The determination of integral solutions of algebraic equations with integral coefficients and with more than one unknown is one of the most challenging problems in number theory - and has concerned mathematicians since ancient times. Covering applications to physics and engineering as well, this relatively elementary discussion will appeal to students and mathematicians from high school level onward. AUTHOR: A. O. Gelfond (1906 1968) was a prominent Soviet mathematician who was on the faculty of Moscow State University. Dover also publishes his Transcendental and Algebraic Numbers.