As originally stated by Gould (1972),
(1)
where GCD is the greatest common divisor and binomial coefficient . This was subsequently
extended by D. Singmaster to
(2)
(Sato 1975), and generalized by Sato (1975) to
(3)
An even larger generalization was obtained by Hitotumatu and Sato (1975), who defined
with
(16)
and showed that each of the twelve binomial coefficients greatest
common divisor .
A second star of David theorem states that if two triangles are drawn centered on a given element of Pascal's triangle as illustrated
above, then the products
The second star of David theorem holds true not only for the usual binomial coefficients , but also for q -binomial
coefficients
(20)
In fact, the theorem holds for the generalized binomial coefficients based on any divisibility sequence, for example, elliptic divisibility sequences (M. Somos, pers. comm., Mar. 24, 2009).
See also Binomial Coefficient ,
Binomial Sums ,
Christmas
Stocking Theorem ,
Pascal's Triangle
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References Ando, S. and Sato, D. "Translatable and Rotatable Configurations which Give Equal Product, Equal GCD and Equal LCM Properties Simultaneously."
In Applications
of Fibonacci Numbers, Vol. 3: Proceedings of the Third International Conference
on Fibonacci Numbers and their Applications held at the University of Pisa, Pisa,
July 25-29, 1988 Ando,
S. and Sato, D. "A GCD Property on Pascal's Pyramid and the Corresponding LCM
Property of the Modified Pascal Pyramid." In Applications
of Fibonacci Numbers, Vol. 3: Proceedings of the Third International Conference
on Fibonacci Numbers and their Applications held at the University of Pisa, Pisa,
July 25-29, 1988 Ando,
S. and Sato, D. "On the Proof of GCD and LCM Equalities Concerning the Generalized
Binomial and Multinomial Coefficients." In Applications
of Fibonacci numbers, Vol. 4: Proceedings of the Fourth International Conference
on Fibonacci Numbers and their Applications held at Wake Forest University, Winston-Salem,
North Carolina, July 30-August 3, 1990 (Winston-Salem, NC, 1990) Ando, S. and Sato, D. "Multiple Color Version of the
Star of David Theorems on Pascal's Triangle and Related Arrays of Numbers."
In Applications
of Fibonacci Numbers, Vol. 6: Proceedings of the Sixth International Research
Conference on Fibonacci Numbers and their Applications held at Washington State University,
Pullman, Washington, July 18-22, 1994 Butterworth,
B. "The Twelve Days of Christmas: Music Meets Math in a Popular Christmas Song."
Inside Science News Service , Dec. 17, 2002. http://www.aip.org/isns/reports/2002/058.html . Gould,
H. W. Not. Amer. Math. Soc. 19 , A-685, 1972. Hitotumatu,
S. and Sato, D. "Expansion of the Star of David Theorem." Abstracts
Amer. Math. Soc. , p. A-377, 1975. Hitotumatu, S. and Sato, D.
"Star of David Theorem. I." Fib. Quart. 13 , 70, 1975. Sato,
D. "Expansion of the Star of David Theorem of H. W. Gould and David
Singmaster." Abstracts Amer. Math. Soc. , p. A-377, 1975. Referenced
on Wolfram|Alpha Star of David Theorem
Cite this as:
Weisstein, Eric W. "Star of David Theorem."
From MathWorld https://mathworld.wolfram.com/StarofDavidTheorem.html
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is a binomial coefficient. This was subsequently
extended by D. Singmaster to
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
has equal greatest
common divisor.
of the three numbers in the associated points of each of the
two stars are the same (Butterworth 2002). This follows from the fact that
