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Problems and results on additive properties of general sequences. V. (English) Zbl 0597.10055
[Part I, cf. Pac. J. Math. 118, 347–357 (1985; Zbl 0569.10032), part IV, cf. Lect. Notes Math. 1122, 85–104 (1985; Zbl 0588.10056).]
A very special case of one of the theorems of the authors states as follows: Let \(1\leq a_1\leq a_2\leq\dots\) be an infinite sequence of integers for which all the sums \(a_i+a_j\), \(1\leq i\leq j\), are distinct. Then there are infinitely many integers \(k\) for which \(2k\) can be represented in the form \(a_i+a_j\) but \(2k+1\) cannot be represented in this form. Several unsolved problems are stated.
Reviewer: P. Erdős

MSC:
11B34 Representation functions
11B75 Other combinatorial number theory
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[1] Erdös, P.: Problems and results in additive number theory. Colloque sur la Théorie des Nombres (CBRM) (Bruxelles, 1956), 127-137.
[2] Erdös, P., andRényi, A.: Additive properties of random sequences of positive integers. Acta Arith.6, 83-110 (1960). · Zbl 0091.04401
[3] Erdös, P., Sárközy, A.: Problems and results on additive properties of general sequences, I. Pacific J. Math.118, 347-357 (1985). · Zbl 0569.10032
[4] Erdös, P., Sárközy, A.: Problems and results on additive properties of general sequences, II. Acta Math. Acad. Sci. Hung. To appear. · Zbl 0669.10078
[5] Erdös, P., Sárközy, A., Sós, V. T.: Problems and results on additive properties of general sequences, III. Studia Sci. Math. Hung. To appear.
[6] Erdös, P., Sárközy, A., Sós, V. T.: Problems and results on additive properties of general sequences, IV. In: Number Theory. Proc., Ootacamund, India. Lect. Notes Math. 1122, pp. 85-104. Berlin-Heidelberg-New York: Springer. 1984.
[7] Halberstam, H., Roth, K. F.: Sequences. Berlin-Heidelberg-New York: Springer. 1983.
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