Continued Fractions and Pell s Equation

Transcription

1 Max Lah Joatha Spiegel May, 06 Abstract Cotiued fractios provide a useful, ad arguably more atural, way to uderstad ad represet real umbers as a alterative to decimal expasios I this paper, we eumerate some of the most saliet qualities of simple cotiued fractio represetatios of real umbers, classify the periodic cotiued fractios as the quadratic irratioals, ad use simple cotiued fractios to fid the iteger solutios x, y) Z to the geeralized Pell s equatio x Dy = ) m To begi, let us defie a fiite simple cotiued fractio { Fiite simple cotiued fractios are a } a method for represetig the ratioal umbers Q = b : a, b Z ; b 0, ad Defiitio Fiite simple cotiued fractio) Let N be a atural umber, ad let a i ) = a 0,, a ) be a fiite sequece of atural umbers a 0,, a N The fiite simple cotiued fractio geerated by a i ), deoted a i, is defied as follows a i = a 0 + a + a Now, we use the Euclidea algorithm to geerate a fiite simple cotiued fractio represetatio for ay ratioal umber Recall that the Euclidea algorithm recursively geerates sequeces q i ) =0 of quotiets q i N ad r i ) =0 of remaiders r i N from two iteger iputs a, b N, ad termiates with r = gcd a, b) ad r = 0 Example Let a b Q be a ratioal umber Let q i) be the fiite sequece of atural umbers q,, q N geerated by the quotiets i the Euclidea algorithm applied to a ad b The For example, let a = 38 ad b = 9 Note that q i = a b 38 = 4) 9) + ; 9 = 4) ) + ; = ) ) + 0

2 So, lettig a i ) 39 = 4, 4, ), we have that 9 = a i = I this way, we have a caoical way of represetig ay ratioal umber a as a fiite simple b cotiued fractio Moreover, it ca be show that every oiteger) ratioal umber has precisely two fiite simple cotiued fractio represetatios, each differig oly i the last two terms of the fiite geeratig sequeces For example, let b i ) 3 = 4, 4,, ), ad ote that 38 9 = a i = = = 3 b i + Defiitio Ifiite simple cotiued fractio) Let a i ) be a ifiite sequece of atural umbers a 0, a, a, N The ifiite simple cotiued fractio geerated by a i ), deoted a i, is defied to be the limit of the partial simple cotiued fractios a i = lim a i ) For ay ifiite sequece a i ) of atural ) umbers a i N, we have defied a ifiite sequece of fiite simple cotiued fractios a i, where a i is called the th coverget However, it =0 remais to show that such a sequece must coverge to a limitig value Covergece of Simple Cotiued Fractios Theorem ) Let a i ) be a ifiite sequece of atural umbers a i N The the sequece a i of covergets a i coverges to a real umber a i R =0 Proof Let p ad q be the umerator ad deomiator of the th coverget a i Lemma The ifiite sequeces p ) =0 ad q ) =0 satisfy the followig recursio relatios a 0, if = 0; p = a a 0 +, if = ; ad ) a p + p, otherwise, ad, if = 0; q = a, if = ; ad a q + q, otherwise 3)

3 Proof By iductio o N Base cases: For the first base case, suppose that = 0 Note that p 0 q 0 = 0 a i = a i = a 0, For the secod base case, suppose that = Note that p q = a i = a 0 + a = a a 0 + a, Iductive step: Suppose for the iductive hypothesis that the lemma holds for all < k, for some atural umber k > Let a i) k be the fiite sequece of atural umbers a i N defied as follows a i) k = a,, a k, a k + ) a k+ Note that k a i = k+ a i By the iductio hypothesis, ) p k+ = p k q k+ q k = a k p k + p a k + k a k+ p k + p k a k q k + = ) q k a k + a k+ q k + q k = a k+ a k p k + p k ) + p k a k+ a k q k + q k ) + q k = a k+p k + p k a k+ q k + q k, Lemma p q p q = ) q q for all atural umbers N Proof By iductio o N Base case: Suppose that = The p 0 p = a 0 q 0 q a a 0 + = =, a a q 0 q Iductive step: Suppose for the iductive hypothesis that The p q p q = ) q q p q p q = ) p q p q = p a q + q ) a p + p ) q = p q p q = ) p q p q = ) q q, 3

4 Therefore, sice q ) =0 is a strictly icreasig sequece of itegers, for all atural umbers m > N, m a i a i = p m p q m q m i=+ approaches 0 as m,, sice q i grows liearly ) p q =0 is a Cauchy sequece of ratioal umbers p i p i q i = q i m i=+ q i q i 0 Therefore, the covergets umber a i R, sice the real umbers R form a complete metric space a i ) =0 a i Q, ad so coverges to some real = Cotiued Fractio Represetatios of Real Numbers I may ways, cotiued fractios are a more atural way to represet real umbers tha decimal expasios As show above, the geeratig sequece of the simple cotiued fractio of the ratio of two umbers is the quotiet sequece costructed by the Euclidea algorithm, ad so the simple cotiued fractio represetatio of a ratioal umber cotais a vast wealth of iformatio about the umber, whereas the decimal expasios ideed, the -ary expasios for ay base N) of may simple fractios obscure such iformatio The Euclidea algorithm techique demostrated above ca be geeralized to fid the cotiued fractio represetatio of a irratioal umber x R \ Q i the same maer However, such a algorithm is ot guarateed to termiate as it is i the case of the ratioal umbers Q I fact, if a umber x R \ Q is irratioal, such a process caot possibly termiate, sice termiatio would imply that there is a fiite simple cotiued fractio represetatio for x, which would imply that x were ratioal As show below, the simple cotiued fractio represetatios of some irratioal umbers have iterestig ad beautiful forms We will see a example of such beauty, ad the we will see a theorem about those irratioal umbers whose simple cotiued fractio represetatios repeat Example Let ϕ = + 5 be the golde ratio ϕ has cotiued fractio represetatio ϕ = = ) Proof Cosider the simple cotiued fractio Sice it repeats with a period of, we have the simple recurrece relatio = +, which follows immediately from Equatio 4) This 4

5 implies that From our defiitio of desired ) 0 = = ± 5 i Equatio 4), we kow that >, so = + 5 = ϕ, as The above techique may be geeralized ad used to show that ay repeatig simple cotiued fractio is a irratioal solutio to a quadratic equatio with iteger coefficiets, by solvig the recurrece relatio geerated by the repetitio We offer the followig theorem without proof Theorem Let a i ) i= be a ifiite sequece of atural umbers a i N If a i ) i= evetually repeats, the α = a i is a irratioal solutio to a quadratic equatio with iteger coefficiets Proof Suppose that a i ) i= evetually repeats It suffices to show that α = i= a i is a quadratic irratioal i the case where a i ) i= is purely periodic Therefore, let a i) i= = a 0,, a ) So α = a 0 + a + + a + α We may reduce this to α = uα + v, for some positive itegers u, v, w, z N So wα + z α wα + z) = uα + v wα + zα = uα + v wα + z u) α v = 0 So α is a solutio to the quadratic equatio wx + z u) x v = 0 Moreover, sice α = a i is a ifiite simple cotiued fractio, α R \ Q is irratioal So α is a quadratic irratioal, as desired 3 Pell s Equatio Let D N be a atural umber, ad suppose that D is ot a perfect square; that is, there does ot exist a iteger k Z such that k = D The Diophatie equatio x Dy = 5) 5

6 is kow as Pell s equatio, ad we ca use the simple cotiued fractio represetatio of D to solve it Theorem 3 Let D N be a atural umber, ad suppose that D is ot a perfect square D has simple cotiued fractio represetatio geerated by the repeatig sequece a 0, a,, a m ) Let p q = m a i The x, y) = p, q) Z are the smallest iteger solutios to the geeralized Pell s equatio, x Dy = ) m 6) Example 3 Cosider Pell s equatio i the case D =, x y = 7) Note that has cotiued fractio represetatio = 4,,,,,, 8 ), so m = 6 m = 5, ad the 5 th coverget is 4,,,,, ) = = 55 + So x, y) = 55, ) is the smallest solutio i itegers to the geeralized Pell s equatio, Equatio 7), x y = ) m =, Moreover, the solutio set S to Pell s equatio is as follows { S = x k, y k ) Z : 55 + ) } k = xk + y k k= Example 4 We may still solve Pell s equatio i the case where m is odd, by squarig the solutios Cosider Pell s equatio i the case D = 9, x 9y = 8) Note that 9 has cotiued fractio represetatio 9 = 5,,,,, 0 ), so m = 5 m = 4, ad the 4 th coverget is 5,,,, ) = = 70 3 So x, y) = 70, 3) is the smallest solutio i itegers to the geeralized Pell s equatio, x 9y = ) m =, So, squarig both sides of the geeralized Pell s equatio, the smallest ) solutio i itegers to Pell s equatio, Equatio 8), is x, y) = ), 70) 3) = 980, 80) Moreover, the solutio set S to Pell s equatio is as follows { S = x k, y k ) Z : ) } k 9 = xk + y k 6 k=

7 Cotiued fractios provide a atural ad useful way to approach represetig real umbers The geeratig sequeces of simple cotiued fractios are iterestigly coected to the Euclidea algorithm, which stems from the atural associatio of ratioal umbers ad greatest commo divisors Moreover, the simple cotiued fractio represetatios of the quadratic irratioals provide a quick ad easy way to solve Pell s equatio, which would otherwise prove difficult Refereces [] Joseph H Silverma, A Friedly Itroductio to Number Theory, Pearso Educatio, Ic, New York, 4th editio, 0 [] Seug Hyu Yag, Cotiued Fractios ad Pell s Equatio, Uiversity of Chicago, 008 [3] Peter houry ad Gerard D offi, Cotiued Fractios ad their Applicatio to Solvig Pell s Equatios, Uiversity of Massachusetts, Bosto, 009 [4] Ahmet Tekca, Cotiued Fractios Expasio of D ad Pell s Equatio x Dy =, Mathematica Moravica, Uiversity of ragujevac, Volume 8-, 04 [5] Eva Dummit, Cotiued Fractios ad Diophatie Equatios, Uiversity of Rochester, New York, Volume, 04 7