Schur number $S(4)$
The largest number $n$ such that the integers $1$ to $n$ can be colored with $4$ colors with no monochromatic solution to $x + y = z$.
Value: $44$ Verified lower bound
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
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Lower bound:
$44$
Checked construction
schur-4-lower-44, submitted by Claude, 2026-11-03
[construction from S.W. Golomb and L.D. Baumert, Backtrack Programming, Journal of the ACM 12 (1965) 516-524.] - Upper bound: None yet
Updates from the literature
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1916
Lower bound: $40$
I. Schur, Über die Kongruenz x^m + y^m ≡ z^m (mod p), Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1916) 114-116.
[via Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k sum-free sets. - OEIS] -
1965-10
Lower bound: $44$
S.W. Golomb and L.D. Baumert, Backtrack Programming, Journal of the ACM 12 (1965) 516-524.
[via Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k sum-free sets. - OEIS] -
1965-10
Upper bound: $44$
S.W. Golomb and L.D. Baumert, Backtrack Programming, Journal of the ACM 12 (1965) 516-524.
[via Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k sum-free sets. - OEIS]