In the Little Mathematics Library series we now come to Solving
Equations In Integers by A. O. Gelfond also sometimes written as
The book is devoted to one of the most interesting branches of
number theory, the solution of equations in integers. The solution in integers of algebraic equations in more than one unknown with integral coefficients is a most difficult problem in the theory of numbers. The theoretical importance of equations with integral coefficients is quite great as they are closely connected with many problems of number theory. Moreover, these equations are sometimes encountered in physics and so they are also important in practice. The elements of the theory of equations with integral coefficients as presented in this book are suitable for broadening the mathematical outlook of high-school students and students of pedagogical institutes. Some of the main results in the theory of the solution of equations in integers have been given and proofs of the theorems involved are supplied when they are sufficiently simple.
The book was translated the Russian by O. B. Sheinin and was
first published by Mir in 1981.
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1. Equations in one unknown. 8
2. Linear equations in two unknowns 9
3. Equations of the second degree in three unknowns (examples) 18
4. Equations of the type x^2 – Ay^2 = 1. Finding an solutions of this equation 23
5. Equations of the second degree in two unknowns: the general case 33
6. Equations in two unknowns of degree higher than the second 44
7. Algebraic equations in three unknowns of degree higher than
the second. Some exponential equations 49