## What is interesting about the number 2009?

This time last year I asked the question ‘What is interesting about the number 2008?‘ and it turned out to be rather popular so I thought I would do the same with 2009. Of course definitions of ‘interesting’ vary but for what what it is worth here are a few things that fit my definition of the term.

- Both 2009 and its reversal 9002 are multiples of 7.
- The 2009th prime (17471) is palindromic.
- You can express 2009 as the sum of 4 positive cubes in exactly 3 ways. (Kudos to anyone who can come up with these 3 ways)

**Non Mathematical fact about 2009:** 2009 is a blue moon year.

**Quiz**: 2009 is (probably) the 44th term of a sequence that begins 1789, 1797, 1801, 1809. What is this sequence?

If you know of any mathematical reasons why 2009 might be considered as interesting then please let me know in the comments.

**Update 31st Dec 2009: **Thanks to the great site Number Gossip, I have discovered that

- 2009 is an Apocalyptic Number.
- 2009 is an Evil Number (for a particular definition of evil at least- there seem to be several)

Oh dear, I fear that numerology cranks are going to have a field day.

10^3 + 10^3 + 2^3 + 1^3 = 12^3 + 6^3 + 4^3 + 1^3 = 10^3 + 9^3 + 6^3 + 4^3

Can you find 4 ways of writing 2009 as a sum of 6 cubes of positive integers?

Can you find 5 ways of writing 2009 as a sum of 7 cubes of positive integers?

Can you write 2009 as the sum of 2 squares?

There are three different ways to express 2009 as the sum of 4 positive cubes:

{1^3, 2^3, 10^3, 10^3} = {1, 8, 1000, 1000}

{1^3, 4^3, 6^3, 12^3} = {1, 64, 216, 1728}

{4^3, 6^3, 9^3, 10^3} = {64, 216, 729, 1000}

Mathematica code:

((Sort/@({x,y,z,k}/.(ToRules/@Reduce[x^3+y^3+z^3+k^3==2009&&x>0&&y>0&&z>0&&k>0,{x,y,z,k},Integers])))//Union)^3

6 cubes:

{{1, 3, 4, 4, 5, 12}, {1, 4, 6, 6, 8, 10}, {3, 4, 4, 5, 9, 10}, {5, 7, 7, 7, 7, 8}}

7 cubes:

{{1, 1, 3, 5, 7, 8, 10}, {1, 3, 3, 4, 6, 7, 11}, {1, 5, 5, 7, 7, 7, 9}, {2, 3, 6, 7, 7, 7, 9}, {4, 4, 4, 4, 8, 8, 9}}

2 squares:

{28,35}

Hey Mike,

I’m reaching here, but some things that catch my eye about numbers are the number of digits and the sum of the digits.

For 2009, you might find it (mildly) interesting that the sum of the digits (11) is also the number of digits required to express 2009 in binary / base-2.

MATLAB code would go something like:

year = ‘2009’;

sum(str2num(year(:))) %gives sum of digits

numel(dec2bin(str2num(year))) %gives number of binary digits

Best,

Rob

Quiz: 2009 is (probably) the 44th term of a sequence that begins 1789, 1797, 1801, 1809. What is this sequence?

the Sequence is +8 +4 +8 …

hence the 44th term of the series is 2009 (11th set of 4 memebers of series)

Animesh

My fault,sorry…I should have given you more of the series since it doesn’t always go +8 +4 +8 +4……

Here is the full series.

1789, 1797, 1801, 1809, 1817, 1825, 1829, 1837, 1841, 1841, 1845, 1849, 1850, 1853, 1857, 1861, 1865, 1869, 1877, 1881, 1881, 1885, 1889, 1893, 1897, 1901, 1909, 1913, 1921, 1923, 1929, 1933, 1945, 1953, 1961, 1963, 1969, 1974, 1977, 1981, 1989, 1993, 2001, 2009

Nice post and comments, Mike! I particularly like what Sol came up with! I’m sure some teachers will be searching for websites that address this topic for the first day back. Despite the observations noted up to this point, let’s propose the following theorem:

There is nothing interesting about the number 2009.

Can one “logically” disprove this statement (rather than providing counterexamples)?

By the way, the logic involved is similar to proving that the set of “interesting numbers” is infinite. (Domain is restricted to positive integers)

Assume that N is the largest “interesting number”. Then N+1 would be the first “non-interesting number” such that all numbers greater than or equal to N are also non-interesting. This would certainly make N+1 “interesting!” This contradicts our assumption. Therefore…

@Rob Thanks for reaching :) One of the fun things about this sort of thing is that no one can agree on what ‘interesting’ means and so some people would read these properties and say ‘So what?’ whereas someone else will read them and say ‘That’s cool / interesting’. In a way, much of mathematics is like thatH

@Dave Thanks for your comments. As for your paradox – how do you define ‘interesting’? ;)

Mike,

I won’t attempt to define “interesting” but I’m sure my former students, my wife and any of my kids would define “interesting” as the opposite of what I find “interesting!

Dave’