Table of integers not exceeding 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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References | Similar Articles | Additional Information

**[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]**-, ``On numbers expressible as the sum of four tetrahedral numbers,'' London Math. Soc.,*Jn.*, v. 20, 1945, p. 3-4. This note includes a table of the exceptional numbers 1000 (erratum: omission of 107). That table is continued up to 2000 in*MTAC*, v. 3, 1949, p. 423-424. MR**0015417 (7:415e)****[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,'' London Math. Soc.,*Jn.*, v. 19, 1944, p. 38-41. MR**0010709 (6:57f)****[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,'' London Math. Soc.,*Jn.*, v. 27, 1952, p. 217-224. This includes the proof of the theorem that every integer is the sum of eight tetrahedrals. MR**0049938 (14:250e)****[15]**L. K. Hua, ``Sur le problème de Waring relatif à un polynome du troisième degré,''*Comptes Rendus*, Academie des Sciences, Paris, v. 210, 1940, p. 650-652. MR**0002343 (2:35f)****[16]**L. K. Hua, ``On Waring's problem with cubic polynomial summands,''*Sci. Report Nat. Tsing Hua Univ.*, (A) v. 4, 1940, p. 55-83. MR**0006194 (3:270b)**

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DOI:
https://doi.org/10.1090/S0025-5718-1958-0099756-3

Article copyright:
© Copyright 1958
American Mathematical Society