It’s 11/11/11 11:11:11.111111… UTC

In honour of this auspicious moment here is some fun binary maths.

What is this sequence 2,3,7,11,13,19,25,… ?

It is actually the sequence of binary dropped primes. If you want a longer list it is Sloane sequence A014580 These are the primes when you multiply in binary but always drop the carry bits. For example 3 x 3 = 112 x 112 = 1012 = 5. I dropped a carry that would normally add 4 to the answer so 5 is not a binary dropped prime. 9 is not prime (I will drop the “dropped” from now on) either, because it is 9 = 3 x 7.

A curious feature of these primes is that if you reverse the digits in binary you always get another prime so 11 = 10112 and 13 = 11012 are both primes. All primes except 3 must have an odd number of non-zero binary digits because otherwise they are a multiple of 3.

In honour of the date and time it seems appropriate to factorise some numbers that consist of all ones in binary

1111 = 11 x11 x 11

111111 = 11 x 111 x 111

1111111 = 1011 x 1101

11111111 = 11 x 11 x 11 x 11 x 11 x 11 x 11 x 11

Others that I missed are prime. You can probably see that a number in this form can only be a binary dropped prime it the number of digits is a normal prime, but the converse is not true because 127 = 11 x 13 in decimal terms

As well as multiplication you can also do binary dropped addition and this gives you a ring which is a unique factorization domain. Don’t worry about subtraction because it is the same as addition. You can also do modulo arithmetic with the rule that a = b (mod n) iff a+b is a multiple of n.

Are these useful or are they just a curiosity for setting math puzzles? In fact they can be used to generate interesting cyclic binary linear codes . Binary linear codes are sets of codewords consisting of sequences of bits that are closed under binary dropped addition. There are always 2k codewords for some k in such a binary linear code and if the block length is n it can be used to code k bits into n bits in a way that is easy to reverse and very efficient for error-correcting. Cyclic codes have the extra property that you can cycle the digits of the code by shifting them left and moving the last digit to the first. In a cyclic code, when you cycle a codeword in this way you always get another codeword in the set.

The simplest non-trivial cyclic codes are parity codes where the number of non-zero bits in a codeword is always even, In this case n = k+1 so encoding consists of adding a parity bit to a k-bit string. Now remember that a number with an even number of non-zero bits is always a multiple of 3 in binary dropped arithmetic, so can we generate other cyclic codes by using multiples of other numbers in binary dropped arithmetic?

(11 minutes and 11 .111 seconds to go, now observing a minutes silence…)

The answer is yes, but it only works when the numbers are factors of the numbers which are sequences of ones. This is because cycling the numbers is equivalent to working modulo such a number. So we can have codes with a block length that are a multiple of three where all codewords are multiples of 111. More interesting examples are codes with 7 bits using multiples of 1011 or 1101. This is the familiar Hamming code Ham(7,4) which is linked to the Fano plane and the octonions. Even better a string of 23 ones must factorize in binary dropped arithmetic into two number of 12 binary digits, because this would be needed for the Golay code, but it is 11.11.1111… so I will have to check the factorization another time.


6 Responses to It’s 11/11/11 11:11:11.111111… UTC

  1. Olaf Schumann says:

    From Germany, especially from Cologne, I only could say: Kölle Alaaf!!!

    See: http://en.wikipedia.org/wiki/Carnival
    In Cologne, it starts at on Nov 11th at 11:11 ;-)

  2. J. S. Markovitch says:

    111.111/3+11.111*3^2=137.036

  3. Hi,

    You have shown me some interesting things I did not know but could have used. But since you are interested in these things I found some rather interesting new things about Fibonacci numbers in my casual study of the primes. Also, I had some general comments on how it seems people do not really understand the complex numbers in relation to tachyon ideas (Kasner, not my idea)

    That symmetry of primes in binary- well, 11 and 13 pairs? I should look into this.

    Which rather large prime was made of 111111… ?

    Happy 1/ .9000009 inverse !

    http://www.pesla.blogspot.com today.

  4. Mike says:

    The first electron shell of the s and p orbitals up to Z = 37 in the periodic table corresponds to an atomic number which is prime:

    Z: Electronic Configuration
    _____________________
    1: 1s1
    3: 2s1
    5: 2p1
    11: 3s1
    13: 3p1
    19: 4s1
    31: 4p1
    37: 5s1

  5. Mike,

    An interesting observation on the so called Magic Numbers of the electron configuration.

    But it is not all just about primes. Of course in the ideal model we have variations due to the temperature of the atoms.

    Note also that from my view in four dimensional (normal) matter the possible number of elements (Z) is equal to 120. But from the current view if there is a limit is would be at 136 of course.

    The PeSla

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