Add new comment

Permalink

I read about this in Parc, S., “50 Visions of Mathematics”, Oxford University Press, (2014) and I wondered if this was true for any number base rather than just base 10.
On trying three digit numbers in octal and hexadecimal I found that they each had an equivalent kernel. This led me to solve the equations for finding the kernel, but using a variable, β, for the number base rather than restricting it to decimal numbers.

The results were as follows:
For two digit numbers there is a kernel if the base is of the form 3*(n – 1) + 5
For three digit numbers there is a kernel if the base is even i.e. of the form 2*n
For four digit numbers there is a kernel if the base is of the form 5*n, though there are kernels for base 2 and base 4.
For five digit numbers there are kernels for bases of the form 3*n

As I was using the computer algebra add-on for Microsoft Word I chose not to solve individually the hundreds of equations for six digit numbers. However, solving the equations based on the base-ten six digit numbers I did find that there are kernels for bases of the form 9*n + 8, 2*n, 15*n + 10, and 2*(n + 2).
In the above n takes the values 0, 1, 2, ...

I haven’t a proof, but it does seem likely that for any value with 3*n digits and an even base, there is a kernel which has n digits of value β – 1, β/2, and β/2 – 1. This is true for three digit numbers, six digit numbers, and some nine and twelve digit numerical examples I’ve tried.

I’ve not seen any references to this so I thought I would post these results as they may be of interest to other readers.

Philip Hickin
UK

Filtered HTML

  • Web page addresses and email addresses turn into links automatically.
  • Allowed HTML tags: <a href hreflang> <em> <strong> <cite> <code> <ul type> <ol start type> <li> <dl> <dt> <dd>
  • Lines and paragraphs break automatically.
  • Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.

  • What do chocolate and mayonnaise have in common? It's maths! Find out how in this podcast featuring engineer Valerie Pinfield.

  • Is it possible to write unique music with the limited quantity of notes and chords available? We ask musician Oli Freke!

  • How can maths help to understand the Southern Ocean, a vital component of the Earth's climate system?

  • Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues from SBIDER about the COVID models that fed into public policy.

  • PhD student Daniel Kreuter tells us about his work on the BloodCounts! project, which uses maths to make optimal use of the billions of blood tests performed every year around the globe.