The generalized Schur number 
associated with the linear
equation
 |
is the least positive integer 
such that every assignment of 
colors to the integers 1, ...,
contains a solution with 
, 
, and 
of the same color (Khamseh 2020). The parameters 
and 
are positive integers,
and the variables in a solution need not be distinct.
For 
,
this least-endpoint convention gives 
, where 
is the Schur number defined
as the largest interval admitting a coloring without a monochromatic solution.
Khamseh and Safareh (2026) proved
 |
for every integer 
. Equivalently, for 
, the least endpoint for the equation 
is 
.
The values for 
,
4, ... are 43, 94, 173, 286, 439, ... (OEIS A083074).
If 
denotes the OEIS indexing starting at 
, then 
. The proof combines an explicit coloring for the
lower bound with a finite symbolic case analysis for
the upper bound. The accompanying checking program
has not been independently replayed here.
."
J. Math. 2020, 7069730, 2020. 
:
A Proof for All 
from a Critical Set of 33 Polynomials." 1 Nov 2026.