BAKER S EXPLICIT ABC-CONJECTURE AND APPLICATIONS

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1 BAKER S EXPLICIT ABC-CONJECTURE AND APPLICATIONS SHANTA LAISHRAM AND T. N. SHOREY Dedicated to Professor Andrzej Schinzel on his 75th Birthday Abstract. The conjecture of Masser-Oesterlé, popularly known as abc-conjecture have many consequences. We use an explicit version due to Baker to solve a number of conjectures.. Introduction The well known conjecture of Masser-Oesterle states that Conjecture.. Oesterlé and Masser s abc-conjecture: For any given ɛ > 0 there exists a constant c ɛ depending only on ɛ such that if () a + b = c where a, b and c are coprime positive integers, then c c ɛ p p abc +ɛ. It is known as abc-conjecture; the name derives from the usage of letters a, b, c in (). For any positive integer i >, let N = N(i) = p i p be the radical of i, P (i) be the greatest prime factor of i and ω(i) be the number of distinct prime factors of i and we put N() =, P () = and ω() = 0. An explicit version of this conjecture due to Baker [Bak94] is the following: Conjecture.2. Explicit abc-conjecture: Let a, b and c be pairwise coprime positive integers satisfying (). Then c < 6 (log N)ω N 5 ω! where N = N(abc) and ω = ω(n). We observe that N = N(abc) 2 whenever a, b, c satisfy (). We shall refer to Conjecture. as abc conjecture and Conjecture.2 as explicit abc conjecture. Conjecture.2 implies the following explicit version of Conjecture Mathematics Subject Classification. D4, D45, D6. Key words and phrases. ABC Conjecture, Fermat-Catalan Equation, Nagell-Ljungrenn Equation, Goormaghtigh Equation.

2 2 SHANTA LAISHRAM AND T. N. SHOREY Theorem. Assume Conjecture.2. Let a, b and c be pairwise coprime positive integers satisfying () and N = N(abc). Then we have (2) c < N Further for 0 < ɛ 3 4, there exists ω ɛ depending only ɛ such that when N = N(abc) N ɛ = p p ωɛ p, we have where κ ɛ = c < κ ɛ N +ɛ 6 5 2π max(ω, ω ɛ ) 6 5 2πω ɛ with ω = ω(n). Here are some values of ɛ, ω ɛ and N ɛ. 3 ɛ ω ɛ N ɛ e 37.0 e e e e e e Thus c < N 2 which was conjectured in Granville and Tucker [GrTu02]. We present here some consequences of Theorem. Nagell-Ljunggren equation is the equation y q = xn (3) x in integers x >, y >, n > 2, q >. It is known that 2 = 35 3, 202 = 74 7, 73 = 83 8 which are called the exceptional solutions. Any other solution is termed as nonexceptional solutions. For an account of results on (3), see Shorey [Sho99] and Bugeaud and Mignotte [BuMi02]. It is conjectured that there are no non-exceptional solutions. We prove in Section 4 the following. Theorem 2. Assume Conjecture.2. There are no non-exceptional solutions of equation (3) in integers x >, y >, n > 2, q >. (4) Let (p, q, r) Z 2 with (p, q, r) (2, 2, 2). The equation x p + y q = z r, (x, y, z) =, x, y, z Z is called the Generalized Fermat Equation or Fermat-Catalan Equation with signature (p, q, r). An integer solution (x, y, z) is said to be non-trivial if xyz 0 and primitive if x, y, z are coprime. We are interested in finding non-trivial primitive integer solutions of (4). The case p = q = r is the famous Fermat s equation which is completely solved by Wiles [Wil95]. One of known solution p = 3 2 of (4) comes from Catalan s equation. Let χ = + +. The parametrization of nontrivial primitive integer p q r solutions for (p, q, r) with χ 0 is completely solved ([Beu04], [Coh07]). It was shown by Darmon and Granville [DaGr95] that (4) has only finitely many solutions

3 BAKER S EXPLICIT ABC-CONJECTURE AND APPLICATIONS 3 in x, y, z if χ < 0.When 2 {p, q, r}, there are some known solutions. So, we consider p 3, q 3, r 3. An open problem in this direction is the following. Conjecture.3. Tijdeman, Zagier: There are no non-trivial solutions to (4) in positive integers x, y, z, p, q, r with p 3, q 3 and r 3. This is also referred to as Beal s Conjecture or Fermat-Catalan Conjecture. This conjecture has been established for many signatures (p, q, r), including for several infinite families of signatures. For exhaustive surveys, see [Beu04], [Coh07, Chapter 4], [Kra99] and [PSS07]. Let [p, q, r] denote all permutations of ordered triples (p, q, r) and let Q = {[3, 5, p] : 7 p 23, p prime} {[3, 4, p] : p prime}. We prove the following in Section 5. Theorem 3. Assume Conjecture.2. There are no non-trivial solutions to (4) in positive integers x, y, z, p, q, r with p 3, q 3 and r 3 with (p, q, r) Q. Further for (p, q, r) Q, we have max(x p, y q, z r ) < e Another equation which we will be considering is the equation of Goormaghtigh x m (5) x = yn integers x >, y >, m > 2, n > 2 with x y. y We may assume without loss of generality that x > y > and 2 < m < n. It is known that 3 = 53 5 = 25 2 and 89 = = 23 (6) 2 are the solutions of (5) and it is conjectured that there are no other solutions. A weaker conjecture states that there are only finitely many solutions x, y, m, n of (5). We refer to [Sho99] for a survey of results on (5). We prove in Section 6 that Theorem 4. Assume Conjecture.2. Then equation (5) in integers x >, y >, m > 2, n > 3 with x > y implies that m 6 and further 7 n 7, n / {, 6} if m = 6; moreover there exists an effectively computable absolute constant C such that max(x, y, n) C. Thus, assuming Conjecture.2, equation (5) has only finitely many solutions in integers x >, y >, m > 2, n > 3 with x y and this improves considerably Saradha [Sar, Theorem.4]. 2. Notation and Preliminaries For an integer i > 0, let p i denote the i th prime. For a real x > 0, let Θ(x) = p x p and θ(x) = log(θ(x)). We write log 2 i for log(log i). We have Lemma 2.. We have

4 4 SHANTA LAISHRAM AND T. N. SHOREY ( log x (i) π(x) x for x >. log x (ii) p i i(log i + log 2 i ) for i (iii) θ(p i ) i(log i + log 2 i ) for i (iv) θ(x) <.00008x for x > 0 (v) 2πk( k e )k e k+ k! 2πk( k e )k e k. ) Here we understand that log 2 =. The estimates (i) and (ii) are due to Dusart, see [Dus99b] and [Dus99a], respectively. The estimate (iii) is [Rob83, Theorem 6]. For estimate (iv), see [Dus99b]. The estimate (v) is [Rob55, Theorem 6]. 3. Proof of Theorem Let ɛ > 0 and N be an integer with ω(n) = ω. Then N Θ(p ω ) or log N θ(p ω ). Given i, we observe that is an increasing function for log M i. Let ɛ M ɛ (log M) i X 0 (i) = log i + log 2 i Then θ(p i ) ix 0 (i) by Lemma 2. (iii). Observe that X 0 (i) > for i 5. Let ω 5 be smallest i such that (7) ɛx 0 (i) log X 0 (i) for all i ω. Note that ɛx 0 (i) for i ω implying log N θ(p ω ) ωx 0 (ω) ω ɛ when ω ω by Lemma 2. (iii). Therefore ω!n ɛ (log N) ω!θ(p ω) ɛ ω (θ(p ω )) ω ω!eɛωx0(ω) eɛωx0(ω) (ωx 0 (ω)) > 2πω( ω ω e )ω (ωx 0 (ω)) when ω ω. ω Thus for ω ω, we have from (7) that ( ω!e ɛωx 0 ) (ω) log > log 2πω + ω(log(ω) ) + ɛωx (ωx 0 (ω)) ω 0 (ω) ω(log ω + log X 0 (ω)) > log 2πω + ω(ɛx 0 (ω) log X 0 (ω) ) log 2πω implying ω!n ɛ (log N) ω!θ(p ω) ɛ ω (θ(p ω )) > 2πω when ω ω ω. Define ω ɛ be the smallest i ω such that θ(p i ) i ɛ and i!θ(p i) ɛ (8) > 2πi for all ω (θ(p i )) i ɛ i ω by taking the exact values of i and θ. Then clearly ω!n ɛ (log N) ω!θ(p ω) ɛ (9) ω (θ(p ω )) > 2πω when ω ω ω ɛ. Here are values of ω ɛ for some ɛ values. 3 ɛ ω ɛ

5 BAKER S EXPLICIT ABC-CONJECTURE AND APPLICATIONS 5 Let ω < ω ɛ and N Θ(p ωɛ ). Then log N θ(p ωɛ ) ωɛ ɛ. Therefore ω!n ɛ (log N) ω!θ(p ω ɛ ) ɛ ω (θ(p ωɛ )) ω Combining this with (9), we obtain (0) (log N) ω ω! Further we now prove < = ω ɛ!θ(p ωɛ ) ɛ ω! (θ(p ωɛ )) ωɛ ω ɛ! (θ(p ω ɛ )) ωɛ ω > ω!ωɛ ωɛ ω 2πω ɛ ω ɛ! N ɛ 2π max(ω, ωɛ ) N ɛ 2πωɛ when N Θ(p ωɛ ). 2πω ɛ. (log N) ω < 5N 3 4 () for N. ω! 6 For that we take ɛ = 3. Then ω 4 ɛ = 4 and we may assume that N < Θ(p 4 ). Then ω < 4. Observe that N Θ(p ω ) and N 3 4 (log N) ω is increasing for log N 4ω 3. For 4 ω < 4, we check that θ(p ω ) 4ω 3 and ω!θ(p ω) 3 4 (θ(p ω )) ω > 6 5 implying () when 4 ω < 4. Thus we may assume that ω < 4. We check that () ω!n 3 4 (log N) ω > 6 5 4ω at N = e 3 for ω < 4 implying () for N e 4ω 3. Thus we may assume that N < e 4ω 3. Then N {2, 3} if ω =, N {6, 0,, 4} if ω = 2 and N {30, 42} if ω = 3. For these values of N too, we find that () is valid implying (). Clearly () is valid when N =. We now prove Theorem. Assume Conjecture.2. Let ɛ > 0 be given. Let a, b, c be positive integers such that a + b = c and gcd(a, b) =. By Conjecture.2, c 6 (log N)ω N where N = N(abc). Now assertion (2) follows from (). Let 5 ω! 0 < ɛ 3 and N 4 ɛ = Θ(p ωɛ ). By (0), we have c < 6N +ɛ 5 2π max(ω, ω ɛ ). The table is obtained by taking the table values of ɛ, ω ɛ given after (9) and computing N ɛ for those ɛ given in the table. Hence the Theorem. 4. Nagell-Ljungrenn equation: Proof of Theorem 2 Let x >, y >, n > 2 and q > be a non-exceptional solution of (3). It was proved by Ljunggren [Lju43] that there are no further solutions of (3) when q = 2. Thus we may suppose that q 3. Further it has been proved that 4 n by Nagell [Nag20], 3 n by Ljunggren [Lju43] and 5 n, 7 n by Bugeaud, Hanrot and Mignotte [BHM02]. Therefore n. From (3), we get + (x )y q = x n.

6 6 SHANTA LAISHRAM AND T. N. SHOREY Then y < x n n q x 3 since q 3 implying N = N(x(x )y) < x 2 y < x 2+ n 3. From (2) in Theorem, we obtain This gives n 8 which is a contradiction. x n < N 7 4 < x n implying n < n. 5. Fermat-Catalan Equation We may assume that each of p, q, r is either 4 or an odd prime. Let [p, q, r] denote all permutations of ordered triple (p, q, r). The Fermat s Last Theorem (p, p, p) was proved by Wiles [Wil95]; [3, p, p], [4, p, p] for p 7 by Darmon and Merel [DaGr95] and [3, 5, 5], [4, 5, 5] by Poonen; [4, 4, p] by Bennett, Ellenberg, Ng [BEN0]. The signatures [3, 3, p] for p 0 9 was solved by Chen and Siksek [ChSi09], [3, 4, 5] by Siksek and Stoll [SiSt] and [3, 4, 7] by Poonen, Schefer and Stoll [PSS07]. Hence we may suppose (p, q, r) is different from those values. We may assume that x >, y >, z >. Then x < z r p, y < z r q. Given ɛ > 0, by Theorem, we have { z r N 7 4 < ɛ if N(xyz) < N (3) ɛ N(xyz) +ɛ (xyz) +ɛ if N(xyz) N ɛ. In particular, taking ɛ = 3, we get 4 implying (4) z r < (xyz) 7 4 < z 7 4 (+ r p + r q ) 4 7 < p + q + r. Thus we need to consider [3, 3, p] for p > 0 9 and (p, q, r) Q. Let ɛ = First assume that N(xyz) N ɛ. Then implying z r < (xyz) +ɛ < z (+ɛ)(+ r p + r q ) p + q + r > + ɛ = 7 05 = Therefore we may suppose that N(xyz) < N 34. Then from (3) that max(x p, y q, z r ) < 7 e implying x, y, z, p, q, r are all bounded. This will imply that [3, 3, p] N with p > 0 9 does not have any solution. Hence the assertion.

7 BAKER S EXPLICIT ABC-CONJECTURE AND APPLICATIONS 7 Let d =gcd(x, y). From (5), we have 6. Goormaghtigh Equation x m + + x = y n + + y implying ord p (x) =ord p (y) for all primes p d. Further { } m m (x i y i x i y i ) = (x y) + = y n + + y m x y which is i= + m i=2 i=2 x i y i x y = We observe that d is coprime to yn m y ym y n m x y y. and also to the left hand side. Therefore ord p (x y) = m ord p (x) = m ord p (y) = m ord p (d) for every prime p d. Let d 2 =gcd(y, x, x y) and d 3 be given by x y = d m d 2 d 3. We observe that d 2 d 3 = if n = m + and d 2 d 3 (y + ) if n = m + 2. We now rewrite (5) as (5) Let (y )x m + d d m 3 = d 2 (x )yn d m d 2. N = N( xm y n (x )(y )d 3 ) N(xy(x )(y )d d 2m d 2 3 ) xy(x )(y )d δ dd 2 where δ = 0 if 2 dd 2 and otherwise. Recall that d =gcd(x, y) and d 2 (x ). Let ɛ < 3. We obtain from (5) and Theorem and x y = 4 dm d 2 d 3 that (6) max{ (y )xm d 3 (x y), (x )yn d 3 } < x y { N 7 4 ɛ Assume that N N ɛ. Then we obtain using (6) that if N < N ɛ N +ɛ if N N ɛ. (7) x m <x 2+2ɛ y +2ɛ (x y) d ɛ 3 < (2 δ x4+5ɛ dd 2 ) +ɛ (8) y n <x +2ɛ y +ɛ (y ) +ɛ (x y) (2 δ dd 2 ). +ɛ since y < x and d 3 x y < x. We observe that from (5) that x m < 2y n implying x < 2 n m y m. This together with (8), d3 x y < x and 2 δ dd 2 2 gives (9) y n <2 2+3ɛ m ɛ n 2+2ɛ+ y m (2+3ɛ). From (7), we obtain m < 4+5ɛ and further from (9), we get n < 2+2ɛ+ n m (2+3ɛ) if m > 3. Let ɛ = 3 4 and N ɛ =. Then m 7 and further 7 n 7 if m = 6 and n {8, 9} if m = 7. Let m = 7, n = m + = 8. Then d 2 d 3 = and we get from d ɛ 3

8 8 SHANTA LAISHRAM AND T. N. SHOREY the first inequality of (7) and y < x that x m < x 4+4ɛ = x 7 implying 7 = m < 7, a contradiction. Let m = 7, n = m+2 = 9. Then d 2 d 3 y+ and we get from (8) with x < 2 n m y m, d3 (y ) < y 2 and 2 δ dd 2 2 that y n < 2 2+2ɛ m ɛ n 2+3ɛ+ y m (2+2ɛ) < y 9 which is a contradiction again. Let m = 6 and n {, 6}. From Nesterenko and Shorey [NeSh98], we get y 8, 5 when n =, 6, respectively. For 2 y 5 and y + x ( yn )) m y, we check that (5) does not hold. Therefore n / {, 6} when m = 6. Hence we have the first assertion of Theorem 4. Now we take ɛ =. Since m 7 and G < x, we get an explicit bound of x, y, m, n 8 from (6) if N < N, implying Theorem 4 in that case. Thus we may suppose that 8 N N. Then we obtain from (7) with ɛ = that m < 4+5ɛ implying m {3, 4} 8 8 and further from (9) that n < 5 if m = 4. This is a contradiction for m = 4 since n > m and n Z. (20) Let m = 3. We rewrite (5) as (2x + ) 2 = 4(y n + + y) + By [NeSh98], we may assume that n 5. Let n = 4 and denote by f(y) the polynomial on the right hand side of (20). Let f (α) = 0. Then α = ± 2i and we check 3 that f(α) 0. Therefore the roots of f are simple. Now we apply Baker [Bak69] to conclude that y and hence x are bounded by effectively computable absolute constant. Let n 6. Now we rewrite (5) as (2) 4y n = (y )(2x + ) 2 + (3y + ). Let G =gcd(4y n, (y )(2x + ) 2, 3y + ). Then G = 4, 2, according as 4 (y ), 4 (y 3) and 2 y, respectively and we get from (2) that (22) Let 4 G yn = y G (2x + )2 + 3y + G. 4y(y )(2x + )(3y + ) N = N( ) G 3 Let ɛ =. We obtain from Theorem with ɛ = that (23) 4y n G < N 7 4 N + y(y )(2x + )(3y + G if N < N if N N. < 6xy3 G. If N < N, then y n < N 7 4 implying the assertion of Theorem 4. Hence we may suppose that N N and further y is sufficiently large. Then we have from x 2 < 2y n that Therefore n 3(n + 5) 24 4y n < (6 2y n+5 2 ) +. < 3 log(6 2) log 4 log y < 24

9 BAKER S EXPLICIT ABC-CONJECTURE AND APPLICATIONS 9 since y is sufficiently large. This is not possible since n 6. Hence the assertion Remarks The examples in this paper show that in applications of the abc conjecture to diophantine equations, it is sufficient to assume that ɛ is not very near to 0. Sometimes it is sufficient to use abc with ɛ = or 3 or even larger. See also the paper of Browkin 2 4 [Bro08], where the minimal sufficient values of ɛ are discussed for some diophantine equations. In general they are large. From this point of view it is probably irrelevant what the abc conjecture says in the case of ɛ near to 0. Acknowledgments We thank the referee for his useful remarks and suggestions in an earlier draft of this paper. The second author would like to thank ISI, New Delhi where this work was initiated during his visit in July 20. References [BEN0] M.A. Bennett, J.S. Ellenberg and N.C. Ng, The Diophantine equation A δ B 2 = C n, International Journal of Number Theory, 6 (200), no. 2, [Bak94] A. Baker, Experiments on the abc-conjecture, Publ. Math. Debrecen 65(2004), [Bak69] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc. 65 (969), [Beu04] F. Beukers, The Diophantine equation Ax p + By q = Cz r, Lectures held at Institut Henri Poincare, September 2004, http : // ermatlectures.pdf Publ. Math. Debrecen 65(2004), [BHM02] Y. Bugeaud, G. Hanrot and M. Mignotte, Sur l equation diophantienne xn x = yq, Proc. London Math. Soc. 84(2002), [Bro08] J. Browkin, A weak effective abc conjecture, Functiones Approx., 39(2008), no., 03. [BuMi02] Y. Bugeaud and and M. Mignotte, L equation de Nagell-Ljunggren xn x = yq, Enseign. Math. 48(2002), [BuMi07] Y. Bugeaud and P. Mihailescu, On the Nagell-Ljunggren equation xn x = y q, Math. Scand. 0(2007), [ChSi09] I. Chen and S. Siksek, Perfect powers expressible as sums of two cubes, Journal of Algebra 322 (2009), [Coh07] H. Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM 240, Springer- Verlag, [DaGr95] H. Darmon and A. Granville, On the equations z m = F (x, y) and Ax p + By q = cz r, Bull. London Math. Soc. 27 (995), [Dus99a] P. Dusart, The k th prime is greater than k(ln k + ln ln k ) for k 2, Math. Comp. 68 (999), 4 45 [Dus99b] P. Dusart, Inégalitiés explicites pour ψ(x), θ(x), π(x) et les nombres premiers, C. R. Math. Rep. Acad. Sci. Canada 2()(999), [Elk9] N. Elkies, ABC implies Mordell, Int. Math. Res. Not. 7 (99), [Ch 9]. [GrTu02] A. Granville and T. J. Tucker, It s as easy as abc, Notices of the AMS, 49(2002), [Kra99] A. Kraus, On the Equation x p + y q = z r : A Survey, Ramanujan Journal 3 (999),

10 0 SHANTA LAISHRAM AND T. N. SHOREY [Lju43] W. Ljunggren, Noen Setninger om ubestemte likninger av formen (x n )/(x ) = y q, Norsk. Mat. Tidsskr.. Hefte 25(943), [Nag20] T. Nagell, Note sur l équation indéterminée (x n )/(x ) = y q, Norsk. Mat. Tidsskr. 2 (920), [NeSh98] Yu.V. Nesterenko and T. N. Shorey, On an equation of Goormaghtigh, Acta Arith. 83 (998), [PSS07] B. Poonen, E.F. Schaefer and M. Stoll, Twists of X(7) and primitive solutions to x 2 + y 3 = z 7, Duke Math. J. 37 (2007), [Rob55] H. Robbins, A remark on Stirling s formula, Amer. Math. Monthly 62 (955), [Rob83] G. Robin, Estimation de la fonction de Tchebychef θ sur le k-ieme nombre premier et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n, Acta Arith. 42 (983), [Sar] N. Saradha, Applications of Explicit abc-conjecture on two Diophantine Equations, Acta Arith. 5 (20), no. 4, [Sho02a] T. N. Shorey, An equation of Goormaghtigh and diophantine approximations, Current Trends in Number Theory, edited by S.D.Adhikari, S.A.Katre and B.Ramakrishnan, Hindustan Book Agency, New Delhi (2002), [Sho99] T. N. Shorey, Exponential diophantine equations involving products of consecutive integers and related equations, Number Theory ed. R.P. Bambah, V.C. Dumir and R.J. Hans-Gill, Hindustan Book Agency (999), [SiSt] S. Siksek and M. Stoll, Partial descent on hyperelliptic curves and the generalized Fermat equation x 3 + y 4 + z 5 = 0, Bulletin of the LMS, 44 (20), [TaWi95] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Annals of Mathematics 4 (995), [Wil95] A. Wiles, Modular elliptic curves and Fermat s Last Theorem, Annals of Mathematics 4 (995), [ABC3] ABC triples, page maintained by Bart de Smit at http : // desmit/abc/index.php?sort =, see also http : //rekenmeemetabc.nl/synthese resultaten, http : // r/ nitaj/abc.html. address: shanta@isid.ac.in Stat Math Unit, Indian Statistical Institute, 7 SJS Sansanwal Marg, New Delhi 006, India address: shorey@math.iitb.ac.in Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai , India