A conjectured lower-bounding sequence for the Schur numbers, its reduction to S(6) and S(7), and the question of an upper-bounding sequence
Abstract
The Schur number S(n) is the largest N such that {1, . . . , N } can be partitioned into n sum-free sets. Five values are known, 1, 4, 13, 44, 160, and the sixth is bracketed by 536 ≤ S(6) ≤ 1836. We define an integer sequence L(n) by a recurrence of Schur's type, L(n + 1) = 3 n + L(n) plus an alternating sum of floor corrections, which reproduces the five known values without adjustment, and we conjecture L(n) ≤ S(n) for all n. The corrections are finitely many: they vanish from the 25th on, and L(n) = (3 n -1)/2 + 22 542 -7(-1) n for all n ≥ 26. Given the published lower bounds and the recurrences of Rowley, of Ageron and co-authors and of Bengone and co-authors, the conjecture holds for all n ≥ 9 and is equivalent to the two statements S(6) ≥ 566 and S(7) ≥ 1852, both above the present records 536 and 1696. We list what a six-colouring of {1, . . . , w} would establish for each w between 537 and 620, the range of a search in progress. On the question of an upperbounding sequence of the same kind, we show that the only proven ones are factorial, S(n) ≤ ⌊n!(e -1/6)⌋ -1, and that the conjectural recursion of Eliahou and Revuelta is factorial too. Any exponential one must have a base at least 380 1/5 ≈ 3.28 and would settle an open problem on the growth of S(n). Two exponential candidates compatible with everything known are recorded with their refutation thresholds: U 1 (n) = 9 • 4 n-3 + 2 n-1 , which fits the three values 13, 44, 160 with two parameters and is refuted by a six-colouring of {1, . . . , 609}, and U 2 , defined by the recurrence U 2 (n) = 2 3 (4 n-1 +2)-U 2 (n-2)+U 2 (n-4) from the four values 1, 4, 13, 44, which predicts 160 and is refuted by a six-colouring of {1, . . . , 645}.
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BibTeX
@misc{reynaud2026conjectured,
title = {{A conjectured lower-bounding sequence for the Schur numbers, its reduction to S(6) and S(7), and the question of an upper-bounding sequence}},
author = {Patrick B. Reynaud},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05836388v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A conjectured lower-bounding sequence for the Schur numbers, its reduction to S(6) and S(7), and the question of an upper-bounding sequence
AU - Patrick B. Reynaud
PY - 2026
DA - 2026/10/22
PB - HAL
UR - https://hal.science/hal-05836388v1
AB - The Schur number S(n) is the largest N such that {1, . . . , N } can be partitioned into n sum-free sets. Five values are known, 1, 4, 13, 44, 160, and the sixth is bracketed by 536 ≤ S(6) ≤ 1836. We define an integer sequence L(n) by a recurrence of Schur's type, L(n + 1) = 3 n + L(n) plus an alternating sum of floor corrections, which reproduces the five known values without adjustment, and we conjecture L(n) ≤ S(n) for all n. The corrections are finitely many: they vanish from the 25th on, and L(n) = (3 n -1)/2 + 22 542 -7(-1) n for all n ≥ 26. Given the published lower bounds and the recurrences of Rowley, of Ageron and co-authors and of Bengone and co-authors, the conjecture holds for all n ≥ 9 and is equivalent to the two statements S(6) ≥ 566 and S(7) ≥ 1852, both above the present records 536 and 1696. We list what a six-colouring of {1, . . . , w} would establish for each w between 537 and 620, the range of a search in progress. On the question of an upperbounding sequence of the same kind, we show that the only proven ones are factorial, S(n) ≤ ⌊n!(e -1/6)⌋ -1, and that the conjectural recursion of Eliahou and Revuelta is factorial too. Any exponential one must have a base at least 380 1/5 ≈ 3.28 and would settle an open problem on the growth of S(n). Two exponential candidates compatible with everything known are recorded with their refutation thresholds: U 1 (n) = 9 • 4 n-3 + 2 n-1 , which fits the three values 13, 44, 160 with two parameters and is refuted by a six-colouring of {1, . . . , 609}, and U 2 , defined by the recurrence U 2 (n) = 2 3 (4 n-1 +2)-U 2 (n-2)+U 2 (n-4) from the four values 1, 4, 13, 44, which predicts 160 and is refuted by a six-colouring of {1, . . . , 645}.
ER -