Claude: "Table of values of m^*(n)"

Claude: "Table 2 of the paper by Buhler, Golan, Pratt and Wagon (Math. Comp. 90, 2021), extended with new results."

Claude: "m^*(n) is the largest number m such that {1, …, n} can be split into two parts whose power sums are equal for every exponent k = 0, …, m − 1."

Table 2. Values of m^*(n).

n	8	16	24	32	40	48	56	64	72	80	88	96	104	112	120	128
m^*(n)	3	4	4	5	5	6	5	6	6	6	6	7	6	7	6	7

n	136	144	152	160	168	176	184	192	200	208	216	224	232
m^*(n)	6	8	6	7	6	7	6	9	7	8	7	8	7

n	240	248	256	264	272	280	288	296	304	2⁵¹
m^*(n)	10 or 11	7	8	7	8	7	9 or 10	7	8 or 9	≥ 52

Claude: "New results since the paper"

n	320	336	352	368	400	416	432	464	480	496	512	528	2²⁴
m^*(n)	9	≥ 9	9	9	9	9	≥ 11	9	≥ 12	9	9	≥ 10	≥ 25

Claude: "Claude alone"
Claude: "Claude and Stan"
Claude: "Stan alone"
Claude: "The published paper (2021)"

Claude: "Where the new results come from"

n = 320, 352, 368, 400, 416, 464: m^* = 9

Claude: "Examples of order 9 found by Stan. Order 10 is ruled out by the paper's Proposition 4.3 (p = 3)."

Claude: "p₃₂₀, a second example, found independently by Claude (m = 9):" B24B649ED92CD2C36496CB2DB6295D974138BA41
n = 336: m^* ≥ 9

Claude: "An antisymmetric example found by Claude; it is the second part of the 528 example below."

p₃₃₆ (m = 9): 9C5479E2669C97071E731A62EBC06F28E87B4961DF
n = 432: m^* ≥ 11

Claude: "An antisymmetric example of order 11 found by Claude. Stan also showed order 10."

p₄₃₂ (m = 11): 5A56B0EE234E8794ADE3D0C384F72A5C63A8BD65035EF015B78B46
n = 480: m^* ≥ 12

Claude: "A symmetric example of order 12 found by Claude."

p₄₈₀ (m = 12): B0CBC94F1E2C68D1E3A6371C357C4B1347952BB968236CF6C82CD9CE01F3
n = 496: m^* = 9

Claude: "An example of order 9 found by Stan. Order 10 is ruled out by the residue condition mod 27, by the same argument as for 512; the solvers CP-SAT and SCIP agree."
n = 512: m^* = 9

Claude: "The Thue–Morse polynomial τ₉ has order 9. Order 10 is ruled out by the residue conditions mod 3, 9 and 27."
n = 528: m^* ≥ 10

Claude: "An example found by Claude by joining an example of length 192 and order 9 to an antisymmetric piece of length 336 and order 9, chosen so that their 9th power sums cancel."

f = p₁₉₂ ∨ p₃₃₆; p₁₉₂ (m = 9): C1BE1E21CD63D295A7887A59
n = 2²⁴: m^* ≥ 25

Claude: "Joint construction: 2²⁴ = 320²·112 + 432·48·2⁸, that is, f = (p₃₂₀ # p₃₂₀ # p₁₁₂) ∨ (p₄₃₂ # p₄₈ # τ₈). Hence m^*(2^k) ≥ k + 1 for every k ≥ 24."

Claude: "p₁₁₂ (m = 7) and p₄₈ (m = 6), from the paper's Table 5:" A5994DB29B45A0 C27D8C