DIOPHANTINE EQUATIONS. About the middle of the third century, A.D., the Greek mathematician Diophantus wrote an epoch-making book entitled Arithmetica. It was important in at least three respects :—Essentially an algebra rather than an arithmetic, it was the first book to be recognized as an algebra; it contained problems requiring rational solutions which have proved to be too difficult for all mathematicians from the time of Diophantus to the present; it exhibited very ingenious methods of attacking many difficult problems of which it gave only a partial solution. The efforts of Pierre De Fermat, Leonhard Euler and other mathematicians to solve completely the difficult problems of Diophantus and other similar ones have given rise to a large part of what is called the analytic theory of numbers. For this reason, all indeterminate equations, with rational co efficients, for which a rational solution is required are called Diophantine equations.
Diophantine Equations, in Practical Problems.—If a banker wishes to change a dcllar by means of five-cent coins (nickels) and twenty-five-cent coins (quarters) by including in the change at least one coin of each kind, he may ask himself in how many ways this can be done, and, by mental arithmetic, he may discover that the answer is three; that is, he can select 5 nickels and 3 quarters, o nickels and 2 quarters, or 15 nickels and I quarter. In Diophantine form, this problem is to find all positive integral solutions of the equation 5x+25y= ioo, or its equivalent x+5y=2o, where x and y represent the numbers of nickels and quarters, respectively. Applying to x+5y= 20 the known theory of the equation ax+by=c (see NUMBER, THEORY ) one finds that all solutions of xd-5y= 2o, and only these solu tions, are given by the formulae x=5+5t, t being an ar bitrary constant commonly called a parameter. As (x, y) are both positive only when t=o, 1, or 2, the solutions sought are as stated above (5, 3), (1o, 2) and (15, 1). This problem not only suggests that an indeterminate equation of the first degree may be useful, but it also exemplifies two important facts in the theory of such equations; viz., if a system of Diophantine equa tions of the first degree with specified coefficients has a solution, it can be found directly by use of known theory and, when one solution of such a system is known, general formulae can be obtained which give without repetition all solutions of the system, and only these solutions, in terms of the known solution and certain parameters that appear only to the first degree.
Diophantine Equations of Higher Degree.—Nearly all Diophantine equations of degree higher than the first are difficult to solve, the difficulty increasing with the degree of the equations. On account of this fact and because of the practical and mathe matical importance of solving indeterminate equations of the second degree, these equations have received more attention than have any other types of Diophantine equations. Consequently, the literature of indeterminate quadratics is extensive; it and the associated theory of third and fourth degree equations consti tute the most important part of the classical Diophantine theory of to-day, which is one of the great treasures of number theorists.
One regrettable feature of this theory, however, is that it contains scarcely a principle that can be applied to more than one type of problem. Mathematicians still have to follow the example of Diophantus and devise a special mode of attack for nearly every Diophantine problem. There is a difference, how ever, between the procedure of Diophantus and that of mathe maticians of to-day. For example, in Problem 8, Book II., of his Arithmetica (see T. L. Heath, Diophantus of Alexandria, io), Diophantus solves the problem of dividing a given square num ber into the sum of two squares essentially as follows. Let the given square number be 16 and let x2 be one of the required squares. Then i6—x2 must be equal to a square. Take a square of the form (mx-4)2, where m is an arbitrary integer and 4 is the number which is the square root of 16; for example, take (2x-4)2 and equate it to i6—x2. One thus obtains 4e—I6xd-I6= 16—x2, so that x=16/5. Hence the required squares are 256/ 25 and 144/25. The customary treatment of this problem at present would be to let N2 be the given square num ber and to write (mx—N)2=N2—x2. Since x is not zero, this equation implies that x=2mN/(i-km2), so that x2= 4m2N2/( IH-m2) and the other required square is N2— x2 =N2 ( —m2)2/0-1-m2)2.
As m is an arbitrary integer, the formulae just written give infinitely many solutions of the problem. This example not only illustrates one of Diophantus's processes, namely that of introducing an arbitrary parameter as a direct means to a de sired solution, but it also shows how mathematicians since the time of Fermat have generalized both the statements and the solutions of many Diophantine problems. Indeed it was to the above problem that Fermat himself appended the famous the orem : x"-Fy"=z" has no solution in positive integers when n is an integer greater than two. No rigorous proof of this theorem has yet been published; nor has it been disproved.
In vivid contrast with Diophantus's desire for a single rational solution, is the modern mathematician's search for all solutions of the type that he seeks, which may be, for example, the rational solutions, the integral solutions, or the positive integral solutions. Indeed a single solution of a Diophantine equation is no longer of interest unless it has some special significance; e.g., that of containing the largest integer that can appear in any positive integral solution of a given equation. The present tendency is to make as complete an analysis of Diophantine equations as is possible. Thus, in a given problem, one desires first to find all rational solutions, next all integral solutions, then all positive integral solutions, and the final touch is to discover any special properties which individual solutions possess.
The following well-known. theoren3s may increase one's under standing of the nature of the results that have been obtained in classical Diophantine theory:— All solutions of the equation x2d-y2=z2 in relatively prime posi tive integers x, y, z are given by the formulae, x=2mn, y=m2—n2, z=m2-I-n2, where m and n are relatively prime positive integers and in is greater than n; furthermore, every set of num bers x, y, z defined by these formulae, when m and n are as just de scribed, is a solution in positive integers of The equation x2-1-y2-kz2-F w2=u, where u is a positive integer, has a solution in which x, y, z, w are non-negative integers.
The equation x3--Fy3=z3 has no solution in which x, y, z are all positive integers.
Every prime number of the form 8n+ or 80-3 is represent able in the form x2+2y2; n, x and y being positive integers.