at most 19 fourth powers. That was established from 1909 to 1912 by Wieferich and A. J. Kempner, in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, in 1964 by Chen Jingrun, and in 1940 by Pillai. Let and respectively denote the integral and fractional part of a positive real number . Given the number , only and can be used to represent ; the most economical representation requires terms of and terms of . It follows that is at least as large as . This was noted by J. A. Euler, the son of Leonhard Euler, in about 1772. Later work by Dickson, Pillai, Rubugunday, Niven and many others has proved that No value of is known for which . Mahler proved that there can only be a finite number of such , and Kubina and Wunderlich have shown that any such must satisfy . Thus it is conjectured that this never happens, that is, for every positive integer . The first few values of are: 1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, ... . The number G(k) From the work of Hardy and Littlewood, the related quantity G(k) was studied with g(k). G(k) is defined to be the least positive integer s such that every sufficiently large integer (i.e. every integer greater than some constant) can be represented as a sum of at most s positive integers to the power of k. Clearly, G(1) = 1. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that . Since for all k, this shows that . Davenport showed that in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12). The exact value of G(k) is unknown for any other k, but there exist bounds. Lower bounds for G(k) The number G(k) is greater than or equal to {| | 2r+2 || if k = 2r with r ≥ 2, or k = 3 × 2r; |- | pr+1 || if p is a prime greater than 2 and k = pr(p − 1); |- | (pr+1 −