Pollock's conjecture (Pollock 1850) asserts that every positive integer is the sum of at most five tetrahedral
numbers (Dickson 2005, p. 23; incorrectly described as "pyramidal numbers"
and incorrectly dated to 1928 in Skiena 1997, p. 43). The conjecture
remains unproved.
The known positive integers that are not the sum of at most four tetrahedral numbers begin
17, 27, 33, 52, 73, ... (OEIS A000797). Salzer
and Levine (1958) found 241 such exceptions up to 1043999, the largest being 343867.
It is conjectured that there are no further exceptions.
Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover,
2005.Numberphile. "343867 and Tetrahedral Numbers." May 21,
2024. https://www.youtube.com/watch?v=CK3_zarXkw0.Pollock,
F. "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." Abs. Papers Commun. Roy.
Soc. London5, 922-924, 1843-1850.Salzer, H. E. and
Levine, N. "Table of Integers Not Exceeding 1000000 That Are Not Expressible
as the Sum of Four Tetrahedral Numbers." Math. Comput.12, 141-144,
1958. https://doi.org/10.1090/S0025-5718-1958-0099756-3.Skiena,
S. S. The
Algorithm Design Manual. New York: Springer-Verlag, pp. 43-45, 1997.Sloane,
N. J. A. Sequence A000797/M5033
in "The On-Line Encyclopedia of Integer Sequences."
