On sequences of integers containing no arithmetic progression
نویسندگان
چکیده
منابع مشابه
On sequences of positive integers containing no p terms in arithmetic progression
We use topological ideas to show that, assuming the conjecture of Erdös [4] on subsets of positive integers having no p terms in arithmetic progression (A. P.), there must exist a subset Mp of positive integers with no p terms in A. P. with the property that among all such subsets, Mp maximizes the sum of the reciprocals of its elements.
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In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with b...
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A subsequence of the sequence (1, 2, ..., n) is called a 3-AP -free sequence if it does not contain any three term arithmetic progression. By r(n) we denote the length of the longest such 3-AP -free sequence. The exact values of the function r(n) were known, for n 6 27 and 41 6 n 6 43. In the present paper we determine, with a use of computer, the exact values, for all n 6 123. The value r(122)...
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Let k and n be positive integers, and let d(n;k) be the maximum density in f0;1;2 : : : ;kn 1g of a set containing no arithmetic progression of k terms with first term a = ∑aik and common difference d = ∑εik, where 0 ai k 1, εi = 0 or 1, and εi = 1 ) ai = 0. Setting βk = limn!∞ d(n;k), we show that limk!∞ βk is either 0 or 1. Throughout, we shall use the notation [a;b) = fa;a+ 1;a+ 2; : : : ;b ...
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In this paper, we are concerned with calculating r(k, n), the length of the longest k-Ap free subsequences in 1, 2, . . . , n. We prove the basic inequality r(k, n) 6 n − bm/2c, where n = m(k − 1) + r and r < k − 1. We also discuss a generalization of a famous conjecture of Szekeres (as appears in Erdős and Turán [4]) and describe a simple greedy algorithm that appears to give an optimal k-AP f...
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ژورنال
عنوان ژورنال: Časopis pro pěstování matematiky a fysiky
سال: 1938
ISSN: 1802-114X
DOI: 10.21136/cpmf.1938.122006