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For the past 21 years one of the authors has been concerned with empirical theorems expressing certain classes of positive integers as the sum of four tetrahedral numbers, or "tetrahedrals" T„ = n(n + l)(w + 2)/6, n > 0, the results being contained in short notes and abstracts, as well as unpublished tables [1 ]~[9]We use 2P to denote a sum of p Tn's. Among the more interesting past findings were : every square < 10  a 24 [7], [8], every Tn < 10  a S4 other than the trivial decomposition Tn — Tn + 0 + 0 + 0 [3], [4], every multiple of 5 < 1  a 24 [9] (see also [11]), and verification of Pollock's conjecture that every integer is a 26 for the first 20000 integers [5], [10]. Incidentally all investigations in [l]-[9] were done by hand, employing at the most a desk calculator. The exceptional numbers, which by definition are those not expressible as 24, were tabulated previously only up to 2000. But even that far interesting features turned up, such as 1314 being the only exceptional number ending in 4, and the very few ending in 1 or 9 [4]. Then for numbers ending in 6 and < 20000, 6186 turned out to be the only exceptional one [6]. Also there was no striking difference between the density of exceptional numbers in the first thousand and in the second thousand brackets, decreasing from around 4|% to 3%, so that it was interesting to speculate upon the approximate density in the neighborhood of say 10 . Then finally, the verification of the conjectures that every integer is a S5 and that every integer m = 5r is a 24 for the first 20000 cases, while every lOr + 6, r = 0, 1, 2, • • •, 617, is a 24 until 6186, posed the question as to whether an exception might occur to either of the former two empirical theorems even after verification in those first 20000 cases. Thus it was felt that tabulation of the exceptional numbers < 10  would afford a much clearer picture as to their distribution and density, as well as stronger evidence for the truth of Pollock's conjecture and the author's (and Richmond's [11]) conjecture that every m = 5r is a 24. This table of exceptional numbers < 10 43999 presents a great surprise in its picture of their distribution which is entirely different from that envisioned from those < 2000 [4]. Most strikingly unexpected is the decrease in the density of exceptional numbers from several percent in the neighborhood of 2000 to what appears to be practically zero near 10 . The scarcity of exceptional numbers in the higher ranges is in accordance with Hua's result that "almost all" positive integers are expressible as a 24 [15], [16]. But even more, this table shows that the likelihood of a given number m being exceptional falls off so rapidly with increasing m that it appears to be a plausible conjecture that there might be some Wo sufficiently large such that every m > m<¡ is a 24 (a conjecture unlikely to suggest itself from the exceptional numbers among only the first few thousand).

Table of integers not exceeding 1000000 that are not expressible as the sum of four tetrahedral numbers