## What is interesting about the number 2008?

December 28th, 2007 | Categories: general math | Tags:

While tidying up the attic I came across my copy of the 2nd edition of The Penguin Dictionary of Curious and Interesting Numbers by David Wells – it’s at the 4th edition now so an update of my library is well overdue. I love this book and have wasted many an hour flicking through it’s pages – which is why my attic is still such a mess. Since it is almost 2008 I wondered if there were any interesting facts about the number 2008 and moved to the relevant section in the book – nothing! The last year before 2008 that was listed was 1980:

“1980-0891=1089 which is one of only 5 patterns in which subtracting a 4 digit number from its reversal leaves the digits rearranged.”

The next entry was 2025 which implies that we have a lot of rather dull years ahead – (only numerically speaking I hope). Of course, as everyone knows, there is no such thing as an uninteresting number (check out the wikipedia page for The Interesting Number Paradox to find a “proof”) and so I embarked on a search to find out why the number 2008 was interesting (to me at least).

I failed! Here is the best I could do (thanks to this site):

2008 is a Kaprekar constant in base 3

So there you have it – 2008 is a boring number and to make things worse my attic is still untidy. Someone out there please try and prove me wrong – find something interesting about this number and let me know in the comments. If it passes my own personal test of ‘interestingness’ (loosely defined as anything that makes me say “thats cool”) then I will update this page with the results

Update

I have had a few great responses to this post most of which can be found in the comments section. My favourites are:

2008 is a Happy Number

There are exactly 1000 numbers less than 2008 that share no factors in common with 2008. That is, Euler’s totient function phi(2008) = 1000 (check here.) Would it be correct to say that 2008 is the beginning of the next Euler totient millenium then?

2008 is the prime number 251 multiplied by the sum its own digits.

Update 2 (Jan 8th 2008)

I have just discovered another blog post that has some fun facts about 2008 over at 360 – my favorite of which is

2008 can be written as the sum of of 16 consecutive positive integers: 118+119+120+…+132+133

1. I looked for sequences (by running a search engine, I didn’t really do any work) and here are a few notable bits.

If we start a Fibonacci sequence with 1, 8, 9,… what will the 14th term be?
If we count all the 4 X 4 matrices with nonnegative integer entries and row and column sums equal to 4?
If we count all the necklace structures using 9 beads of no more than 5 different colors?
If we look for the first prime number of the form 2^n + 3, where n is greater than 2000….

Slim pickings indeed.

2. Hmm, indeed.

2008 is the prime number 251 multiplied by the sum its own digits.

That’s all I could think of.

3. The sum of 251 is 8 which can be represented as interms of prime numbers
2^3

4. 2008 is a 336-gonal number.

5. There are exactly 1000 numbers less than 2008 that share no factors in common with 2008.

That is, Euler’s totient function phi(2008) = 1000 (check here.

6. Thanks for all of the comments so far – 2008 is slowly but surely starting to turn out to be more interesting than I first thought.

Here is another one – 2008 is a Happy Number.

7. Unfortunately, you can’t really say that 2008 is the beginning of the next “Euler totient millenum”, since it is actually the fifth number with a totient of 1000. In particular, phi(1111) = phi(1255) = phi(1375) = phi(1875) = phi(2008) = 1000. In Mathematica:

In:= Select[ Range, EulerPhi[#] == 1000& ]

Out= {1111, 1255, 1375, 1875, 2008}

But it’s still pretty cool. =)

8. Hi Brent

ack! I should have checked. Oh well – never mind. Thanks for your comment. I see you are a Mathematica fan too – have you got version 6?

9. Oh, I can’t believe I didn’t check to see if it was a Happy number, I’d just watched that episode of Doctor Who as well. That’s my favourite. That 2008 is provably a Happy New Year! *grin*

10. Related to the (1,8) Fibonacci Sequence mentioned above, and to supplement it a bit, 2008 is the 12th indexed number in the Fibonacci progression 8,9 (where 8 = index # 0). The progression goes:

8, 9, 17, 26, 43, 69, 112, 181, 293, 474, 767, 1241, 20008

You can create Fibonacci 8, 9 numbers (I personally call this progression the “Ionian Scale”) by adding together any alternating two Golden Scale numbers, much in the same manner one adds alternating Fibonacci Numbers to get a Lucas Number.

The Golden Scale?

2, 5, 7, 12, 19, 31, 50, 81… etc.

2 + 7 = 9, 5 + 12 = 17, etc.

It’s related to music theory. The pentatonic scale, for instance has 5 notes per octave. The diatonic scale has 7. Our standard issue chromatic scale has 12 (Do, Re, Mi, Fa….) If you do a search for:

music theory “2, 5, 7, 12”