Star of David Theorem

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star-of-davidHanukkah starts today at sundown, so in honor of the holiday here is the Star of David Theorem.  In simplest terms, this theorem says that the greatest common divisor of  {{n-1} \choose k}, {n \choose {k-1}}, and {{n+1} \choose {k+1}} is equal to the greatest common divisor of  {{n-1} \choose {k-1}}, {n \choose {k+1}}, and {{n+1} \choose k}.  To see why it’s called the Star of David, look at the following visual.  the greatest common divisor of the blue corners and the greatest common divisor of the purple corners are equal.  Together, the two triangles form the Star of David.

star-of-david

For example, if n=4 and k=2, this says that the greatest common divisor of  {3 \choose 2}, {4 \choose 1}, and {5 \choose 3} is equal to the greatest common divisor of  {3 \choose 1}, {4\choose 3}, and {5 \choose 2}.  As it turns out, this is one of the less interesting examples because both sides simplify to gcd(3,4,10), which is 1.  So let’s look for another example.

Where there are binomial coefficients, Pascal’s triangle can’t be too far behind.  Sadly, when the above Theorem is placed visually into Pascal’s triangle, it ends up looking kind of turned and squooshed.

pascal-star-1Visually, the top star illustrates the previous not-so-interesting example of how gcd(3,4,10) is equal to gcd(3,4,10).  But the lower star illustrates that that gcd(36,210,165) is equal to gcd(84,45,330), and this is a little less obvious.  In this second case, but greatest common divisors are equal to 3.

According to Wolfram’s Mathworld, the Star of David Theorem was first stated by H. W. Gould in 1972, and there were several generalization in the years immediately following.  Apprently the association with Pascal’s triangle wasn’t noticed until 6 years ago, however, by B. Butterworth in this article (which is originally about using Pascal’s triangle to illustrate the song “The Twelve Days of Christmas”).

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8 Responses to “Star of David Theorem”

  1. Star of David theorem — The Endeavour Says:

    [...] 360 blog has a post today about the Star of David Theorem. See the original post for an explanation of the figure below and to learn how this theorem relates [...]

  2. Michael Croucher Says:

    Hello

    This looks like a great post to add to the end of year Christmas carnival – is that OK? Feel free to submit others too.

    Merry Xmas.
    Mike

  3. Ξ Says:

    Mike — thanks, that’d be great! I’m glad to know you’re hosting the Carnival on Sunday!

  4. De stelling van de Davidsster at QED Says:

    [...] deze leuke stelling: de stelling van de Davidsster. Ik las hierover enkele dagen geleden op de blog 360. De stelling zegt: Om te zien waar de stelling zijn naam vandaan haalt, helpt het om de [...]

  5. Star of David Theorem « 7 Days and 8 nights Says:

    [...] source : threesixty360 [...]

  6. Star of David Theorems in Pascal Triangle | CTK Insights Says:

    [...] now known under the moniker of The Star of David Theorem. They were mentioned in the blogosphere here and [...]

  7. Deborah Says:

    How I perfectly numbered the Star of David using the 2 mystery numbers, #137 and #9

  8. I.Am.A.Jew Says:

    Cool!! I never knew you could do that. Why is it starring on Christmas??? O.O I’m going to go there.

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