Abstract
Let k≥3 be an integer, the Schur number Sk(3) is the least positive integer, such that for every 3-coloring of the integer interval [1,Sk(3)] there exists a monochromatic solution to the equation x1+⋯+xk=xk+1, where xi, i=1,…,k need not be distinct.In 1966, a lower bound of Sk(3) was established by Znám (1966). In this paper, we determine the exact formula of Sk(3)=k3+2k2−2, finding an upper bound which coincides with the lower bound given by Znám (1966). This is shown in two different ways: in the first instance, by the exhaustive development of all possible cases and in the second instance translating the problem into a Boolean satisfiability problem, which can be handled by a SAT solver.
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