Peter Ramcharitar
Diophantine Equations
Diophantine equations are polynomial equations that have two or more variables. The answers can only be an integer value. These equations were named after Diophantus of Alexandria even though these equations were first solved by Hindu Mathematicians. There are three classes of these equations, equations with no solutions, equations with finite solutions, and equations with infinitely many solutions.
Diophantus of Alexandria was a Hellenistic mathematician. He is known as the father of Algebra. Diophantus was also the writer of books called Arithmetica. One of his books was read by Pierre De Fermat, and he saw an equation that Diophantus said had no solution. He was the first person to bring symbols into algebraic equations. The symbols that Diophantus used were scribal abbreviations ( the abbreviations that were used by the scribes who wrote in Latin or Greek). The study of these equations by Diophantus is now called Diophantine Analysis.
Aryabhata was the first person to solve a Diophantine equation systematically. He was very interested to find an integer value to Diophantine equations that are in the form ax+by=c.He needed to find out the number that when divided by 7 has a remainder of 1, that same number has a remainder of 5 when divided by 8, and this same number has to have a remainder of 4 when divided by 9. The equation that he used to do this was N=8x+5=9y+4=7z+1.The smallest integer value for the value of N was 85. The process that Aryabhata took to get this answer was very difficult and took an extensive period of time for him to reach his goal because there were many steps that he had to do to get his final answer. . The method that Aryabhata used is called the kuṭṭaka. This is the process of using a recursive equation to convert the original numbers into smaller numbers. The equation that he used became the standard that was used for solving linear equations.
A linear Diophantine equation is an equation that is in the general form of ax+by=c.a, b, and c are given and the solutions are the integer values of x and y that make the equation true. The theorem that describes this states “This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if (x, y) is a solution, then the other solutions have the form (x + kv, y − ku), where k is an arbitrary integer, and u and v are the quotients of a and b (respectively) by the greatest common divisor of a and b.” The proof for this states “If d is this greatest common divisor, Bézout's identity asserts the existence of integers e and f such that ae + bf = d. If c is a multiple of d, then c = dh for some integer h, and (eh, fh) is a solution. On the other hand, for every pair of integers x and y, the greatest common divisor d of a and b divides ax + by. Thus, if the equation has a solution, then c must be a multiple of d. If a = ud and b = vd, then for every solution (x, y), we have a(x + kv) + b(y − ku) = ax + by + k(av − bu) = ax + by + k(udv − vdu) = ax + by, showing that (x + kv, y − ku) is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one deduces that u(x2 − x1) + v(y2 − y1) = 0. As u and v are coprime, Euclid's lemma shows that v divides x2 − x1, and thus that there exists an integer k such that x2 − x1 = kv and y2 − y1 = −ku. Therefore, x2 = x1 + kv and y2 = y1 − ku, which completes the proof.” The Chinese remainder theorem describes an important class of linear Diophantine systems of equations: let n1, …, nk be k pairwise coprime integers greater than one, a1, …, ak be k arbitrary integers, and N be the product n1 ··· nk.”(diophantine) The Bézout’s identity theorem says that a and b are integers that have a greatest
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common divisor that is d. Euclid’s lemma says that if a prime number p divides the product ab of the two integers a and b, then the prime number p has to divide at least one of the integer a and b.
In a more general form, any system of linear Diophantine equations can be solved by calculating the Smith normal form ( the normal form that is able to be defined by any matrix with the entries in a principal ideal domain.) but it has to be in a way that is similar to using the reduced row echelon ( A matrix will be in echelon form if it has the shape that is a result of a Gaussian elimination.) to solve the system over a field. When matrix notation is used, any system of linear Diophantine equations can be written as ax=c where c will be an m1 column matrix of integers, x will be an n1 column matrix of unknowns, and a will be an mn matrix of integers. Shown in figure 1 is are matrices.
Figure 1
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Homogeneous Diophantine equations are Diophantine equations that have a homogeneous polynomial ( a polynomial whose nonzero terms will all have the same degree).
An example of this equation is the one from Fermat’s last theorem: xd+yd-zd=0. This homogeneous polynomial in n indeterminates defines a hypersurface ( the generalization of the concepts in a hyperplane, surface, and plane curve.) in the projective space of dimension n – 1. The process of solving a homogeneous Diophantine equation is identical to finding the rational points of a projective hypersurface. It is very difficult to solve a homogeneous Diophantine equation even in its simplest form. It is easier to solve these problems in a degree that is higher than 1.
A general form for an exponential diophantine equation is ax2+cy2=k where the integer values of a,c, and k are given and the unknowns are x and y. The second order equation which most people work with is ax2+bxy+cy2=k. According to Kiyoshi Itô, this second order equation can be solved by just using solutions to the Pell equation. All of the solutions of
ax2+bxy+cy2=1are part of the convergents to the continued fractions of the roots of ax2+bx+c. If there are more than two variables in the equation, the form of the equation would be x2-Dy2=1 as to where D is an integer in a pell equation. A pell equation is an equation in the form that is shown before where D>0. In this equation, it is shown that the continued fraction of a quadratic surd will always become periodic at ar+1as to wherear+1=2a0.
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To calculate the continued fractions convergents to D, always use the recurrence relations which are shown in these equations. a0=[D],p0=a0,p1=a0a1+1pn=anpn-1+pn+2,q0=1,q1=a1,qn=anqn-1+qn-2. The table shown in figure 2 shows all of the smallest integer solutions to the Pell equation.
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Figure 2
There are also forms which give a prime solution to regular exponential diophantine equations that aren’t as complicated as the pell equations. These forms are shown in Figure 3
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Figure 3
Diophantine equations are used in chemistry. They are used to balance chemical equations. For example, these equations are able to balance an equation like the one in figure 5.
Figure 4
From this equation, chemists are immediately able to find the equations that are shown in figure 5 by breaking down the equation shown in figure 4.
Figure 5
The system that is shown in figure 6 can be simplified to 5×1+2×3-4x'4=0. This equation is now a linear Diophantine equation. From this equation, you can tell that the solution is [x1, x3, x'4]=[5,2,-4]. So when these numbers are substituted into the chemical equation, the equation becomes what is shown in figure 6.
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Figure 6
But this is not the only solution to this equation. The set S of all positive integer solutions of
is infinite and can be written in the form shown in figure 7.
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Figure 7
Diophantine equations have been used since 400A.D. There are many forms of these equations including exponential, linear, and homogeneous Diophantine equations. They are used in to balance out chemistry equations in their linear form.
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Works Cited