## The Sphenic Numbers

April 16th, 2008 | Categories: general math, Wolfram Demonstrations | Tags:

In the recent 30th edition of the Carnival of Mathematics, the author mentioned that 30 is the first Sphenic number. I had never heard the term before so I thought I would investigate them a bit – nothing serious, just a bit of googling. The Wikipedia page was the first hit and this told me that Sphenic Numbers are positive integers that are the product of three distinct primes.

The article also demonstrated that Sphenic Numbers have exactly 8 divisors and gave some other bits of trivia such as the largest known example of a Sphenic Number (the product of the 3 largest known primes). There was also a link to the sequence of sphenic numbers on The On-Line Encyclopedia of Integer Sequences and that was about it. Oddly for something so elementary – there was nothing about Sphenic Numbers on Mathworld although this may change now that my Wolfram Demonstration on Sphenic Numbers has been published.

I was curious about what had written about Sphenic numbers in the literature so I googled the term on Google Scholar – amazingly there was not a single hit. So it seems that these numbers were considered important enough by someone to give them their own name but no one has then used that name in the literature….EVER!

How about books? Searching for “Sphenic Numbers” in Google Books also results in no hits. “Sphenic Number” results in one hit for a book called “Worlds of If” – No preview is available and the only quote you can get is “A sphenic number is one with unequal factors”

So – Where on earth did the name come from? Wikipedia says that its from the old greek word ‘sphen’ which means wedge but what I really want to know is who first gave these numbers this name? Why are there no references to the name in the literature? Does anyone know any interesting theorems concerning Sphenic Numbers?

Finally, is there a name for numbers that are the product of 4 distinct primes, or 5, or 6?

1. I don’t know what use an appreciation of Sphenic Numbers is, however I would point out that if you multiplied 4 primes then you would have 16 divisors (2^4), and if you multiplied 5 primes then you would have 32 divisors (2^5); and presumably so on!

Going backwards this would prove that the product of 2 primes can only be factored by themselves because there are only 2^2 divisors = 4 (and we can ignore the product itself and 1, which leaves 2 divisors – the 2 primes); and this will be a great relief to encryption experts!

Come on maths guys – what use is this knowledge?