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Star of David theorem

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Mathematical result on arithmetic properties of binomial coefficients

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

TheStar of David theorem is a mathematical result onarithmetic properties ofbinomial coefficients. It was discovered byHenry W. Gould in 1972.

Statement

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Thegreatest common divisors of the binomial coefficients forming each of the two triangles in theStar of David shape inPascal's triangle are equal:

{\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

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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

Forn=9,k=3 orn=9,k=6, the element 84 (circled bold) is surrounded by, in sequence, the elements 28, 56, 126, 210, 120 and 36 (bold). Taking alternating values, we have gcd(28, 126, 120) = 2 = gcd(56, 210, 36).

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

Generalization

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

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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

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

References

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^^a ^bWeisstein, Eric W."Star of David Theorem".mathworld.wolfram.com. Retrieved2024-12-31.

H. W. Gould, "A New Greatest Common Divisor Property of The Binomial Coefficients",Fibonacci Quarterly 10 (1972), 579–584.
Star of David theorem, fromMathForum.
Star of David theorem, blog post.