Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE GREATEST COMMON DIVISOR OF BINOMIAL COEFFICIENTS

  • Sunben Chiu (School of Mathematical Sciences South China Normal University) ;
  • Pingzhi Yuan (School of Mathematical Sciences South China Normal University) ;
  • Tao Zhou (School of Mathematical Sciences South China Normal University)
  • Received : 2022.03.02
  • Accepted : 2023.04.21
  • Published : 2023.07.31
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Abstract

Let n ⩾ 2 be an integer, we denote the smallest integer b such that gcd {(nk) : b < k < n - b} > 1 as b(n). For any prime p, we denote the highest exponent α such that pα | n as vp(n). In this paper, we partially answer a question asked by Hong in 2016. For a composite number n and a prime number p with p | n, let n = ampm + r, 0 ⩽ r < pm, 0 < am < p. Then we have $$v_p\(\text{gcd}\{\(n\\k\)\;:\;b(n)1\}\)=\{\array{1,&&a_m=1\text{ and }r=b(n),\\0,&&\text{otherwise.}}$$

Keywords

Acknowledgement

Supported by National Science Foundation of China (No. 12171163).

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