# Star of David theorem

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

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:

{\displaystyle {\begin{aligned}&\gcd \left\{{\binom {n-1}{k-1}},{\binom {n}{k+1}},{\binom {n+1}{k}}\right\}\\[8pt]={}&\gcd \left\{{\binom {n-1}{k}},{\binom {n}{k-1}},{\binom {n+1}{k+1}}\right\}.\end{aligned}}}

## Examples

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

The above greatest common divisor also equals ${\displaystyle \gcd \left({n-1 \choose k-2},{n-1 \choose k-1},{n-1 \choose k},{n-1 \choose k+1}\right).}$[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

The two sets of three numbers which the Star of David theorem says have equal greatest common divisors also have equal products.[1] 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

${\displaystyle {a \choose b}={\frac {a!}{(a-b)!b!}}.}$