Schur's theorem

Schur's theorem states that every finite coloring of the positive integers contains positive integers x, y, and x+y that all have the same color.

Finite coloring: A partition of a set into finitely many classes, each assigned a color. The theorem applies to every partition of the positive integers into finitely many classes.
Issai Schur: A German mathematician whose work included foundational results in algebra and number theory. Schur proved the theorem in 1916 while studying Fermat's last theorem modulo primes.
Positive integer: An integer greater than zero. The theorem's variables range over positive integers, excluding zero and negative values.
Van der Waerden's theorem: A theorem stating that every finite coloring of the positive integers contains arbitrarily long monochromatic arithmetic progressions. It generalizes the search for forced monochromatic patterns beyond a single additive equation.
Pigeonhole principle: A counting principle stating that placing more objects than boxes forces at least one box to contain multiple objects. Finite colorings force repeated colors among strategically chosen integer sums.
Fermat's last theorem: The assertion that no positive integers satisfy xⁿ+yⁿ=zⁿ for any integer n greater than two. Schur's original motivation was to derive a finite-field consequence of this equation.
Addition: The arithmetic operation that combines quantities into their sum. The required pattern is precisely that two positive integers and their sum share a color.
Rado's theorem: A theorem characterizing which finite systems of linear equations have monochromatic solutions under every finite coloring of the positive integers. Schur's equation x+y=z is a basic example within the broader theory of partition-regular systems.
Cyclic group: A group generated by a single element, whose elements are successive powers of that generator. A finite cyclic group supplies the residue structure used in a standard proof.
Finite fields: Fields with finitely many elements, whose arithmetic includes addition and multiplication modulo a prime power. Schur used finite-field arithmetic to connect hypothetical Fermat solutions with a monochromatic sum.

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