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A GCD Property on Pascal’s Pyramid and the Corresponding LCM Property of the Modified Pascal Pyramid

In: Applications of Fibonacci Numbers

Author

Listed:
  • Shiro Ando
  • Daihachiro Sato

Abstract

Concerning the six binomial coefficients A1, A2,…, A6 A2 A3 surrounding any entry A inside Pascal’s triangle, Hoggatt and A1 A A4 Hansell [1] proved the identity (1) $${A_1}{A_3}{A_5} = {A_2}{A_4}{A_6},$$ which has been generalized to the case of multinomial coefficients by Hoggatt and Alexanderson [2]. Meanwhile, Gould [3] found the remarkable property (2) $$\gcd \left( {{A_1},{A_3},{A_5}} \right) = \gcd \left( {{A_2},{A_4},{A_6}} \right),$$ which was established by Hillman and Hoggatt [4] for the generalized binomial coefficients defined by (16) for m=2. He also showed that the equality (3) $$lcm\left( {{A_1},{A_3},{A_5}} \right) = lcm\left( {{A_2},{A_4},{A_6}} \right)$$ does not always hold.

Suggested Citation

  • Shiro Ando & Daihachiro Sato, 1990. "A GCD Property on Pascal’s Pyramid and the Corresponding LCM Property of the Modified Pascal Pyramid," Springer Books, in: G. E. Bergum & A. N. Philippou & A. F. Horadam (ed.), Applications of Fibonacci Numbers, pages 7-14, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-1910-5_2
    DOI: 10.1007/978-94-009-1910-5_2
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