10,010,101_{[10]} = 100,110,001,011,110,111,110,101_{[2]}; 10,010,101_{[2]} = 149_{[10]} is also prime.

10,011,101_{[10]} = 100,110,001,100,000,111,011,101_{[2]}; 10,011,101_{[2]} = 157_{[10]} is also prime.

10,100,011_{[10]} = 100,110,100,001,110,100,101,011_{[2]}; 10,100,011_{[2]} = 163_{[10]} is also prime.

10,101,101_{[10]} = 100,110,100,010,000,101,101,101_{[2]}; 10,101,101_{[2]} = 173_{[10]} is also prime.

10,110,011_{[10]} = 100,110,100,100,010,000,111,011_{[2]}; 10,110,011_{[2]} = 179_{[10]} is also prime.

10,111,001_{[10]} = 100,110,100,100,100,000,011,001_{[2]}.

11,000,111_{[10]} = 101,001,111,101,100,100,101,111_{[2]}; 11,000,111_{[2]} = 199_{[10]} is also prime.

11,100,101_{[10]} = 101,010,010,101,111,111,000,101_{[2]}; 11,100,101_{[2]} = 229_{[10]} is also prime.

11,110,111_{[10]} = 101,010,011,000,011,011,011,111_{[2]}.

11,111,101_{[10]} = 101,010,011,000,101,010,111,101_{[2]}.

100,011,001_{[10]} = 101,111,101,100,000,101,111,111,001_{[2]}; 100,011,001_{[2]} = 281_{[10]} is also prime.

100,100,111_{[10]} = 101,111,101,110,110,100,000,001,111_{[2]}.

100,111,001_{[10]} = 101,111,101,111,001,001,010,011,001_{[2]}; 100,111,001_{[2]} = 313_{[10]} is also prime.

101,001,001_{[10]} = 110,000,001,010,010,011,100,101,001_{[2]}.

101,001,011_{[10]} = 110,000,001,010,010,011,100,110,011_{[2]}; 101,001,011_{[2]} = 331_{[10]} is also prime.

101,001,101_{[10]} = 110,000,001,010,010,011,110,001,101_{[2]}.

101,100,011_{[10]} = 110,000,001,101,010,100,111,101,011_{[2]}.

101,101,001_{[10]} = 110,000,001,101,010,110,111,001,001_{[2]}.

101,101,111_{[10]} = 110,000,001,101,010,111,000,110,111_{[2]}; 101,101,111_{[2]} = 367_{[10]} is also prime.

101,110,111_{[10]} = 110,000,001,101,101,000,101,011,111_{[2]}.

101,111,011_{[10]} = 110,000,001,101,101,010,011,100,011_{[2]}; 101,111,011_{[2]} = 379_{[10]} is also prime.

101,111,111_{[10]} = 110,000,001,101,101,010,101,000,111_{[2]}; 101,111,111_{[2]} = 383_{[10]} is also prime.

110,010,101_{[10]} = 110,100,011,101,001,111,011,110,101_{[2]}.

110,100,101_{[10]} = 110,100,011,111,111,111,010,000,101_{[2]}; 110,100,101_{[2]} = 421_{[10]} is also prime.

110,101,001_{[10]} = 110,100,100,000,000,001,000,001,001_{[2]}.

110,110,001_{[10]} = 110,100,100,000,010,010,100,110,001_{[2]}; 110,110,001_{[2]} = 433_{[10]} is also prime.

110,111,011_{[10]} = 110,100,100,000,010,100,100,100,011_{[2]}; 110,111,011_{[2]} = 443_{[10]} is also prime.