DIOPHANTINE EQUATIONS

Submitted by:

Aaron Zerhusen, Chris Rakes, & Shasta Meece

MA 330-002

Dr. Carl Eberhart

February 16, 1999

DIOPHANTINE EQUATIONS

HISTORY:

Because little is known on the life of Diophantus, historians have approximated his birth to be at about 200 AD in Alexandria, Egypt and his death at 284 AD in Alexandria as well. Diophantus married at the age of 33 and had a son who later died at 42, only 4 years before Diophantus' death at 84. He is best known for his work, Arithmetica , which contains 13 books "consisting of 130 problems giving numerical solutions to determinate equations (those with a unique solution) and indeterminate equations" (Diophantus). The method he formulated for solving later became known as Diophantine analysis. From his book, Arithmetica , only 6 of the 13 books have survived. Scholars who studied his works concluded that "Diophantus was always satisfied with a rational number and did not require a whole number" (Diophantus). He did not deal with negative solutions and only required one solution to a quadratic equation, which was what most of the Arithmetica problems led to (Diophantus). Brahmagupta was the first to give the general solution of the linear Diophantine equation ax + by = c (Boyer 221). Diophantus did not use sophisticated algebraic notation. He did, however, introduce an algebraic symbolism that used an abbreviation for the unknown he was solving for (Diophantus). He also gained fame from another book called On Polygonal Numbers . Diophantus' methods of solving problems have had both lasting effects and great benefits for the studies of algebra and number theory.

NUMBERS & SYMBOLS:

Diophantus stated the traditional definition of a number to be a collection of units, but in his problems, he referred to each of his positive rational solutions as a number (Bashmakova 5). He represented the unknown number he was solving for by the symbol resembling the Greek letter (Boyer 181). He introduced what is now known as negative numbers and referred to them as derived from a word meaning to be missing, not to suffice (Bashmakova 5). Positive numbers were called meaning existence, being (Bashmakova 6). Diophantus is the one who came up with the ideas that a negative multiplied by a positive was a negative number and that negative times negative resulted in a positive number. To account for the exponents of the unknowns he was solving for, Diophantus introduced the following notations for the six powers (in algebraic notation: x, , , ..... ):

The first power -

The second power -

The third power -

The fourth power -

The fifth power -

The sixth power -

(Bashmakova 7).

He also denoted the constant term, , by the symbol (Bashmakova 7). A special symbol was created for negative exponents. For example, the negative exponents of and were represented by ^ ( ) and ^ ( ) respectively .

(Bashmakova 7). Diophantus had no symbol for the multiplication operation. He believed it was unnecessary because his coefficients were all definite numbers or fractions (Heath 39). "The results were simply put down without any preliminary step which would call for the use of a symbol" (Heath 39). For addition, no symbol was used to express this operation. Instead, it was expressed by "juxtaposition" (Heath 39). On the other hand, Diophantus used a symbol to represent subtraction. The symbol used to denote this operation was "an inverted with the top shortened" (Heath 41). Scholars discovered that Diophantus did not consider a symbol necessary for division in which the divisor divides the dividend without a remainder (Heath 44). In the other cases, the quotient was expressed as a fraction, whether the divisor is a specific number or contains the variable (Heath 44).

DIOPHANTINE EQUATIONS:

The purpose of any Diophantine equation is to solve for all the unknowns in the problem. When Diophantus was dealing with 2 or more unknowns, he would try to write all the unknowns in terms of only one of them. These equations can fall into two categories: (A) Determinate equations of different degrees or (B) Indeterminate equations (Heath 58). Determinate equations are divided into pure determinate equations, mixed quadratic equations, simultaneous equations involving quadratics, and cubic equations. Pure equations are those that contain only one power of the unknown, whatever the degree (Heath 58). Quadratic equations contain unknowns with degree powers of 2 and 1. Diophantus only dealt with one particular case in Arithmetica concerning cubic equations. Scholars believe it is not possible to judge from that example how far Diophantus was acquainted with the solution of equations of a degree higher than the second (Heath 67). Diophantus did not, in his Arithmetica , work on indeterminate equations of the first degree because he did not think they had any particular significance (Heath 67). Indeterminate equations of the second degree or higher contain two or more unknowns to solve for.

Diophantine equations are equations of polynomial expressions for which rational or integer solutions are sought. Usually, the term implies that we want integer solutions, but in a sense these are equivalent. If a given equation has rational solutions, a corresponding equation with integer solutions can be found by multiplying the first equation by an integer constant, namely, the least common multiple of the denominators of the numbers obtained by raising the solutions to the appropriate power. As the name suggests, many problems that we now call Diophantine equations are addressed in the Arithmetica of Diophantus. However, some of these problems were known well before the time of Diophantus. Also, some of the most famous problems of number theory, such as Fermat's Last Theorem, are Diophantine equations posed by mathematicians living much later.

Two well known results from beginning number theory are examples of Diophantine equations which predate Diophantus. These are linear equations of two variables, that is ax+by=c, and the quadratic equation of three variables, . Both of these problems were known by the Babylonians. Solutions to the second are often known as Pythagorean triangles, or Pythagorean triples, since a geometric interpretation of this is lengths of the sides of a right triangle, and the expression is, of course, the Pythagorean Theorem. Fermat's Last Theorem, that has no solution for n>2, is a generalization of this.

Solutions to linear Diophantine equations of two variables

For Diophantine equations of the type ax+by=c, there exists an infinite number of solutions if (a,b)|c, that is if the greatest common divisor of a and b divides c. If (a,b) does not divide c, then there exist no solutions in the integers. All solutions are of the form +kb/g, +ka/g, where g=(a,b).

Solutions to the Pythagorean theorem

It can be shown that all Pythagorean triples are of the form , where r>s>0 and r, s, and k are all integers.

First, obtain all relatively prime solutions.

Assume that (x,y,z)=1. If not, divide the expression by and the resulting terms are relatively prime. Now we can see that, since they are relatively prime, , , and cannot all be even. Thus x and y cannot both be even, since this would force z to be even as well. If x and y are both odd, then (mod 4) and (mod 4) (using := to represent congruence), so (mod 4). This is a contradiction. Thus, x and y must have opposite parity. Assume without loss of generality that y is even. From , and 4| , we get . Since and are relatively prime and their product is a perfect square, and are perfect squares. Let and . Now by simple arithmetic we obtain the desired identities (Niven 231). Enter values for r, s, and k in the following code and obtain your own Pythagorean triples.

> ptrips := (r,s,k) -> [k*(r^2-s^2),2*k*r*s,k*(r^2+s^2)];

> ptrips(2,1,1);

A generalization and geometric view of one of Diophantus' problems

A common criticism of Diophantus is that he never developed a general method of solutions to his problems. In her book Diophantus and Diophantine Equations Isabella Bashmakova refutes this view. She presents the arguement that many of his techniques were more general than critics think, but are not recognized as such due to limitations of his notation. For example, Diophantus does not introduce additional variables into a problem, but rather introduces an arbitrary integer. From reading the problem I intend to discuss, found in Book Two of Arithmetica , it can be seen that he is aware that any integer will do. Here is the problem, as it appears in Heath:

To divide a given square into a sum of two squares.

To divide 16 into a sum of two squares.

Let the first summand be and thus the second . The latter is to be a square. I form the square of the difference of an arbitrary multiple of x diminished by the root of 16, that is, diminished by 4. I form, for example, the square of 2x-4. It is . I put this expression equal to and subtract 16. In this way I obtain , hence x=16/5.

Thus one number is 256/25 and the other 144/25. The sum of these numbers is 16 and each summand is a square (Heath 145).

In the following commands we present Bashmakova's interpretation of this geometrically. In modern symbols and analytic geometry, the problem can be presented as solving for some integer a, which is the graph of a circle. Diophantus makes the substitution y=kx+a, again with a given k, which is a line through (0,a) and one other point on the circle, which he shows will also be rational. The following code will solve for that other point.

> line := y = k*x +a; circ := x^2 + y^2 = a^2;

> solve({line,circ},{x,y});

Since we know that k and a are rational, and the rational numbers form a field, the coordinates are also rational numbers.

> restart;

> with(plots);

> with(plottools);

> a:=4;

> k:=2;

> c:=circle([0,0], a, color=blue):

> f:=k*x+a:

> pl:=plot(f,x=-a..a):

> display([c,pl]);

Thus we can clearly see that the solutions to the equation are equavalent to the rational points on the circle centered at the origin with radius z, for a fixed z. Also, since we can divide a rational number by another rational and always have a rational as an answer (since, that's right, the rationals are a field) this is equivalent to finding the rational points on the unit circle.

The Impact of Diophantine Mathematics on the Modern World

Diophantus made contributions in mathematics that reverberated throughout history into the present day. Some of the most influential of his work is his number theory, his algebra, and his methods of problem solving. One of the primary factors that aided in preserving his works for so long was the follow-up work done by future mathematicians. Three major scholars that built on his works in some way were Vieta, Fermat, and Poincaré.
Vieta was a French jurist who lived from 1540 to 1603. His only published work was entitled Principes de cosmographie , which deals mostly with astronomy and cosmology. All of his mathematical pursuits resulted as an outgrowth of his scientific endeavors. He used Diophantus' work to compute plane and spherical triangles using general trigonometric relations that exist between components of such triangles.
Vieta's major contribution to modern mathematics arose from his reworking of the general Diophantine method. Vieta made use of Aristotelian ideas such as a determinate in an expression is important only in its character as a given, not intrinsically important in and of itself. Vieta went further and took two Diophantine procedures and united them. Specifically, Diophantus used problematical synthesis, geometric constructions, as the first step in a solution process. He would then take the specific solution and form an indeterminate solution. In other types of problems, he would do the exact reverse and first find the analytic solution and later the problematic solution. Vieta compiled this procedure and hypothesized that these two procedures paralleled each other and called his new procedures logistice speciosa. Vieta understood his procedure to be the "most comprehensive possible 'analytic' art, indifferently applicable to numbers and to geometric magnitudes" (Klein 165). Vieta took his procedures a step further and stated that he was more interested in the finding of the correct finding than in the correct finding itself. This type of thinking, which sprang from followers of Diophantine methods, has filtered down to us today and has been responsible for many of the breakthroughs in mathematics throughout history.
Pierre de Fermat lived contemporaneously with Descartes in the 17th century. Fermat worked with several areas of mathematics, although he was actually not a professional mathematician. He was first a lawyer, then a councilor in the local parlement . He studied analytic geometry, in which he used the notations set down by Vieta, which Vieta originally formulated based on Diophantus. Fermat went on from there and worked with finding tangent lines to curves. He formulated the definition of the derivative, . He also did work with integration, and his work is so remarkable that Laplace refers to him as the "father of differential calculus". As if these accomplishments weren't enough for an amateur mathematician, he went on to develop a theory of numbers based directly on Diophantus' Arithmetica . He developed his method of infinite descent as a method of proof. He used it to prove that the is irrational. The proof goes that if the number is rational it can be written as a ratio of integers / . He then shows that he can continually simplify his beginning equations down to a new a and b. Then he can find a third, fourth, etc. a and b. By the well ordering principle, there must be a least element to the sets that he can write, but since there is no end to the a's and b's , he arrives at a contradiction to his original assumption that the number is rational. He also claims to have found a proof for the theorem, . This equation has no positive integer solutions for n>2. So if I divide through by , this equation then becomes + =1, and this new equation has no positive rational solutions for t>2. The graphs of some of these values of n are interesting:

> fermy := proc(n,clr) local r,s; plots[implicitplot](r^n+s^n=1,r=0..1.0,s=0..(1.0),
scaling=constrained,color=clr) end;

> plots[display]([fermy(100,black),fermy(2,red),fermy(3,green),fermy(10,orange)],numpoints=100^2,scaling=constrained);

>

>

>

An adequate proof for Fermat's last theorem was found by Andrew Wiles at Princeton University within the last couple of years. His later theorem is that is divisible by p if a and p are relatively prime, or that the gcd(a,p)=1.

He proved this theorem using induction. Later, he expanded on this formula by induction that is prime. However, he only used five cases, and this theorem has been proven incorrect.
Poincaré was born in Nancy, France in the mid-nineteenth century. A lot of his work was with partial differential equations, which comes down directly from Fermat, who expanded much of his work from Diophantus. He contributed much to function theory, especially Abelian functions. He also came up with a model of Lobachevskian geometry within a Euclidean framework. The problem is as follows:
"Suppose that a world is bounded by a large sphere of radius R and the absolute temperature at a point within the sphere is , where r is the distance from the center of the sphere. . . To inhabitants of such a world, the universe would appear to be infinite; and rays of light, or 'straight lines', would not be rectilinear, but would be circles orthogonal to the limiting sphere and would appear to be infinite." (Boyer 604).

So, although Diophantus lived thousands of years ago, he made contributions to the field of mathematics, the rewards of which are still being reaped today. His work was followed by Vieta, who took his ideas and came up with a standard for algebraic expression which is still used today. In that since, every work on Algebra since Vieta has owed allegiance to Diophantus. But his work's impression on the world continued when Fermat referenced his work and came up with the basis for differential calculus, which has unlimited applications in the modern world. Fermat's work with prime numbers is the basis of modern encryption techniques, which is actually based on Euler's generalization of Fermat's prime number theories. These theories all began with a study of linear Diophantine equations. Encryption techniques are used on computer systems worldwide, and protect not only military intelligence, but also financial information such as the entire world's economy. Poincaré also made great advances based on Fermat's work in areas such as physics, topology, and differential equations. These examples of Diophantine contributions are sufficient to show that his works have helped shape the world in which we live.

References :
An Introduction to the Theory of Numbers ; Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery,

John Wiley and Sons, Inc.; NY New York; 1991.

Bashmakova, Isabella Grigoryevna. Diophantus and Diophantine Equations . The Mathematical

Association of America. USA, 1997.

Boyer, Carl. A History of Mathematics . John Wiley and Sons, Inc. New York, 1991.
Diophantus of Alexandria . http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Diophantus.html. Feb. 11, 1999.

Gow, James. A Short History of Greek Mathematics . Chelsea Publishing Company.
Cambridge, 1884.
Heath, Sir Thomas L. Diophantus of Alexandria: A Study in the History of Greek Algebra . Dover

Publications, Inc. New York, 1964.
Klein, Jacob. Greek Mathematical Thought and the Origin of Algebra . M.I.T. Press.
Cambridge, 1968.