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On a Conjecture of Graham Concerning a Sequence of Integers

Published online by Cambridge University Press:  20 November 2018

E. Z. Chein
E. Z. Chein*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
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Let 0<a1<…<an be integers and (a, b) denotes the greatest common divisor of a, b. R. L. Graham [1] has conjectured that

for some i and j. In a recent paper Weinstein [2] has improved Winterle's result [3] and has proven the following interesting theorem:

If A is the sequence a1< … <an where ak = P, a prime for some k and , then

.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 3 , 01 September 1978 , pp. 285 - 287
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Graham, R. L., Unsolved problem 5749, Amer. Math. Monthly 77 (1970), 775. Google Scholar
2. Weinstein, G. On a conjecture of Graham concerning greatest common divisors proceedings Amer. Math. Soc. 63 (1977), 33-38.Google Scholar
3. Winterle, Riko, 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, pp. 357-361. MR 42 #3051.Google Scholar
4. Vélez, W. Y., Some remarks on a number theoretic problem of Graham, Acta. Arith. 32 (1977), no. 3, 233-238.Google Scholar