# Greek Number Theory

Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Preview

Number theory is the second large field of mathematics that comes to us from the Pythagoreans via Euclid. The Pythagorean theorem led mathematicians to the study of squares and sums of squares; Euclid drew attention to the primes by proving that there are infinitely many of them. Euclid’s investigations were based on the so-called Euclidean algorithm, a method for finding the greatest common divisor of two natural numbers. Common divisors are the key to basic results about prime numbers, in particular unique prime factorization, which says that each natural number factors into primes in exactly one way. Another discovery of the Pythagoreans, the irrationality of $$\sqrt{2}$$, has repercussions in the world of natural numbers. Since$$\sqrt{2}\neq m/n$$ for any natural numbers m, n, there is no solution of the equation $$x^2 - 2y^2 = 0$$ in the natural numbers. But, surprisingly, there are natural number solutions of $$x^2 - \rm{2}y^2 = 1$$, and in fact infinitely many of them. The same is true of the equation $$x^2 - Ny^2 = 1$$ for any nonsquare natural number N. The latter equation, called Pell’s equation, is perhaps second in fame only to the Pythagorean equation $$x^2 + y^2 = z^2$$, among equations for which integer solutions are sought. Methods for solving the Pell equation for general N were first discovered by Indian mathematicians, whose work we study in Chapter 5. Equations for which integer or rational solutions are sought are called Diophantine, after Diophantus. The methods he used to solve quadratic and cubic Diophantine equations are still of interest. We study his method for cubics in this chapter, and take it up again in Chapters 11 and 16.

## Keywords

Rational Point Rational Solution Integer Solution Diophantine Equation Great Common Divisor
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Bashmakova, I. G. (1981). Arithmetic of algebraic curves from Diophantus to Poincaré. Historia Math. 8(4), 393–416.
2. Burton, D. M. (1985). The History of Mathematics. Boston, MA.: Allyn and Bacon Inc.
3. Cohen, M. R. and I. E. Drabkin (1958). Source Book in Greek Science. Cambridge, MA.: Harvard University Press.Google Scholar
4. Davis, M. (1973). Hilbert’s tenth problem is unsolvable. Amer. Math. Monthly 80, 233–269.
5. Fowler, D. H. (1980). Book II of Euclid’s Elements and a pre-Eudoxan theory of ratio. Arch. Hist. Exact Sci. 22(1-2), 5–36.
6. Gelfond, A. O. (1961). The Solution of Equations in Integers. San Francisco, CA.: W. H. Freeman and Co. Translated from the Russian and edited by J. B. Roberts.Google Scholar
7. Jones, J. P. and Y. V. Matiyasevich (1991). Proof of recursive unsolvability of Hilbert’s tenth problem. Amer. Math. Monthly 98(8), 689–709.
8. Krummbiegel, B. and A. Amthor (1880). Das Problema bovinum des Archimedes. Schlömilch Z. XXV. III. A. 121–136, 153–171.Google Scholar
9. Lenstra, H. W. (2002). Solving the Pell equation. Notices Amer. Math. Soc. 49, 182–192.
10. Nathanson, M. B. (1987). A short proof of Cauchy’s polygonal number theorem. Proc. Amer. Math. Soc. 99(1), 22–24.
11. van der Waerden, B. (1976). Pell’s equation in Greek and Hindu mathematics. Russ. Math. Surveys 31(5), 210–225. 