The Star of David theorem is a mathematical result on

arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...

properties of

binomial coefficients In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...

. It was discovered by Henry W. Gould in 1972.

Statement

The

greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...

s of the binomial coefficients forming each of the two triangles in the

Star of David The Star of David (, , ) is a symbol generally recognized as representing both Jewish identity and Judaism. Its shape is that of a hexagram: the compound of two equilateral triangles. A derivation of the Seal of Solomon was used for decora ...

shape in

Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...

are equal: :

= & \gcd\left\. \end

Examples

Rows 8, 9, and 10 of Pascal's triangle are For ''n''=9, ''k''=3 or ''n''=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

The above greatest common divisor also equals

\gcd \left(, , , \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 :

=\frac.

References

{{reflist * H. W. Gould, "A New Greatest Common Divisor Property of The Binomial Coefficients", ''

Fibonacci Quarterly The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding ...

'' 10 (1972), 579–584.
Star of David theorem
from ''MathForum''.
Star of David theorem
blog post.

External links

Demonstration of the Star of David theorem
in ''

Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...