Star of David theorem

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The Star of David theorem (the rows of the Pascal triangle are shown as columns here).

The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972.

Statement[edit]

The greatest common divisors of the binomial coefficients forming each of the two triangles in the Star of David shape in Pascal's triangle are equal:

Examples[edit]

Rows 8, 9, and 10 of Pascal's triangle are

1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1

For n=9, k=3 or n=9, k=6, the element 84 is surrounded by, in sequence, the elements 28, 56, 126, 210, 120, 36. Taking alternating values, we have gcd(28, 126, 120) = 2 = gcd(56, 210, 36).

The element 36 is surrounded by the sequence 8, 28, 84, 120, 45, 9, and taking alternating values we have gcd(8, 84, 45) = 1 = gcd(28, 120, 9).

Generalization[edit]

The above greatest common divisor also equals [1] Thus in the above example for the element 84 (in its rightmost appearance), we also have gcd(70, 56, 28, 8) = 2. This result in turn has further generalizations.

Related results[edit]

The two sets of three numbers which the Star of David theorem says have equal greatest common divisors also have equal products.[1] So for example, again observing that the element 84 is surrounded by, in sequence, the elements 28, 56, 126, 210, 120, 36, and again taking alternating values, we have 28×126×120 = 26×33×5×72 = 56×210×36. This result can be confirmed by writing out each binomial coefficient in factorial form, using

See also[edit]

References[edit]

  1. ^ a b Weisstein, Eric W. "Star of David Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/StarofDavidTheorem.html

External links[edit]