Star of David theorem



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:



\begin{align} & \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{align} $$

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||
 * }
 * ||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 $$\gcd \left({n-1 \choose k-2}, {n-1 \choose k-1}, {n-1 \choose k}, {n-1 \choose k+1}\right). $$ 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. 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
 * $${a \choose b}=\frac{a!}{(a-b)!b!}.$$