On sequences containing at most 3 pairwise coprime integers

Choi, S. L. G.

doi:10.1090/S0002-9947-1973-0327710-7

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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On sequences containing at most $3$ pairwise coprime integers
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by S. L. G. Choi PDF

Trans. Amer. Math. Soc. 183 (1973), 437-440 Request permission

Abstract:

It has been conjectured by Erdös that the largest number of natural numbers not exceeding n from which one cannot select $k + 1$ pairwise coprime integers, where $k \geq 1$ and $n \geq {p_k}$, with ${p_k}$ denoting the kth prime, is equal to the number of natural numbers not exceeding n which are multiples of at least one of the first k primes. It is known that the conjecture holds for k = 1 and 2. In this paper we establish the truth of the conjecture for k = 3.

References

P. Erdős, Extremal problems in number theory, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 181–189. MR 0174539