Wednesday, September 23, 2009

Name your Number : Great Joy !

Recently I stumbled on site named WikiDump that loosely archives interesting but deleted items from Wikipedia. There I found an an entry on the 'Zuckerman number' that intrigued me.

A Zuckerman number is an integer that is divisible by the product of its digits in a given number base. All integers between 1 and the base number are Zuckerman numbers. No integer with a zero as one or more of its digits in base b can be a Zuckerman number in that base. In base 10, the first few Zuckerman numbers with more than one digit are 11, 12, 15, 24, 36, 111, 112, 115, 128, 132, 135, 144, 175, 212, 216, 224, 312, 315, 384.

And here is the Wikipedia entry that discusses the deletion.
What I considered strange about this is that the definition of this 'special numbers' seems quite valid. Although the usefulness of this (other than making it a programming exercise) seems arbitrary, I see no reason why mr. Zuckerman does not deserve his small area of fame for discovering it. So I decided to dig deeper...
The definition can also be found on PlanetMath, which claims to reference 'J. J. Tattersall, Elementary number theory in nine chapters, p. 86. Cambridge: Cambridge University Press (2005)'
Now this book is by no means a fake or in any way humorous collection of real mathematics. I briefly went over it and checked some facts and biographies, and it all seems pretty legit. The tricky part is that on 'Google Books' the referenced page 86 is not present. Mr Zuckerman himself though is present on several other pages, proving that he is in fact a significant mathematician. In fact he was co-author of 'An Introduction to the Theory of Numbers' (By Niven, Zuckerman and Montgomery) which is a book that is often referred to in math studies.
But when we search for the other author, Ivan Niven, we find that there is actually something called a 'Nivenmorphic' number. And the definition for this is ' an integer that is divisible by the sum of its digits when written in that base'. Which by now sounds familiar. To add a little confusion though, the official name for this seems to be a 'Harshad number'. And Harshad means 'great joy' in Sanskrit. So who said mathematics was no fun...?

Edit 2020: This has been on my blog quite some time, and only now I see my mistake: Zuckerman numbers are divisible by the PRODUCT, and Harshad numbers by the SUM of their digits ! So they are actually completely different. And if you google the 'Zuckerman Number' you will find it is well represented throughout the internet.

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