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A systematic collection of fundamental open problems is presented to chart directions for future mathematical research. Topics range across set theory, the consistency of arithmetic, geometry, transformation groups, number theory, algebraic forms, analysis, the calculus of variations, and the mathematical formulation of physical axioms. Challenges highlighted include questions about the continuum, irrationality and transcendence, the distribution and properties of primes, reciprocity laws, solvability of Diophantine equations, uniformization and boundary-value problems, and the finiteness or structure of function systems. Each problem is posed to emphasize its technical difficulty and its capacity to stimulate new methods and broader theoretical development.
Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.