On a conjecture of Graham
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- by J. W. Sander PDF
- Proc. Amer. Math. Soc. 102 (1988), 455-458 Request permission
Abstract:
Let ${a_1} < {a_2} < \cdots < {a_n}$ be a finite sequence of positive integers containing a prime power ${p^d}$ with the property: ${a_i} \ne {p^k}{a_j}$ for all $i,j$ and $k > 0$. Then ${\max _{i,j}}{a_i}/\left ( {{a_i},{a_j}} \right ) \geq n$.References
- R. D. Boyle, On a problem of R. L. Graham, Acta Arith. 34 (1977/78), no. 2, 163–177. MR 465996, DOI 10.4064/aa-34-2-163-177
- E. Z. Chein, On a conjecture of Graham concerning a sequence of integers, Canad. Math. Bull. 21 (1978), no. 3, 285–287. MR 511574, DOI 10.4153/CMB-1978-050-7
- 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 R. L. Graham, Unsolved problem 5749, Amer. Math. Monthly 77 (1970), 775.
- Rivka Klein, The proof of a conjecture of Graham for sequences containing primes, Proc. Amer. Math. Soc. 95 (1985), no. 2, 189–190. MR 801321, DOI 10.1090/S0002-9939-1985-0801321-2
- J. Marica and J. Schönheim, Differences of sets and a problem of Graham, Canad. Math. Bull. 12 (1969), 635–637. MR 249388, DOI 10.4153/CMB-1969-081-4
- R. J. Simpson, On a conjecture of R. L. Graham, Acta Arith. 40 (1981/82), no. 2, 209–211. MR 649120, DOI 10.4064/aa-40-2-209-211
- William Yslas Vélez, Some remarks on a number theoretic problem of Graham, Acta Arith. 32 (1977), no. 3, 233–238. MR 429708, DOI 10.4064/aa-32-3-233-238
- Gerald Weinstein, On a conjecture of Graham concerning greatest common divisors, Proc. Amer. Math. Soc. 63 (1977), no. 1, 33–38. MR 434941, DOI 10.1090/S0002-9939-1977-0434941-3
- 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
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 455-458
- MSC: Primary 11A05
- DOI: https://doi.org/10.1090/S0002-9939-1988-0928959-9
- MathSciNet review: 928959