Celebratio Mathematica — Bateman

Partitions and Waring’s problem

Works connected to Paul Erdős

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P. T. Bateman and P. Erdős: “Partitions into primes,” Publ. Math. Debrecen 4 (1956), pp. 198–200. Dedicated to the memory of Tibor Szele. MR 0079013 Zbl 0073.03101 article

Tibor Szele	Related
Paul Erdős	Related

P. T. Bateman and P. Erdős: “Monotonicity of partition functions,” Mathematika 3 : 1 (1956), pp. 1–14. MR 0080121 Zbl 0074.03502 article

Let \( A \) be an arbitrary set of positive integers (finite or infinite) other than the empty set or the set consisting of the single element unity. Let \( p(n) = p_A(n) \) denote the number of partitions of the integer \( n \) into parts taken from the set \( A \), repetitions being allowed. In other words, \( p(n) \) is the number of ways \( n \) can be expressed in the form \[ n_1a_1 + n_2a_2 + \cdots ,\] where \( a_1,a_2,\dots \) are the distinct elements of \( A \) and \( n_1 \), \( n_2,\dots \) are arbitrary non-negative integers. In this paper we shall prove that \( p(n) \) is a strictly increasing function of \( n \) for sufficiently large \( n \) if and only if \( A \) has the following property (which we shall subsequently call property \( P_1 \)): \( A \) contains more than one element, and if we remove any single element from \( A \), the remaining elements have greatest common divisor unity.