What is interesting about the number 2009?

December 28th, 2008 | Categories: general math | Tags:

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

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

  1. December 29th, 2008 at 01:13
    Reply | Quote | #1

    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?

  2. VolMike
    December 29th, 2008 at 10:45
    Reply | Quote | #2

    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}

  3. VolMike
    December 29th, 2008 at 10:47
    Reply | Quote | #3

    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

  4. VolMike
    December 29th, 2008 at 16:59
    Reply | Quote | #4

    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}

  5. December 30th, 2008 at 10:01
    Reply | Quote | #5

    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

  6. Animesh
    December 31st, 2008 at 12:33
    Reply | Quote | #6

    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)

  7. Mike Croucher
    December 31st, 2008 at 13:51
    Reply | Quote | #7

    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

  8. December 31st, 2008 at 15:02
    Reply | Quote | #8

    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…

  9. Mike Croucher
    December 31st, 2008 at 15:15
    Reply | Quote | #9

    @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’? ;)

  10. December 31st, 2008 at 16:50

    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’

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