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Star of David Theorem


StarofDavidTheorem

As originally stated by Gould (1972),

 GCD{(n-1; k),(n; k-1),(n+1; k+1)} 
 =GCD{(n-1; k-1),(n; k+1),(n+1; k)},
(1)

where GCD is the greatest common divisor and (n; k) is a binomial coefficient. This was subsequently extended by D. Singmaster to

 GCD{(n-1; k),(n; k-1),(n+1; k+1)} 
=GCD{(n-1; k-1),(n; k+1),(n+1; k)} 
=GCD{(n-1; k-2),(n-1; k-1),(n-1; k),(n-1; k+1)}
(2)

(Sato 1975), and generalized by Sato (1975) to

 GCD{(n; k+2),(n-1; k),(n-2; k-2),(n; k-1),(n+2; k),(n+1; k+1)} 
=GCD{(n-2; k),(n-1; k-1),(n; k-2),(n+1; k),(n+2; k+2),(n; k+1)} 
=GCD{(n-2; k-5+j)|j=1,2,3,4,5,6,7}.
(3)

An even larger generalization was obtained by Hitotumatu and Sato (1975), who defined

M_p={(n-p+1; k-2p+j+1)|j=1,2,...,3p-2},  (p>=1)
(4)
A_p={(n-p+j; k+p-1)|j=1,2,...,3p-2}  (p>=1)
(5)
R_p={(n-p+j; k-2p+j-1)|j=1,2,...,3p-2}  (p>=1)
(6)
Delta_p={(n-p+2t+1; k-p+t+1),(n+p-t-1; k+t),(n-t; k+p-2t-1)|t=1,2,...,p-1}  (p>=2)
(7)
del _p={(n-t; k-p+t+1),(n-p+2t+1; k+t),(n+p-t-1; k+p-2t-1)|t=1,2,...,p-1}  (p>=2)
(8)
U_p= union _(r=1)^pM_r
(9)
V_p= union _(r=1)^pA_r
(10)
W_p= union _(r=1)^pR_r
(11)
D_p= union _(r=1)^pDelta_r
(12)
N_p= union _(r=1)^pdel _r
(13)
B_p=M_p union A_p union R_p
(14)
S_p= union _(r=1)^pB_r
(15)

with

 Delta_1=del _1=(n; k),
(16)

and showed that each of the twelve binomial coefficients M_p, A_p, R_p, Delta_p, del _p, U_p, V_p, W_p, D_p, N_p, B_p, and S_p has equal greatest common divisor.

StarofDavidTheorem2

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 P 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

P=(n-1; k-1)(n+1; k)(n; k+1)
(17)
=(n-1; k)(n; k-1)(n+1; k+1)
(18)
=((n-1)!n!(n+1)!)/((k-1)!k!(k+1)!(n-k-1)!(n-k)!(n-k+1)!).
(19)

The second star of David theorem holds true not only for the usual binomial coefficients, but also for q-binomial coefficients, where the common product is given by

 P=((q^k)_infty(q^(k+1))_infty(q^(k+2))_infty(q^(n-l))_infty(q^(n-k+1))_infty(q_(n+k-2))_infty)/((q)_infty^3(q^n)_infty(q^(n+1))_infty(q^(n+2))_infty).
(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 (Ed. G. E. Bergum, A. N. Philippou and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 15-26, 1990a.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 (Ed. G. E. Bergum, A. N. Philippou and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 7-14, 1990b.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) (Ed. G. E. Bergum, A. N. Philippou and A. F. Horadam). Dordrecht, Netherlands: Kluwer, 9-16, 1991.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 (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 31-45, 1996.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--A Wolfram Web Resource. https://mathworld.wolfram.com/StarofDavidTheorem.html

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