The solution of graham’s greatest common divisor problem

Abstract

The following conjecture of R. L. Graham is verified: Ifnn 0, wheren 0 is an explicitly computable constant, then for anyn distinct positive integersa 1,a 2, ...,a n we have\(\mathop {\max }\limits_{i,j} \) a i /(a i ,a j ) ≧ ≧n, and equality holds only in two trivial cases. Here (a i ,a j ) stands for the greatest cnmmon divisor ofa i anda j .

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References

  1. [1]

    P. Erdős andR. L. Graham,Old and New Problems and Results in Combinatorial Number Theory, Genève, 1980.

  2. [2]

    R. L. Graham, Unsolved problem 5749,Amer. Math. Monthly,77 (1970), 775.

    Article  Google Scholar 

  3. [3]

    D. R. Heat-Brown andH. Iwaniec, On the difference between consecutive primes,Invent. Math. 55 (1979), 49–69.

    Article  MathSciNet  Google Scholar 

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Szegedy, M. The solution of graham’s greatest common divisor problem. Combinatorica 6, 67–71 (1986). https://doi.org/10.1007/BF02579410

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AMS subject classification (1980)

  • 10 A 05
  • 10 A 25