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Table of integers not exceeding $ 10\,00000$ that are not expressible as the sum of four tetrahedral numbers


Authors: Herbert E. Salzer and Norman Levine
Journal: Math. Comp. 12 (1958), 141-144
MSC: Primary 65.00
DOI: https://doi.org/10.1090/S0025-5718-1958-0099756-3
MathSciNet review: 0099756
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  • [1] H. E. Salzer, ``Tetrahedral numbers,'' MTAC, v. 1, 1943, p. 95.
  • [2] -, Table of first two hundred squares expressed as a sum of four tetrahedral numbers,'' Amer. Math. Soc., Bull., v. 49, 1943, p. 688.
  • [3] -, ``New tables and facts involving sums of four tetrahedral numbers,'' Amer. Math. Soc., Bull., v. 50, 1944, p. 55.
  • [4] Herbert E. Salzer, On numbers expressible as the sum of four tetrahedral numbers, J. London Math. Soc. 20 (1945), 3–4. MR 0015417, https://doi.org/10.1112/jlms/s1-20.1.3
  • [5] -, ``Further empirical results on tetrahedral numbers,'' Amer. Math. Soc., Bull., v. 52, 1946, p. 420.
  • [6] -, ``An 'empirical theorem' which is true for the first 618 cases, but fails in the 619th,'' Amer. Math. Soc., Bull., v. 53, 1947, p. 908 (errata on p. 1196).
  • [7] -, ``Table expressing every square up to one million as a sum of four non-negative tetrahedral numbers,'' Amer. Math. Soc., Bull., v. 54, 1948, p. 830.
  • [8] -, ``Representation table for squares as sums of four tetrahedral numbers,'' MTAC, v. 3, 1948, p. 316.
  • [9] -, ``Verification of the first twenty thousand cases of an empirical theorem with the aid of a device for mass computation,'' Amer. Math. Soc., Bull., v. 55, 1949, p. 41. Empirical Theorems of Other Authors :
  • [10] F. Pollock, ``On the extension of the principle of Fermat's Theorem of the polygonal numbers to the higher orders of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders,'' Roy. Soc. London, Proc., S A, v. 5, 1850, p. 922-924.
  • [11] H. W. Richmond, Notes on a problem of the “Waring” type, J. London Math. Soc. 19 (1944), 38–41. MR 0010709, https://doi.org/10.1112/jlms/19.73_Part_1.38
  • [12] L. E. Dickson, History of the Theory of Numbers, v. 2, Diophantine Analysis, Carnegie Institute of Washington, publication no. 256, v. II (reprinted by G. E. Stechert, N. Y., 1934), p. iv-v and Chapter I, especially p. 4, 7, 22-23, 25, 39.
  • [13] L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939, Chap. VII, ``Sums of nine values of a cubic function,'' p. 130-146. This contains the proof of the theorem that every integer is the sum of nine tetrahedrals, with further references to the most advanced work up to that time, that of R. D. James and L. K. Hua on the theorem that every sufficiently large integer is the sum of eight tetrahedrals.
  • [14] G. L. Watson, Sums of eight values of a cubic polynomial, J. London Math. Soc. 27 (1952), 217–224. MR 0049938, https://doi.org/10.1112/jlms/s1-27.2.217
  • [15] Loo-Keng Hua, Sur le problème de Waring relatif à un polynome du troisième degré, C. R. Acad. Sci. Paris 210 (1940), 650–652 (French). MR 0002343
  • [16] Loo-keng Hua, On Waring’s problem with cubic polynomial summands, Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1940), 55–83. MR 0006194

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DOI: https://doi.org/10.1090/S0025-5718-1958-0099756-3
Article copyright: © Copyright 1958 American Mathematical Society