What do the numbers mean on the coat of arms of David Johnston, Canada’s Governor-General?

Official car of the Canadian Governor-General (photo by adbdallahh - CC-BY)

Canada’s Governor-General, David Johnston, like all Governors-General, has an official coat of arms.

David’s coat of arms is quite a busy one and includes stripes (representing family, knowledge, order and organization), a crown (because the G-G represents the Queen), books (representing knowledge and the law), a bookshelf (representing the acquisition of knowledge) with five books (one for each of his daughters), two Latin mottoes, a couple of astrolabes (representing exploration), a pair of winged feet (representing physical activity and communication), a candle (representing learning and enlightenment), and … unicorns!

The Latin mottoes are “Desiderantes Meliorem Patriam”, meaning “they desire a better country”, which is the motto of the Order of Canada of which David Johnston is a member, and “Contemplare Meliora” which means “to envisage a better world”.

At the bottom is a black strip with some golden digits, all of which are zeroes or ones. This forms a binary number 33 bits long (a bit is a binary digit). That’s an unusual length, because the main use for binary numbers is in computing where the most common lengths are 8, 16, 32 or 64 bits.

The official description of the coat of arms simply says this:

The wavy band inscribed with zeros and ones represents a flow of information, digital communication and modern media.

But why was this particular combination of ones and zeroes chosen: 110010111001001010100100111010011 ? David Johnston was president of the University of Waterloo and is a bright guy. This won’t just be some strong of digits that looked good to a graphic designer.

If we convert that number to decimal notation we get 6830770643. On the face of it that’s uninteresting, but it turns out that it is a prime number. In other words, no other numbers can be multiplied together to get that number.

As numbers get larger and larger, prime numbers become more scarce. Nevertheless, it’s not hard to prove that there are an infinite number of prime numbers.

Take another look at that 33-digit binary number, and you’ll see that it reads the same forwards and backwards. That makes it a palindrome (like, for example, the name Hannah). To be more specific, it is a palindromic binary number that is also prime.

These palindromic primes (also known as palprimes) are of interest to mathematicians because they are rare. Almost all palindromic numbers are composite, i.e. non-prime (here’s a proof).

Finally, 6830770643 is approximately the human population of our planet. The global population is estimated to have passed that figure during 2010, the year this coat of arms was adopted.

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