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21 Dec

Star of David Theorems in Pascal Triangle

I am not sure who coined the term "The Star of David Theorem" to designate the identities discovered in the early 1970s. There are in fact two of them, both related to the "Star of David" configuration in Pascal triangle

(The diagram is courtesy wikipedia.org.)

The first result discovered by Hoggatt and Hansell in 1971 comes in a round-about way as a corollary of the following

Theorem 1

For n, m, 0 < n < m , m > 2, the product of the six binomial coefficients surrounding m \choose n is a perfect integer square.

The proof is a little bit strange in my view so that I reproduce it below:

Proof

The six binomial coefficients are:

\displaystyle{m-1 \choose n-1},{m-1 \choose n},{m \choose n+1},{m+1 \choose n+1},{m+1 \choose n},{m \choose n-1}

The product is

\displaystyle{m-1 \choose n-1}\times {m-1 \choose n}\times {m \choose n+1}\times {m+1 \choose n+1}\times {m+1 \choose n}\times {m \choose n-1}

which is equal to

\displaystyle\left[ \frac{(m-1)!m!(m+1)!}{(m-n-1)!(m-n)!(m-n+1)!} \right]^2

Since each binomial coefficient is an integer, the product is an integer, and since the square of a rational number is an integer if and only if the rational number is an integer, it follows that the product is an integer square.

Corollary

Each alternate triad of the six binomial coefficients have equal products.

Or, formally,

\displaystyle{m-1 \choose n-1}\times {m \choose n+1}\times {m+1 \choose n} = {m-1 \choose n}\times {m+1 \choose n+1}\times {m \choose n-1},

which can be verified directly, since each of the two products equals

\displaystyle\frac{(m-1)!m!(m+1)!}{(m-n-1)!(m-n)!(m-n+1)!}.

The round-about derivation may be a curiosity, but I believe that its form led H. W. Gould to a 1972 conjecture which has been proved by Hoggatt and Hillman later that year.

Theorem

\displaystyle\text{gcd}\left({m-1 \choose n-1}, {m \choose n+1}, {m+1 \choose n}\right) = \text{gcd}\left({m-1 \choose n}, {m+1 \choose n+1}, {m \choose n-1}\right).

Both theorems are now known under the moniker of The Star of David Theorem. They were mentioned in the blogosphere here and here.

References

  1. H. W. Gould, A New Greatest Common Divisor Property of The Binomial Coefficients, Fibonacci Quarterly 10 (1972), 579–584. (Part 1, Part 2)
  2. V. E. Hoggatt, Jr., W. Hansell, "The Hidden Hexagon Squares", Fibonacci Quarterly, Vol. 9, No. 2 (1971), pp. 120, 133. (Part 1)
  3. V. E. Hoggatt, Jr., A. P. Hillman, "Proof of Gould's Conjecture on Greatest Common Divisors", Fibonacci Ouarterly. Vol. 10, No. 6 (1972), pp. 565-568. (Part 1, Part 2)

2 Responses to “Star of David Theorems in Pascal Triangle”

  1. 1
    Math Teachers at Play 46 « Let's Play Math! Says:

    [...] Alexander explains the Star of David Theorems in Pascal Triangle. [...]

  2. 2
    Теорема (звезда Давида) | Математика, которая мне нравится Says:

    [...] http://www.mathteacherctk.com/blog/2011/12/star-of-david-theorems-in-pascal-triangle/ [...]

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