On a conjecture of Graham concerning greatest common divisors

Author:
Gerald Weinstein

Journal:
Proc. Amer. Math. Soc. **63** (1977), 33-38

MSC:
Primary 10A25

DOI:
https://doi.org/10.1090/S0002-9939-1977-0434941-3

MathSciNet review:
0434941

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Abstract: Let be a finite sequence of positive integers. R. L. Graham has conjectured that . The following are proved:

(1) If , a prime, for some *l* and , then the conjecture holds.

(2) Given a finite sequence of positive integers , where , a prime, for some *I* and , consider the set of all positive integral multiples of which are . Denote these multiples by . Define to be a set consisting of the integers where *r* is maximal, such that and . Thus

(a) for at most finitely many *n*.

(b) If for then for all positive integers *n*.

**[1]**R. L. Graham,*Unsolved problem 5749*, Amer. Math. Monthly**77**(1970), 775.**[P]**P. Erdős,*Problems and results on combinatorial number theory*, A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) North-Holland, Amsterdam, 1973, pp. 117–138. MR**0360509****[2]**J. Marica and J. Schönheim,*Differences of sets and a problem of Graham*, Canad. Math. Bull.**12**(1969), 635–637. MR**0249388**, https://doi.org/10.4153/CMB-1969-081-4**[3]**Riko Winterle,*A problem of R. L. Graham in combinatorial number theory*, Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1970) Louisiana State Univ., Baton Rouge, La., 1970, pp. 357–361. MR**0268152****[4]**William Yslas Vélez,*Some remarks on a number theoretic problem of Graham*, Acta Arith.**32**(1977), no. 3, 233–238. MR**0429708**, https://doi.org/10.4064/aa-32-3-233-238

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0434941-3

Article copyright:
© Copyright 1977
American Mathematical Society