Pell s Equation is a Diophantine equation in the form: Pell s Equation Claire Larkin The Equation x 2 dy 2 = where x and y are both integer solutions and n is a positive nonsquare integer. A diophantine equation is a polynomial equation where there are two or more unknowns and only the integer solutions are studied. This is important because only the solutions of x and y that are integer solutions are allowed for Pell s Equation. The Pell Equation is extremely important to mathematics because it allows you to approximate rational solutions to d. The equation also helps to find units in quadratic number fields as well as finding solutions to unrelated diophantine problems. There are many times that the equation comes out throughout the history of mathematics. It also has a great history dealing with some of the best mathematicians throughout history. The History of the Pell Equation Pell s Equation was in fact not first studied by John Pell himself, as some believe, and actually has very little to do with him. Brahmagupta was the first person to study this equation in depth, although there may have been others that looked at it before then, there is no evidence to prove it. Mathematicians who may have had contact with it before Brahmagupta was Diophantus and Archimedes. Brahmagupta was the first person to study this equation when he came up with his lemma. Brahmagupta s lemma was discovered in 628 AD, which is now known as the method of composition. From this lemma, he was able to make other discoveries regarding Pell s equation. Brahmagupta s lemma said this: If (a, b) and (c, d) are solutions to a Pell equation in the form x 2 + dy 2 =, then (bc + ad, bd + nac) and (bc ad, bd nac) are both integer solutions to the equation. Another important discovery that he made was that if (a, b) satisfies Pell s equation then so does (2ab, b 2 +na 2 ). Then he saw that the same thing could be applied again, showing that there could be infinitely many solutions to the equation, all starting from one solution. This showed the uniqueness of the Pell equations. The next person to do work on the Pell equation Bhaskara II, an Indian mathematician and astronomer, in 50. He discovered the cyclic method, which was an algorithm used to produce solutions to Pell s equation. He began by realizing that for any m, (, m) satisfies the equation. By using the method of composition and solving for x and y, he found that are solutions to x = (am + b)/k, y = (bm + na) 2 nx 2 + (m 2 n)/k = y 2. If one chooses his values of m wisely, then he is able to solve the Pell equation. Bhaskara had no proof of his method, it is unknown the exact reasons why. The next person to contribute to Pell s equation was Narayana, who gave new examples of the cyclic method that Bhaskara discovered. The Pell s Equation became of real interest when Fermat issued his famous challenge, now know as Fermat s Challenge. The European mathematicians had no knowledge of the Indian mathematical discoveries that had been going on nearly 500 years earlier. In February of 657 Fermat issued the following challenge:
Given a positive integer d, find a positive integer y such that dy 2 + is a perfect square. In simpler terms, this equation came down to x 2 dy 2 = 2, which is now known as Pell s equation. Fermat issued several challenge problems which were attempted by various mathematicians. Some of these mathematicians were Frenicle de Bessy, Brouncker, and Wallis. Brouncker discovered a method that is similar to the continued fractions method, which was later developed by Lagrange. Frenicle de Bessy then challenged Brouncker to solve 33x 2 + = y 2, because he claimed that he could solve any example of the equation. He found the smallest solutions to be x = 893805856460, y = 32882082934849 Later on in 658, Rahn published an algebra book which contained an example of a Pell equation. The book was written with the help of John Pell, and that is the only known connection between Pell and the equation that is now named Pell s equation. During this time that Pell s equation was being studied in the 7th century, there was a vast amount of mathematics occurring throughout the world and especially in Europe. The Scientific Revolution was also going on throughout the 7th century, which brought forth new ideas in astronomy, mathematics, and philosopphy. Pierre Fermat is said to have single-handedly invented modern number theory. He is credited for advancements that led to the development of early calculus as well as early progress in probability theory. He did much of his math through communication with other mathematicians which had little or no proof of his theorems. His Little Theorem and Fermat s Last Theorem are two of his most well known theorems, although Fermat s Last Theorem was not solved until almost 350 years later. Although Pell s Equation seems small in value compared to many of the advancements during this time, they are extremely important in number theory and arise in various other areas of mathematics. Figure : John Pell, the man the equation is named after. 2
Figure 2: Indian mathematician Brahmagupta, one of the first people to study what is now known as Pell s equation. The Cattle Problem The Archimedes Cattle Problem is one of the most well-known historical problems of polynomial equations with integer solutions. In 769 the Herzog August Library in Germany published translations of some of the manuscripts that were in Greek and Latin, one of these being a Greek poem of 44 lines, which is now known as the Cattle Problem. It asks the reader to find the number of cattle in the herd of the god of the sun. The first part of the problem goes as follows: The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted (dappled), and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd? In addition, the first part of the problem can be set up as a system of equations. W = 5 6 B + Y B = 9 20 D + Y D = 3 42 W + Y 3
w = 7 2 (W + Y ) b = 9 (D + d) 20 d = (Y + y) 30 y = 3 (W + w) 42 The first three linear equations can easily be solved. We have Doing the arithmetic, we find that W = 5 6 B + Y = 5 6 ( 9 20 D + Y ) + Y = 5 6 ( 9 20 (3 42 W + Y ) + Y ) + Y 4455W = 30Y, which gives general solutions of W = 30t, Y = 4455t, D = 7900t, and B = 800t. All of these can be divided by 5, so we are given the following: W = 2226t, Y = 89t, D = 580t, B = 602t After taking into account the last four equations from above, we can get the equations for the other four variables, w, b, d, y The second part of the problem says: But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom. This is saying that the number of White and Black bulls is a square number and the Yellow and Dappled cows is a triangular number. This gives you two equations: and B + W = (2 2 )(3)()(29)(4657)k D + Y = t2 + t 2 By solving for t and then substituting D+Y and k and finding something that makes the discriminant a perfect square, it ends as a Pell equation as follows: p 2 + (4)(609)(7766)(4657 2 )q 2 = The solution to the problem was first solved by AmthorThis is the first known time that a Pell equation appeared, but it is unknown whether Archimedes made the connection between his question coming down to an equation of this form and it being an equation that appears in many other places. This was the first time 4
in history that it is known to appear, but it was not studied as thoroughly as it was later on by Brahmagupta and then others after him. Figure 3: IThe Greek mathematician who wrote the Cattle Problem which was not solved until many years after it was written. Continued Fractions To understand how a Pell s Equation is solved, it is critical to understand continued fractions. The definition of a continued fraction expansion of a real number is the following: The expression of the form where the a i s are integers. a 0 + a + a 2 + a 3 + If the starting number is rational, then it will produce a finite continued fraction representation of the number. Continued fractions are used to represent rational numbers as fractions as well as irrational numbers. For irrational numbers, the continued fraction provides rational approximations to the number, which is called the convergent of the continued fraction. Here is an example of the continued fraction of 89 37 : 89 37 = 2 + 5 37 = 2 + 37 2 + 2 + 2 + 7 5 = 5
The continued fraction of this would then be expressed as : [2; 2, 2, 7]. When dealing with infinite continued fractions, it is also important to understand what periodic continued fractions are. Periodic continued fractions are those whose terms repeat from some point onwards. The minimal number of repeating terms is called the period of the continued fraction. This will be used later on when using continued fractions to solve Pell s Equation. Figure 4: This is another representation of continued fractions through the use of rectangles. Solving Pell s Equation through Continued Fractions To begin, we must understand the theorem which shows that continued fractions give solutions to Pell s equation. It is easy to say that the continued fraction is a solution, but it is much more complicated to understand why they give solutions. The first theorem that is shown explains that every irrational number can be represented with as an infinite continued fraction Theorem (): Every irrational number has a unique representation as an infinite continued fraction, the representation being obtained from the continued fraction algorithm. Corollary: if pn q n is the nth convergent to the irrational number x, then x p n q n < < q n + q n qn 2 Theorem () says that every irrational number has an infinite continued fraction representation, and then the corollary shows that the convergent of the irrational number also follows the said properties. 6
Theorem (2): Let x be an arbitrary irrational number. If the rational number a where b > and b gcd(a, b) =, satisfies x a b < 2b 2 then a b is one of the convergents of p n/q n in the continued fraction representation of x. This is an important theorem to state because it will be directly used in the next theorem that shows how continued fractions are used to find solutions to Pell s Equations. Theorem (3): If is a positive solution of, then p, q x 2 dy 2 = p q is convergent of the continued fraction expansion of d. Proof: In light of the hypothesis that p 2 dq 2 =, we have This implies that p > q as well as that As a result, 0 < p q d < If we then take into consideration the fact that p d p + q d = p q d = q(p + q d d q(q d + q d) = x a b < 2b 2 d 2q 2 d = 2q 2 tells us a b is one of the convergents of p n q n, then that indicates that p/q must be a convergent of d. Although the converse of this theorem is not always true, that all of the convergents of p n q n The next theorem Theorem: If p/q is a convergent of the continued fraction expansion of d, then x = p, y = q is a solution of one of the equations x 2 dy 2 = k where k < + 2 d. 7
Proof: If p/q is a convergent of d, then the corollary from Theorem () guarantees that and therefore if you multiply both sides by q, d p q < q 2 This being so, we have p q d < q p + q d = (p q d) + 2q d < p q d + 2q d < q + 2q d < ( + 2 d)q These two inequalities combine to yield p + q d = p q d p + q d < q ( + 2 d)q = + 2 d Which shows that the x = p, y = p is a solution to one of the equations. These proofs show that the solution to a Pell equation can be found through the convergence of a continued fraction of d. Therefore, the way to find the solution to Pell s equation is by finding the continued fraction representation of d, then finding the convergent of the equation at the last point in the period, which will give you p = x, q = y Norms of Real Quadratic Fields To understand how Pell s equation is used in Norms of Real quadratic fields, there are many things that must first be defined. A complex number is a number that can be expressed in the form of a + bi where a aand b are real numbers and i is the imaginary unit. A Quadratic Integer is a complex number which is a solution of an equation in the form of. It is an algebraic integer in a quadratic field. x 2 + Bx + C = 0 Every square-free integer D defines a quadratic integer ring, which is the integral domain of the algebraic integers contained in Q( D). In simpler terms, the Ring of Integers are all of the quadratic integers belonging to a given quadratic field. The Number Field is defined as: 8
Subset, F, of C that is i) Closed under all operations and ii) There exists a complex number a such that each element of F is a rational combination of (, a, a 2,..., a m ) for some m. A quadratic integer in Q( D) may be written as a + b D, where a and b are integers. The Norm of a quadratic integer is then defined as: N(a + b D) = a 2 + b 2 D where, the norm of a quadratic integer is always an integer. A Unit is a quadratic integer in the ring of integers of Q( D) if the norm is equal to or -. This is where Pell s equation comes into play. Remember that Pell s Equation is defined as x 2 my 2 = If a unit is when the norm, N(a + b D) = a 2 + b 2 D, is equational to or -, then you are essentially solving Pell s Equation when you find the units of the real quadratic fields. This is because the Norm is equal to a 2 + b 2 D, which is the left side of Pell s Equation. Then to find the Unit of it the norm has to equal or - for it to be a unit. This is then Pell s Equation, a 2 + b 2 D =. Pell s Equation appears in many areas throughout mathematics, and the Norms of Real Quadratic Fields is one of those. By studying Pell s Equation and the ways to solve it, it is much easier to solve the norms of the real quadratic fields. Because we already know how to solve Pell s Equation through continued fractions, if you are given a Quadratic field Q( D) and asked to find solutions of it that have a norm of, then you can solve the Pell Equation a 2 + b 2 =. One example of this would be if you are given the quadratic field Q( 2) and asked to find the solutions of it that have a norm of. Since this is a real quadratic field, we know that. Therefore, the norm of the equation would be Q( 2) = (a + b D : a, b Q) N(a + b 2) = a 2 + b 2 2. If we are trying to find when it has a norm of, we would set the norm equal to, a 2 + b 2 2 =. This then is a Pell Equation. It can be solved through continued fractions. The continued fraction expansion is: 2 = [; 2]. Because the period, the part that repeats, is 2, then the convergent of 2 is 3 2. This tells us that p q = 3 2, and p = x and q = y. Therefore, the solution to the quadratic field Q( 2) with a norm of is (3, 2). In addition, this shows how Pell s equation can have infinitely many solutions because if there is a single solution which is the unit of the ring, then you can raise it to any power. If you raise a unit to any power then it will still be a unit. This also shows how Pell s equation can have extremely large solutions. 9
Conclusion Pell s Equation first appeared when the mathematician Archimedes created the infamous Cattle Problem that was solved many years after it was first written. It was first studied intensely in India by mathematicians Brahmagupta and Bhaskara II as well as Narayana. In Europe it was first introduced by Fermat, who challenged fellow mathematicians to solve equations in the form of a Pell equation. However, it was falsely named after mathematician John Pell, because he had very little to do with solving the equation. It can be solved through the use of continued fractions. It also appears many times throughout number theory and appears in much larger mathematical concepts as a much smaller concept. References http://www.maths.surrey.ac.uk/hosted-sites/r.knott/fibonacci/cfintro.html#section. http://archimedespalimpsest.org/about/history/archimedes.php http://wstein.org/edu/2007/spring/ent/ent-html/node65.html https://issuu.com/tsangkinfun/docs/_david_m._burton elementary_number/353 http://www.maths.surrey.ac.uk/hosted-sites/r.knott/fibonacci/cfintro.html http://mathworld.wolfram.com/pellequation.html http://www.historyguide.org/intellect/lecture7a.html http://www.storyofmathematics.com/7th_descartes.html http://www.pballew.net/mathbooks.html http://www-history.mcs.st-and.ac.uk/histtopics/pell.html https://en.wikipedia.org/wiki/archimedes _cattle_problem 0