There are only finitely many Diophantine quintuples

There are only finitely many Diophantine quintuples Andrej Dujella Department of Mathematis, University of Zagreb, Bijenička esta 30 10000 Zagreb, Croatia E-mail: duje@math.hr Abstrat A set of m positive integers is alled a Diophantine m-tuple if the produt of its any two distint elements inreased by 1 is a perfet square. Diophantus found a set of four positive rationals with the above property. The first Diophantine quadruple was found by Fermat (the set {1, 3, 8, 120}). Baker and Davenport proved that this partiular quadruple annot be extended to a Diophantine quintuple. In this paper, we prove that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. 1 Introdution A set of m positive integers {a 1, a 2,..., a m } is alled a Diophantine m-tuple if a i a j + 1 is a perfet square for all 1 i < j m. Diophantus first studied the problem of finding four numbers suh that the produt of any two of them inreased by unity is a square. He found a set of four positive rationals with the above property: {1/16, 33/16, 17/4, 105/16}. However, the first Diophantine quadruple, {1, 3, 8, 120}, was found by Fermat. Euler was able to add the fifth positive rational, 777480/8288641, to the Fermat s set (see [6], pp. 513 520). Reently, Gibbs [13] found examples of sets of six positive rationals with the property of Diophantus. A folklore onjeture is that there does not exist a Diophantine quintuple. The first important result onerning this onjeture was proved in 1969 by Baker and Davenport [3]. They proved that if d is a positive integer suh that {1, 3, 8, d} forms a Diophantine quadruple, then d = 120. This problem was stated in 1967 by Gardner [12] (see also [17]). In 1979 Arkin, Hoggatt and Strauss [1] proved that every Diophantine triple an be extended to a Diophantine quadruple. More preisely, let {a, b, } be a Diophantine triple and ab + 1 = r 2, a + 1 = s 2, b + 1 = t 2, where r, s, t are positive integers. Define d + = a + b + + 2ab + 2rst. 0 Mathematis Subjet Classifiation (2000): Primary 11D09, 11D25, 11D48; Seondary 11B37, 11J68, 11J86. 1

There are only finitely many Diophantine quintuples 2 Then {a, b,, d + } is a Diophantine quadruple. Indeed, ad + + 1 = (at + rs) 2, bd + + 1 = (bs + rt) 2, d + + 1 = (r + st) 2. There is a stronger version of the Diophantine quintuple onjeture. Conjeture 1 If {a, b,, d} is a Diophantine quadruple and d > max{a, b, }, then d = d +. Conjeture 1 was proved for ertain Diophantine triples [16, 20] and for some parametri families of Diophantine triples [7, 8, 10]. In partiular, in [10] it was proved that the pair {1, 3} annot be extended to a Diophantine quintuple. A Diophantine quadruple D = {a, b,, d}, where a < b < < d, is alled regular if d = d +. Equivalently, D is regular iff (1) (a + b d) 2 = 4(ab + 1)(d + 1) (see [14]). The equation (1) is a quadrati equation in d. One root of this equation is d +, and other root is d = a + b + + 2ab 2rst. It is easy to hek that all small Diophantine quadruples are regular; e.g. there are exatly 207 quadruples with max{a, b,, d} 10 6 and all of them are regular. Sine the number of integer points on an ellipti urve (2) y 2 = (ax + 1)(bx + 1)(x + 1) is finite, it follows that, for fixed a, b and, there does not exist an infinite set of positive integers d suh that a, b,, d is a Diophantine quadruple. However, bounds for the size [2] and for the number [19] of solutions of (2) depend on a, b, and aordingly they do not immediately yield an absolute bound for the size of suh set. The main result of the present paper is the following theorem. Theorem 1 There are only finitely many Diophantine quintuples. Moreover, this result is effetive. We will prove that all Diophantine quintuples Q satisfy max Q < 10 1026. Hene we almost ompletely solve the problem of the existene of Diophantine quintuples. Furthermore, we prove Theorem 2 There does not exist a Diophantine sextuple. Theorems 1 and 2 improve results from [9] where we proved that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples. As in [9], we prove Conjeture 1 for a large lass of Diophantine triples satisfying some gap onditions. However, in the present paper these gap onditions are muh weaker than in [9]. Aordingly, the lass of Diophantine triples for whih we are able to prove Conjeture 1 is muh larger. In fat, in an arbitrary Diophantine quadruple, we may find a triple for whih we are able to prove Conjeture 1.

There are only finitely many Diophantine quintuples 3 In the proof of Conjeture 1 for a triple {a, b, } we first transform the problem into solving systems of simultaneous Pellian equations. This redues to finding intersetions of binary reurrene sequenes. In Setion 5 we almost ompletely determine initial terms of these sequenes, under assumption that they have nonempty intersetion whih indues a solution of our problem. This part is a onsiderable improvement of the orresponding part of [9]. This improvement is due to new gap priniples developed in Setion 4. These gap priniples follow from the areful analysis of the elements of the binary reurrene sequenes with small indies. Let us mention that in a joint paper with A. Pethő [10] we were able to determine initial terms, in a speial ase of triples {1, 3, }, using an indutive argument. Applying some ongruene relations we get lower bounds for solutions. In obtaining these bounds we need to assume that our triple satisfies some gap onditions like b > 4a and > b 2.5. Let us note that these onditions are muh weaker then onditions used in [9], and this is due to more preise determination of the initial terms. Comparing these lower bounds with upper bounds obtained from the Baker s theory on linear forms in logarithms of algebrai numbers (a theorem of Matveev [18]) we prove Theorem 1, and omparing them with upper bounds obtained from a theorem of Bennett [5] on simultaneous approximations of algebrai numbers we prove Theorem 2. In the final steps of the proofs, we use again the above mentioned gap priniples. 2 Systems of Pellian equations Let us fix some notation. Let {a, b, } be a Diophantine triple and a < b <. Furthermore, let positive integers r, s, t be defined by ab + 1 = r 2, a + 1 = s 2, b + 1 = t 2. In order to extend {a, b, } to a Diophantine quadruple {a, b,, d}, we have to solve the system (3) ad + 1 = x 2, bd + 1 = y 2, d + 1 = z 2, with positive integers x, y, z. Eliminating d from (3) we get the following system of Pellian equations (4) (5) az 2 x 2 = a, bz 2 y 2 = b. In [9], Lemma 1, we proved the following lemma whih desribes the sets of solutions of the equations (4) and (5). Lemma 1 There exist positive integers i 0, j 0 and integers z (i) j = 1,..., j 0, with the following properties: (i) (z (i) 0, x(i) 0 ) and (z(j) 1, y(j) 1 0, x(i) 0, z(j) 1, y(j) ) are solutions of (4) and (5), respetively. 1, i = 1,..., i 0,

There are only finitely many Diophantine quintuples 4 (ii) (6) (7) (8) (9) z (i) 0, x(i) 0, z(j) 1, y(j) 1 satisfy the following inequalities a( a) s + 1 2(s 1) < 2 1 x (i) 0 < 0.841 4 a, 1 z (i) (s 1)( a) 0 < 2a 2 a < 0.421, 1 y (j) 1 b( b) t + 1 2(t 1) < < 0.783 4 b, 2 1 z (j) (t 1)( b) 1 < 2b 2 b < 0.32. (iii) If (z, x) and (z, y) are positive integer solutions of (4) and (5) respetively, then there exist i {1,..., i 0 }, j {1,..., j 0 } and integers m, n 0 suh that (10) (11) z a + x = (z (i) 0 z b + y = (z (j) 1 (i) a + x 0 )(s + a) m, )(t + b) n. b + y (j) 1 Let (x, y, z) be a solution of the system (4) & (5). From (10) it follows that z = v (i) m for some index i and integer m 0, where (12) v (i) 0 = z (i) 0, v(i) 1 = sz (i) 0 + x (i) 0, v(i) m+2 = 2sv(i) m+1 v(i) m, and from (11) we onlude that z = w (j) n for some index j and integer n 0, where (13) w (j) 0 = z (i) 1, w(j) 1 = tz (j) 1 + y (j) 1, w(j) n+2 = 2tw(j) n+1 w(j) n. Let us onsider the sequenes (v m ) and (w n ) modulo 2. From (12) and (13) it is easily seen that (14) (15) v (i) 2m z(i) 0 (mod 2), v (i) 2m+1 sz(i) 0 + x (i) 0 (mod 2), w (j) 2n z(j) 1 (mod 2), w (j) 2n+1 tz(j) 1 + y (j) 1 (mod 2). We are searhing for solutions of the system (4) & (5) suh that d = (z 2 1)/ is an integer. Using (14) and (15), from z = v m = w n we obtain [z (i) 0 ]2 v 2 m z 2 1 (mod ), [z (j) 1 ]2 w 2 n z 2 1 (mod ). Therefore we are interested only in equations v m = w n satisfying [z (i) 0 ]2 [z (j) 1 ]2 1 (mod ).

There are only finitely many Diophantine quintuples 5 We will dedue later more preise information on the initial terms z (i) 0 and z (j) 1 (see Setion 5). Let us mention now a result of Jones [15], Theorem 8, whih says that if < 4b then z (j) 1 = 1. As a onsequene of Lemma 1 and the relations (14) and (15), we obtain the following lemma. From now on, we will omit the supersripts (i) and (j). Lemma 2 1) If the equation v 2m = w 2n has a solution, then z 0 = z 1. 2) If the equation v 2m+1 = w 2n has a solution, then z 0 z 1 < 0 and x 0 s z 0 = z 1. In partiular, if b > 4a and > 100a, then this equation has no solution. 3) If the equation v 2m = w 2n+1 has a solution, then z 0 z 1 < 0 and y 1 t z 1 = z 0. 4) If the equation v 2m+1 = w 2n+1 has a solution, then z 0 z 1 > 0 and x 0 s z 0 = y 1 t z 1. Proof. See [9], Lemma 3. 3 Relationships between m and n In [9], Setion 4, we proved that v m = w n implies m n if b > 4a, > 100b and n 3. We also proved that m 3 2n, provided {a, b, } satisfies some rather strong gap onditions. In this setion we will first prove an unonditional relationship between m and n, and then we will improve that result under various gap assumptions. Lemma 3 If v m = w n, then n 1 m 2n + 1. Proof. We have the following estimates for v 1 : v 1 = sz 0 + x 0 x 0 s z 0 = 2 a z 2 0 x 0 + s z 0 > 2 2 4 2 a 2x 0 > 4x 0 > 3.364 4 a, Hene (16) If > 4b, then v 1 < 2x 0 < 1.682 4 a. 3.364 4 a (2s 1)m 1 < v m < 1.682 4 a(2s) m 1 for m 1. w 1 2 b z 2 1 y 1 + b z 1 > 4y 1 > If < 4b then, by [15], Theorem 8, z 1 = ±1, y 1 = 1 and w 1 t = a + r = s > a > Furthermore, w 1 < 2y 1 < 1.566 4 b and therefore (17) 3.132 4 b. 3.132 4 b. 3.132 4 b (2t 1)n 1 < w n < 1.566 4 b(2t) n 1 for n 1.

There are only finitely many Diophantine quintuples 6 We thus get (18) Sine (19) (2s 1) m 1 < 5.269 4 ab 2 (2t) n 1. 2s 1 = 2 a + 1 1 > 1.767 a and 2t = 2 b + 1 < 2.042 b, it follows that (2s 1) 2 > 3.12a > 2t. It implies and m 2n + 1. On the other hand, we have (2s 1) m 1 < 2.635 (2t) n < (2t) n+0.43 < (2s 1) 2n+0.86 (2t 1) n 1 < 5.269 4 ab 2 (2s) m 1 < 5.269 4 ab 2 (2t 1) m 1 < (2t 1) m+0.489 and n m + 1. Thus we proved the lemma for m, n 0. It remains to hek that v 0 < w 2 and w 0 < v 2. Indeed, 2t w 2 = 2tw 1 w 0 > 3.132 4 b ( 4 ) b 2 b > 1.566 1 > 1.093 > v 0, v 2 = 2sv 1 v 0 > 2s 3.364 4 a 2 b ( 4 ) a 2 a > 1.682 1 > 0.579 > w 0. 2 a Lemma 4 Assume that > 10 10. If v m = w n and m, n 2, then 1) > b 4.5 = m 11 9 n + 7 9, 2) > b 2.5 = m 7 5 n + 3 5, 3) > b 2 = m 3 2 n + 1 2, 4) > b 5/3 = m 8 5 n + 3 5. Proof. As in the proof of Lemma 3, assuming > max{b 5/3, 10 10 }, we have v m > 0.999 2x 0 (2s 1) m 1 > w n < 1.415 4 b(2t) n 1, 1.416 4 a (2s 1)m 1, for m, n 1. Hene (2s 1) m 1 < 2.004 4 ab 2 (2t) n 1 and (20) 1.999 m 1 a (m 1)/2 (m 1)/2 < 2.004 n b (n 1)/2+1/4 (n 1)/2+1/4 a 1/4. If m 2 and > b ε, then (20) shows that at least one of the following two inequalities holds: (21) m 1 < ( n 2 2 1 4 ) 1 ε + n 2, (22) m 1 < 1.004n. The inequality (22) implies all statements of the lemma. For ε = 4.5, (21) implies m < 11 9 n + 8 9 and m 11 9 n + 7 9. Similar arguments apply to all other statements of the lemma.

There are only finitely many Diophantine quintuples 7 4 Gap priniples In [9], Lemma 14, we proved that if {a, b,, d} is a Diophantine quadruple and a < b < < d, then d 4b. The proof was based on the fat, proved by Jones [15], Lemma 4, that = a + b + 2r or 4ab + a + b. In this setion we will develop a stronger and more preise gap priniple by examining the equality v m = w n for small values of m and n. Let us note that sine (4ab + 1) < d + < 4(ab + 1), we may expet an essential improvement only if we assume that d d +. Lemma 5 Let v m = w n and define d = (v 2 m 1)/. If {0, 1, 2} {m, n}, then d < or d = d +. Proof. From the proof of Lemma 3 we have: v 0 < w 2, w 0 < v 2, v 1 < w 3, w 1 < v 4, v 2 < w 4, w 2 < v 6. Therefore, the ondition {0, 1, 2} {m, n} implies (m, n) {(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (2, 3), (3, 2), (4, 2), (5, 2)}. If 0 {m, n}, then d <. If (m, n) = (1, 1), then d < for z 0 < 0, and d = d + for z 0 > 0 (see [9], proof of Theorem 3). Assume that (m, n) = (1, 2). We have v 1 = sz 0 + x 0, w 2 = z 1 + 2(bz 1 + ty 1 ). By Lemma 2, if z 1 > 0 then z 0 < 0 and x 0 + sz 0 = z 1. Hene w 2 > v 1 = w 0. If z 1 < 0 then z 0 > 0 and (23) x 0 sz 0 = z 1. Inserting (23) into the relation v 1 = w 2 we obtain (24) bz 1 + ty 1 = x 0. From (23), (24) and the system (4) & (5), we obtain (b a)t = z0 2 (b a). Therefore, z 0 = t, x 0 = r, z 1 = st r, y 1 = rt bs. It implies v 1 = st + r and d = v2 1 1 = 1 (ab2 + a + b + 1 + 2rst + ab 2 + 2 1) = d +. Assume now that (m, n) = (2, 1). In the same manner as in the ase (m, n) = (1, 2), we obtain z 1 = s, y 1 = r, z 0 = st r, x 0 = rs at and d = d +. Let (m, n) = (2, 2). We have v 2 = z 0 + 2(az 0 + sx 0 ), w 2 = z 1 + 2(bz 1 + ty 1 ), and sine z 0 = z 1, we obtain az 0 + sx 0 = bz 0 + ty 1 and (b a)(y 2 1 x 2 0 + b a) = ax 2 0 + x 2 0 by 2 1 + y 2 1 2stx 0 y 1.

There are only finitely many Diophantine quintuples 8 Therefore (b a) 2 = (sy 1 tx 0 ) 2 and sine sy 1 = tx 0, we have (25) tx 0 sy 1 = b a. Furthermore, and (a + 1)(bx 2 0 + a b) = a(bx 2 0 + x 2 0 + (b a) 2 2t(b a)x 0 ) (b a)(x 2 0 + 2atx 0 + a 2 t 2 ) = (b a)(ab + 1)(a + 1) = (b a)r 2 s 2. Finally, x 0 = rs at, y 1 = rt bs, z 0 = st r. Now we obtain and d = d +. v 2 = st r + 2(ast ar + rs 2 ast) = st + r Let (m, n) = (3, 1). We may assume that {a, b, } {1, 3, 8}. Then a 15 and b 48. It implies 2s 1 > 1.807 a and 2t < 2.021 b. From (18) we obtain whih implies 6, a ontradition. (1.807 a) 2 < 5.269 4 ab 2 < 5.269 4 a 3, Let (m, n) = (3, 2). Assume that z 0 > 0, z 1 < 0. We have v 1 > 2sz 0, v 2 > (4s 2 1)z 0, v 3 > (4s 2 1)(2s 1)z 0 > 7(a) 3/2 z 0, and w 1 < 2 2t z 1, w 2 < 2tw 1 < 2 z 1. We have also z 1 > 4x 0 and therefore w 2 < 4x 0. Sine x 0 <, we obtain w 2 < 4 3/2 < 7(a) 3/2 z 0 < v 3. Assume now that z 0 < 0, z 1 > 0. Then z 1 = sz 0 + x 0 and the ondition v 3 = w 2 implies x 0 + 2az 1 = bz 1 + ty 1 > 2bz 1 and x 0 > 4z 1 > x 0, a ontradition. Let (m, n) = (2, 3). If z 0 > 0, z 1 < 0, then v 2 = w 3 implies y 1 + 2bz 0 = az 0 + sx 0 < 2az 0 + z 0. Therefore z0 2 < 4. On the other hand, z2 0 1 (mod ). Hene, z 0 = 1. But, if > 4b then z 0 > 4y 1 > 1. If < 4b then z 1 = 1, y 1 = 1, and Lemma 2 implies = b + 2, whih ontradits the fat that = a + b + 2r. Assume that z 0 < 0, z 1 > 0. As in the ase (m, n) = (3, 2), we have v 2 < 2 z 0 and w 3 > 7.3(b) 3/2 z 1. If > 4b then z 0 > and sine y 1 <, we obtain 4y 1 v 2 < 4 3/2 < 7.3(b) 3/2 z 1 < w 3. If < 4b then w 3 > 7.3b 1.5 1.5 > 0.9 3 > 2 > v 2. Let (m, n) = (4, 2). We have v 4 = z 0 + 4(2az 0 + sx 0 ) + 8a 2 (az 0 + sx 0 ), and the relation v 4 = w 2 implies (26) bz 0 + ty 1 = 4az 0 + 2sx 0 + 4a(az 0 + sx 0 ).

There are only finitely many Diophantine quintuples 9 It holds ty 1 b z 0 < z 0. Hene, if z 0 > 0 then the left hand side of (26) is 2bz 0 + z 0 < 3z 0, while the right hand side is > 4a 2 z 0 > 3z 0. If z 0 < 0, then the left hand side of (26) is z 0, while the right hand side is 4a 4a z 0 > a. Let (m, n) = (5, 2). It follows from (18) that and 10.66a 2 2 < 10.65b, a ontradition. (1.807 a) 4 < 5.269 b 2.021 b We now an prove the following gap prinipal, whih we will improve again in Proposition 1 below. Lemma 6 If {a, b,, d} is a Diophantine quadruple and a < b < < d, then d = d + or d 1.16 2.5 b 1.5. Proof. By Lemma 5, if d d + then m 3 and n 3. From (17) it follows that and w 3 > (2t 1) 2 3.132 4 b > 81 4 b 24 3.132 3 3 d 1.161b1.5 3.5 1 > 1.16 2.5 b 1.5. Using the gap priniple from Lemma 6 we an prove the following lemma. Lemma 7 Under the notation from above, we have v 3 w 3. Proof. If z 0, z 1 > 0 then define z := x 0 sz 0 = y 1 tz 1, and if z 0, z 1 < 0 then define z := x 0 + sz 0 = y 1 + tz 1. Define also d 0 = (z 2 1)/. Then d 0 is an integer. Furthermore, d 0 + 1 = z 2, ad 0 + 1 = 1 (a2 x 2 0 2asx 0 z 0 + a 2 z 2 0 + az 2 0 a + ) = (ax 2 0 2asx 0 z 0 + a 2 z 2 0 + x 2 0) = (sx 0 az 0 ) 2, bd 0 + 1 = 1 (b2 y 2 1 2bty 1 z 1 + b 2 z 2 1 b + ) = (ty 1 bz 1 ) 2. We have Therefore z = 2 a z 2 0 x 0 + s z 0 > d 0 > 0.088 a 1 3.364 4 a and z <. > 0.043 > 0, a

There are only finitely many Diophantine quintuples 10 and thus the set {a, b,, d 0 } forms a Diophantine quadruple. Sine d 0 <, by Lemma 6, we have two possibilities: the quadruple {a, b,, d 0 } is regular or (27) > 1.16d 2.5 0 b 1.5. Assume that {a, b,, d 0 } is regular, i.e. d 0 = d. Then z = r st. From (x 0 r) = s( z 0 t) and gd(, s) = 1 it follows that z 0 t (mod ), and sine z 0 <, t <, we onlude that z 0 = t and x 0 = r. We an proeed analogously to prove that z 1 = s and y 1 = r. The ondition v 3 = w 3 implies sz 0 + 3x 0 + 4a(x 0 + sz 0 ) = tz 1 + 3y 1 + 4b(y 1 + tz 1 ) and we obtain a = b, a ontradition. Hene we may assume that the quadruple {a, b,, d 0 } is not regular. But it means that > 10 6, whih implies and z = 2 a z 2 0 x 0 + s z 0 > 0.749 2x 0 > 0.749 1.415 4 a > 0.529 4 a d 0 > 0.2798 a 1 > 0.2797. a Thus from (27) we obtain > 0.0479 1.25 a 1.25 b 1.5 > 0.0479 1.25 a 0.25 and a 189958, whih ontradits the assumption that > 10 6. Now we an prove the following strong gap priniple, whih is the main improvement to [9] and whih we will use several times later. Observe that espeially the dependene on is muh better than in the gap priniple in [9]. Proposition 1 If {a, b,, d} is a Diophantine quadruple and a < b < < d, then d = d + or d > 2.695 3.5 a 2.5. Proof. From Lemmas 5 and 7 it follows that m 4 or n 4. By (16), we have v 4 Therefore, if m 4 then 3.364 4 a (2s 1)3 125 8 8 3.36 4 a 5 5. Similarly, from (17) it follows w 4 and for n 4 we obtain d 2.696a2.5 4.5 1 3.132 4 b (2t 1)3 d 3.919b2.5 4.5 1 > 2.695 3.5 a 2.5. 729 24 24 3.132 4 b 5 5, > 3.918 3.5 b 2.5 > 2.695 3.5 a 2.5.

There are only finitely many Diophantine quintuples 11 Corollary 1 If {a, b,, d, e} is a Diophantine quintuple and a < b < < d < e, then e > 2.695 d 3.5 b 2.5. Proof. Assume that {b,, d, e} is a regular Diophantine quadruple. Then e 4d(b + 1) < d 3. The quadruple {a,, d, e} is not regular and, by Proposition 1, we have e > 2.695d 3.5 a 2.5 > d 3. Therefore the quadruple {b,, d, e} is not regular and hene e > 2.695d 3.5 b 2.5. 5 Determination of the initial terms Using the gap priniples developed in the previous setion we will improve Lemma 2 and obtain more speifi information on the initial terms of the sequenes (v m ) and (w n ). Lemma 8 1) If the equation v 2m = w 2n has a solution, then z 0 = z 1. Furthermore, z 0 = 1 or z 0 = r st or z 0 < min{0.869 a 5/14 9/14, 0.972 b 0.3 0.7 }. 2) If the equation v 2m+1 = w 2n has a solution, then z 0 = t, z 1 = r st and z 0 z 1 < 0. 3) If the equation v 2m = w 2n+1 has a solution, then z 0 = r st, z 1 = s and z 0 z 1 < 0. 4) If the equation v 2m+1 = w 2n+1 has a solution, then z 0 = t, z 1 = s and z 0 z 1 > 0. Proof. 1) From Lemma 2 we have z 0 = z 1. Define d 0 = (z 2 0 1)/. Then d 0 is an integer and d 0 + 1 = z 2 0, ad 0 + 1 = 1 (az2 0 a + ) = x 2 0, bd 0 + 1 = 1 (bz2 1 b + ) = y 2 1. Hene, we have three possibilities: d 0 = 0 or {a, b,, d} is a regular Diophantine quadruple or {a, b,, d} is an irregular Diophantine quadruple. If d 0 = 0 then z 0 = 1. If the quadruple {a, b,, d} is regular, then (sine d 0 < ) we have d 0 = d and z 0 = r st. Otherwise we may apply Proposition 1 to obtain 2.695d 3.5 0 a2.5. Sine z 0 1, we have z0 2 + 1. We may assume that > 106 and therefore d 0 = z2 0 1 z2 ( 0 1 1 ) > 0.999 z2 0 + 1. Hene we obtain 4.5 > 2.685 z 7 0 a2.5 and z 0 < 0.869a 5/14 9/14. Analogously, from Lemma 6, we obtain z 0 < 0.972b 0.3 0.7. 2) Let z = z 1 = x 0 + sz 0 if z 1 > 0, and z = z 1 = x 0 sz 0 if z 1 < 0. Define d 0 = (z 2 1)/. Then d 0 is an integer and d 0 + 1 = z 2, ad 0 + 1 = (sx 0 ± az 0 ) 2, bd 0 + 1 = y 2 1. In the proof of Lemma 7 it is shown that d 0 > 0 and d 0 <. Therefore {a, b,, d} is a Diophantine quadruple, and by the proof of Lemma 7, it must be regular. It means that d 0 = d and z = r st. It implies z 1 = r st and z 0 = t.

There are only finitely many Diophantine quintuples 12 3) Let z = z 0 = y 1 + tz 1 if z 0 > 0, and z = z 0 = y 1 tz 1 if z 0 < 0, and define d 0 = (z 2 1)/. Then d 0 + 1 = z 2, ad 0 + 1 = x 2 0, bd 0 + 1 = (ty 1 ± bz 1 ) 2 and 0 < d 0 <. If the quadruple {a, b, d 0, } is not regular, then from Lemma 6 we have (28) We an assume that > 10 6. If > 4b then and if < 4b then Therefore, z = 2 b z 2 1 y 1 + t z 1 1.16d 2.5 0 b 1.5. > 0.749 2y 1 > 0.749 1.415 4 b > 0.529 4 b, z > a > 0.529 4 b. d 0 > 0.2798 b 1 > 0.279. b Now (28) implies > 1 20.97 5/4 b 1/4 and b 193372, a ontradition. It follows that the quadruple {a, b,, d} is regular, i.e. d 0 = d and z = r st. Hene z 0 = r st and (y 1 r) = t( z 1 s). Sine gd(t, ) = 1 we have z 1 s (mod ), whih implies z 1 = s. 4) Let z = x 0 s z 0 = y 1 t z 1 and d 0 = (z 2 1)/. In the proof of Lemma 7 we have shown that {a, b, d 0, } is a regular Diophantine quadruple, and that this fat implies z 0 = t and z 1 = s. 6 Standard Diophantine triples In [9] we proved Conjeture 1 for triples satisfying some gap onditions like b > 4a and > max{b 13, 10 20 }. In Setion 7 we will prove Conjeture 1 under ertain weaker assumptions. This results will suffie for proving Theorem 1 sine we will show that every Diophantine quadruple ontains a triple whih satisfies some of our gap assumptions. Definition 1 Let {a, b, } be a Diophantine triple and a < b <. Diophantine triple of We all {a, b, } a the first kind if > b 4.5, the seond kind if b > 4a and > b 2.5, the third kind if b > 12a and b 5/3 < < b 2, the fourth kind if b > 4a and b 2 < 6ab 2. A triple {a, b, } is alled standard if it is a Diophantine triple of the first, seond, third or fourth kind.

There are only finitely many Diophantine quintuples 13 The Diophantine triples of the first and the seond kind appear naturally when we try to modify results from [9] using Lemma 8. They orrespond to triples with properties b > 4a, > b 13 and b > 4a, > b 5, onsidered in [9]. On the other hand, triples of the third and the fourth kind ome from the analysis what kind of triples a regular Diophantine quadruple may ontain. Note also that these four ases are not mutually exlusive. We now use the improved gap priniple (Proposition 1) again to show that the set of all standard Diophantine triples is large. Proposition 2 Every Diophantine quadruple ontains a standard triple. Proof. Let {a, b,, d} be a Diophantine quadruple. If it is not regular, then by Proposition 1 and [9], Lemma 14, we have d > 3.5 and > 4a. Hene {a,, d} is a triple of the seond kind. Assume that {a, b,, d} is a regular quadruple. Then (4ab + 1) < d < 4(ab + 1). If b > 4a and b 1.5, then d > b 2.5 and we see that {a, b, d} is a triple of the seond kind. If b > 4a and < b 1.5, then we have two possibilities: if = a + b + 2r then < 4b, 2 < d < 4b 2 and therefore {b,, d} is a triple of the fourth kind; if 4ab + a + b then d < 2, d > b > 5/3 and it follows that {a,, d} is a triple of the third kind. We may now assume that b < 4a. By [15], Theorem 8, we have = k (k 1) or = k (k 2), where the sequenes ( k ) and ( k ) are defined by 0 = 0, 1 = a + b + 2r, k = (4ab + 2) k 1 k 2 + 2(a + b), 0 = 0, 1 = a + b 2r, k = (4ab + 2) k 1 k 2 + 2(a + b). If > b 2.5 then d > 4ab > b 4.5 and we see that {a, b, d} is a triple of the first kind. Sine 2 = 4r(a + r)(b + r) > 4ab 2 > b 3, the ondition b 2.5 implies = 1 or = 2. If = 1 then a + b + 4 3 ab = ab ( a b + b a + 4 3 ) ab(0.5 + 2 + 4 3 ) < 4.81 ab and 4r. Hene d > 4ab > 0.172 3 > 2 and d < 4r 2 2 r < 2a 2. Therefore the triple {a,, d} is of the fourth kind. If b 2.5 and = 2 4ab + 2a + 2b (we may assume b > a + 2, sine otherwise 2 = 1 ), then d < 2 and d > 4ab > b 2 > 1.8 > 5/3. Therefore the triple {a,, d} is of the third kind. 7 Lower bounds for solutions The main tool in obtaining lower bounds for m and n satisfying v m = w n (m, n > 2) is the ongruene method introdued in the joint paper of the author with A. Pethő [10].

There are only finitely many Diophantine quintuples 14 Lemma 9 1) v 2m z 0 + 2(az 0 m 2 + sx 0 m) (mod 8 2 ) 2) v 2m+1 sz 0 + [2asz 0 m(m + 1) + x 0 (2m + 1)] (mod 4 2 ) 3) w 2n z 1 + 2(bz 1 n 2 + ty 1 n) (mod 8 2 ) 4) w 2n+1 tz 1 + [2btz 1 n(n + 1) + y 1 (2n + 1)] (mod 4 2 ) Proof. See [9], Lemma 4. If v m = w n then, of ourse, v m w n (mod 4 2 ) and we an use Lemma 9 to obtain some ongruenes modulo. However, if a, b, m and n are small ompared with, then these ongruenes are atually equations. It should be possible to prove that these new equations are in ontradition with the starting equations v m = w n. This will imply that m and n annot be too small. We will prove a lower bound for n (and therefore also for m by Lemma 3) depending on and we will do this separately for Diophantine triples of the first, seond, third and fourth kind in the following four lemmas. Lemma 10 Let {a, b, } be a Diophantine triple of the first kind and > 10 100. If v m = w n and n > 2, then n > 0.01. Proof. Assume that n 0.01. By Lemma 17, we have max{ m/2 +1, n/2 +1} < n 0.01. Aording to Lemma 8, we will onsider six ases. 1.1) v 2m = w 2n, z 0 = 1 From Lemma 9 we have (29) ±am 2 + sm ±bn 2 + tn (mod 4). Sine > b 4.5, we have am 2 < 0.243 <, sm < 0.623 <, bn 2 < 0.243 <, tn < 0.623 <. Therefore we may replae by = in (29): (30) ±am 2 + sm = ±bn 2 + tn. From (30), squaring twie, we obtain [(am 2 bn 2 ) 2 m 2 n 2 ] 2 4m 2 n 2 (mod ). Sine 4m 2 n 2 < 0.047 < and [(am 2 bn 2 ) 2 m 2 n 2 ] 2 < 0.969 <, we therefore have [(am 2 bn 2 ) 2 m 2 n 2 ] 2 = 4m 2 n 2, and (31) am 2 bn 2 = ±m ± n. From (30) and (31) we obtain (32) m(s ± 1) = n(t ± 1). Inserting (32) into (30) we obtain (33) n = (s ± 1)[t(s ± 1) (t ± 1)s] ±[a(t ± 1) 2 b(s ± 1) 2 ] = (s ± 1)(±t s) ±(±2at + 2a ± 2bs 2b).

There are only finitely many Diophantine quintuples 15 Sine (s ± 1)(±t s) (s 1)(t s) = (s 1)(b a) t + s > 2(s 1) 2 b a > 2 b and ± 2at + 2a 2bs 2b 4bs + 4b < 6b a, (33) implies that n > ontradition. 1.2) v 2m = w 2n, z 0 = r st We may assume that b 4. Then we have 12 b > 0.377, a z 0 = z 1 = 2 a b 1 r + st > 64 63 2 2r > 2.155 ab, and from z 1 < 2 it follows < b 5.4a2 b < 5.4b 3 < b 4.5, a ontradition. 1.3) v 2m = w 2n, z 0 1, r st From Lemma 9 we obtain (34) az 0 m 2 + sx 0 m bz 0 n 2 + ty 1 n (mod 4). By Lemma 8, we have Therefore, (35) az 0 m 2 < 0.806 <, sx 0 m < (a z 0 + z 0 )m < 0.8 <, bz 0 n 2 < 0.876 <, ty 1 n < (b z 1 + z 1 )n < 0.87 <. az 0 m 2 + sx 0 m = bz 0 n 2 + ty 1 n. Assume now that b > 4a. Sine z 0 1, we have z0 2 max{ + 1, 3/a}. This implies 0 sx 0 a z 0 1 = x2 0 + a a2 a z 0 (sx 0 + a z 0 ) 1.001a 2a 2 z 2 0 < 0.1669, 0 ty 1 b z 1 1 = y2 1 + b b2 b z 1 (ty 1 + b z 1 ) 1.001b 2b 2 z 2 0 < 0.0418. If z 0 > 1 then, by (35), we have az 0 m(m + 1.1669) bz 0 n(n + 1), and sine m, n 2, we obtain a 1.5835m 2 > bn 2 and m n 1.589. But now Lemma 4 implies n 1, a ontradition. If z 0 < 1 then (35) implies a z 0 m(m 1)) b z 0 n(n 1.0418). Hene, by Lemma 4, we onlude that 4n(n 1.0418) < ( 11 9 n + 7 )( 11 18 9 n 11 ) < 1.4939n 2 0.2716n 0.2376, 18 whih implies 2.5061n 2 3.8956n + 0.2376 < 0 and n 1, a ontradition.

There are only finitely many Diophantine quintuples 16 (36) Assume now that b < 4a. Squaring twie the relation (35) we obtain By Proposition 1, we have [(am 2 bn 2 ) 2 x 2 0m 2 y 2 1n 2 ] 2 4x 2 0y 2 1m 2 n 2 (mod ). y 2 1 < b z2 0 + 1 < 4a a 5/7 9/7 0.754 + 1 < 3.018(a) 2/7 < 0.364. This implies that the both sides of the ongruene (36) are less than. Indeed, the left hand side is bounded above by max{ 8 9 +0.08, 8 9 +0.08, 0.728+0.006+0.04 } < 0.969 <, while 4x 2 0 y2 1 m2 n 2 < 0.006+0.728+0.04 = 0.774 <. Therefore we have an equality in (36), and this implies (37) am 2 bn 2 = ±x 0 m ± y 1 n. From (35) and (37) we obtain x 0 m(s ± 1) = y 1 n(t ± 1) and n = A/B, where A = x 2 0y 1 (s ± 1)(±t s), B = z 0 [ab(y 2 1 x 2 0) + 2(a b) ± 2aty 2 1 2bsx 2 0]. We have the following estimates These estimates yield A x 2 0y 1 (s + 1)(t + s) < 2.005x 2 0y 1 ab, B > z 0 [ab(2y 1 1) 2b 4aty1 2 2s(b a)] [ > z 0 y 1 ab 2 1 2 4ty 1 y 1 ay 1 b 2s ] > 1.569 z 0 y 1 ab. ay 1 n < 2.005x2 0 y 1 ab 1.569 z 0 y 1 ab < 1.278 x 2 0 z 0 ab < 1.278 az0 2 0.999 z 0 ab < 1.28 z 0 < 1/4 < 1, a ontradition. 2) v 2m+1 = w 2n The impossibility of this ase is proven in 1.2). 3) v 2m = w 2n+1 By Lemma 8 and 1.2), z 0 > 5.4b 3 < b 4.5, a ontradition. 4) v 2m+1 = w 2n+1 From Lemmas 8 and 9 we obtain (38) 2.155, and from z ab 0 < 2 a it follows < 5.4ab2 < ±astm(m + 1) + rm ±bstn(n + 1) + rn (mod 2), where m = m, n = n if z 0 < 0, and m = m + 1, n = n + 1 if z 0 > 0. Multiplying (38) by s and t, respetively, we obtain (39) (40) ±atm(m + 1) + rsm ±btn(n + 1) + rsn (mod 2), ±asm(m + 1) + rtm ±bsn(n + 1) + rtn (mod 2). We have btn(n + 1) < 0.855 <, rtn < 0.845 <. Therefore we have equalities in (39) and (40). This implies rm = rn and am(m + 1) = bn(n + 1), and finally m = n = 0.

There are only finitely many Diophantine quintuples 17 Lemma 11 Let {a, b, } be a Diophantine triple of the seond kind and > 10 100. v m = w n and n > 2, then n > 0.04. If Proof. Assume that n 0.04. Then max{ m/2 + 1, n/2 + 1} < n 0.04. 1.1) v 2m = w 2n, z 0 = 1 Sine am 2 < 0.408, sm < 0.701, bn 2 < 0.408 and tn < 0.701, (29) implies (30). Furthermore am 2 sm < m a < 0.06 < 0.001, bn 2 tn < n b < 0.06 < 0.001. Hene m n 0.999 1.001 t s > 0.999 b 1.001 2 a > 1.99, ontrary to Lemma 4. 1.2) v 2m = w 2n, z 0 = r st We have z 0 = z 1 = 2 a b 1 > 2 5 4 1.4 r + st 2r > 5 4 0.4 2.13 ab, 2 b it follows < 5.25a2 b < 1.32ab 2. Hene {a, b, } is a Diophantine and from z 1 < triple of the fourth kind, and this ase will be treated in Lemma 13. 1.3) v 2m = w 2n, z 0 1, r st By Proposition 1, we have > b 3.5, and Lemma 8 implies az 0 m 2 < 0.907 <, sx 0 m < 0.871 <, bz 0 n 2 < 0.98 <, ty 1 n < 0.944 <. Therefore, equation (35) holds again. As in Lemma 10, we obtain a ontradition with Lemma 4. 2) v 2m+1 = w 2n This ase is impossible by Lemma 2. Namely, if 100a then a 10 98 and a 1.03, a ontradition. 3) v 2m = w 2n+1 As in 1.2), we obtain < 5.25ab 2 and again {a, b, } is a triple of the fourth kind. 4) v 2m+1 = w 2n+1 Relation (38) implies [am(m + 1) bn(n + 1)] 2 r 2 (n m) 2 (mod 2). Sine a 2 m 2 (m + 1) 2 < 0.96, b 2 n 2 (n + 1) 2 < 0.96, and r 2 m 2 < 0.48, we obtain (41) bn(n + 1) am(m + 1) = ±r(m n), whih implies (42) 2n + 1 a 2m + 1 b = ±4r(m n) + b a [(2n + 1) b + (2m + 1) a] b(2m + 1).

There are only finitely many Diophantine quintuples 18 By Lemma 4, the right hand side of (42) is 4r(m n) b(2n + 1)(2m + 1) + Therefore, (43) b b(2n + 1)(2m + 1) < 2n + 1 2m + 1 < 103 150 < 0.687. 4 b 2 4n+5 10 b(2n + 1)(2m + 1) + 1 15 < 3 25 + 1 15 = 14 75. If n = 1 then m = 1 (by Lemma 4), ontrary to (43). If n = 2 then m = 2 or m = 3, whih both ontradit the relation (43). Hene we may assume that n 3, m 3, and now (42) implies 2n+1 2m+1 < 0.59. On the other hand, from Lemma 4 we see that 2n+1 2m+1 5 7 3 7(2m+1) 0.653, a ontradition. Lemma 12 Let {a, b, } be a Diophantine triple of the third kind and > 10 100. v m = w n and n > 2, then n > 0.15. If Proof. Assume that n 0.15. Then max{ m/2 + 1, n/2 + 1} < n 0.15. 1.1) v 2m = w 2n, z 0 = 1 Sine am 2 < 0.9, sm < 0.951, bn 2 < 0.9 and tn < 0.951 <, the proof is idential to that of Lemma 11. 1.2) v 2m = w 2n, z 0 = r st From (34) we have (44) ±astm(m 1) + rm ±bstn(n 1) + rn (mod ). Multiplying (44) by 2st we obtain (45) ±2[am(m 1) bn(n 1)] 2rst(n m) (mod 2). Let α be the absolutely least residue of 2rst modulo 2, and let A = (2rst 2r 2 +)(st+r). Then α (st + r) A and A = 2ar + 2br + 2r + st 2 r < 2ar + 2br + 2r. Sine r st = 2 b a 1 r+st < 2 2 < b b < br, we obtain A < 2r(a+b+1) < 13 ab 6 br and α < 13br 12 < 1.15b. We have 2am(m 1) < ab 0.904, 2bn(n 1) < 0.904, α(m n) < 0.651. Therefore ±2[am(m 1) bn(n ± 1)] = α(n m), and it implies (46) 2n 1 a 2m 1 b = ±2α(m n) + b a [(2n 1) b + (2m 1) a] b(2m 1). By Lemma 4, the right hand side of (46) is 1.15b 6n+3 5 b(2n 1)(2m 1) + 1 (2n 1)(2m 1) < 0.495,

There are only finitely many Diophantine quintuples 19 and therefore (47) 2n 1 2m 1 < 0.784. If n = 2 then Lemma 4 implies m = 2 or 3. The ase m = 2 ontradits the inequality (47), while for m = 3 relation (46) implies 2n 1 2m 1 < 0.586. But for n = 2, m = 3 we have 2n 1 2m 1 { 3 5, 5 7 }, a ontradition. Hene n 3, and sine n = m = 3 ontradits (47), we have also m 4. Now, from (46) we obtain 2n 1 2m+1 2m 1 < 0.456. But for m 4 we have 2n+1 > 2n 1 2m 1 > 5 8 3 4(2m 1) > 0.517, a ontradition. 1.3) v 2m = w 2n, z 0 1, r st Proposition 1 implies > b 3.5 and therefore this ase is impossible. 2) v 2m+1 = w 2n The impossibility of this ase is shown in Lemma 11. 3) v 2m = w 2n+1 From Lemmas 8 and 9 it follows that and (48) ±2astm(m 1) + r(2m ± 1) ±2bstn(n + 1) + r(2n + 1) (mod 4) ±2[am(m 1) bn(n + 1)] 2rst(n m + δ) (mod 2), where δ {0, 1}. Let α be defined as in 1.2). As in 1.2), we obtain and (49) ±2[am(m 1) bn(n + 1)] = α(n m + δ) 2n + 1 a 2m 1 b = The right hand side of (49) is and therefore (50) ±2α(m n δ) + b a [(2n + 1) b + (2m 1) a] b(2m 1). 1.15b 6n+11 5 b(2n 1)(2m 1) + 1 (2n 1)(2m 1), 2n + 1 2m 1 < 0.861, 2n + 1 2m + 1 < 0.617, respetively. If n = 1, then by Lemmas 4 and 7 we have m = 2. This is learly impossible if we have 2n+1 2n+1 2m 1 on the left hand side of (49), while for 2m+1 we obtain 3 5 < 0.509, a ontradition. If n = 2 then, by Lemma 17, we have m = 2, 3 or 4, and (50) implies m = 4. Sine m = n = 3 is also impossible by (50), we onlude that n 2 and m 4. This implies 2n+1 2m 1 < 0.469, 2n+1 2m+1 < 0.429, respetively. On the other hand, we have 2n+1 2m 1 > 2n+1 2m+1 5 8 1 2m+1 > 0.513, a ontradition. 4) v 2m+1 = w 2n+1

There are only finitely many Diophantine quintuples 20 Let α be defined as in 1.2). From (38) we obtain whih yields ±2[am(m + 1) bn(n + 1)] = α(n m), (51) 2n + 1 a 2m + 1 b = ±2α(m n) + b a [(2n + 1) b + (2m + 1) a] b(2m + 1). From (51) it follows that 2n+1 2m+1 a ontradition. < 0.494. But Lemma 4 implies 2n+1 2m+1 5 8 3 8(2m+1) 0.55, Lemma 13 Let {a, b, } be a Diophantine triple of the fourth kind and > 10 100. v m = w n and n > 2, then n > 0.2. If Proof. Assume that n 0.2. Then max{ m/2 + 1, n/2 + 1} < n 0.2. 1.1) v 2m = w 2n, z 0 = 1 The proof is idential to that of Lemma 11. 1.2) v 2m = w 2n, z 0 = r st The first part of the proof is idential to that of Lemma 12. In partiular, the relation (45) is valid. Estimating r st, we get r st < 2 ab < 6ab2 2 ab < 3b ab < 3br. Therefore A < 2r(a + b + 1) < 2.5br and α < 2.5br 2 < 1.33b. Note that ab α 2 4r 2 s 2 t 2 4r 2 (mod 2), 4r 2 4.5ab < 1.125b 2 < 2 and α 2 < 1.77b 2 < 2. Hene, α 2 = 4r 2 and α = ±2r. Sine am(m + 1) < 0.9, bn(n + 1) < 0.9, r(m n) < 0.7, we have (52) bn(n + 1) am(m + 1) = ±r(m n). Inserting α = ±2r in (46) and using Lemma 4 and the estimate α = 2r < b 2 + 4 < 1.001b, we obtain (53) 2n 1 2m 1 < 1 2 + 1.001n + 1 (2n 1)(2m 1) < 0.834. If (m, n) = (3, 2), then (52) implies 6b 12a = ±r and (4b 9a)(9b 16a) = 1, whih is learly impossible for b > 4a. Hene n 3, m 4 and (53) yields 2n 1 2m 1 < 0.615. On the other hand, 2n+1 2m+1 > 2n 1 2m 1 2 3 1 3(2m 1) > 0.619, a ontradition. 1.3) v 2m = w 2n, z 0 1, r st Proposition 1 implies > b 3.5, and we have < 6ab 2 < 1.5b 3. Therefore, this ase is impossible. 2) v 2m+1 = w 2n

There are only finitely many Diophantine quintuples 21 The impossibility of this ase is shown in Lemma 11. 3) v 2m = w 2n+1 The first part of the proof is the same as in Lemma 12. As in 1.2), we onlude that α = ±r. We have (54) am(m 1) bn(n + 1)] = ±r(m n δ) and (55) 2n + 1 a 2m 1 b = The relation (55) implies ±4r(m n δ) + b a [(2n + 1) b + (2m 1) a] b(2m 1). (56) 2n + 1 2m ± 1 < 1 2 + 2r(n + 2 2δ) b(2n + 1)(2m 1) + 1.001r(2m 1) 2 + 1 (2n + 1)(2m 1) < 1 1.001(n + 2 2δ) + 1 + < 0.945. 2 (2n + 1)(2m 1) If n = 1, then m = 2 and we must have the sign + in (56). But then (54) implies 6a 2b = ±r and (9a 4b)(4a b) = ±1, a ontradition. If (m, n) = (3, 2), then we obtain 12a 6b = ±r, whih has no solution by 1.2). If (m, n) = (4, 2), we have two possibilities: 12a 6b = ±r or 20a 6b = ±2r. We have to onsider only the seond possibility, and it implies (25a 9b)(4a b) = 1, whih has no integer solution. Hene n 3, m 4 and 2n+1 2m 1 < 0.623. We see that (m, n) (4, 3), whih implies m 5 and 2n+1 2m+1 < 0.55, 2n+1 2n+1 2m 1 < 0.545, respetively. On the other hand, 2m 1 > 2n+1 2m+1 2 3 1 2m+1 > 0.576, a ontradition. 4) v 2m+1 = w 2n+1 As in the proof of Lemmas 11 and 12, using α = ±2r, we obtain (42). It implies (57) 2n + 1 2m + 1 < 1 1.001(n + 1) + 1 + 2 (2n + 1)(2m + 1) < 0.834. If (m, n) = (2, 1), then (41) implies 2b 6a = ±r, whih is impossible (see 2)). Hene we have n 2, m 3, and (57) atually gives 2n+1 2m+1 < 0.615. On the other hand 2n+1 2m+1 2 3 1 3(2m+1) > 0.619, a ontradition. 8 Linear forms in three logarithms Solving reurrenes (12) and (13) we obtain (58) (59) v m = 1 2 a [(z 0 a + x0 )(s + a) m + (z 0 a x0 )(s a) m ], w n = 1 2 b [(z 1 b + y1 )(t + b) n + (z 1 b y1 )(t b) n ]. Using standard tehniques (see e.g. [3, 11]) we may transform the equation v m = w n into an inequality for a linear form in three logarithms of algebrai numbers. In [9], Lemma

There are only finitely many Diophantine quintuples 22 5, we proved that (assuming > 4b, but this assumption an be replaed by > b +, whih is satisfied for any triple {a, b, }) if m, n 0, then (60) 0 < m log(s + a) n log(t + b) + log b(x0 + z0 a) a(y1 + z1 b) < 8 3 a(s + a) 2m. Thus, we have everything ready for the appliations of the Baker s theory of linear forms in logarithms of algebrai numbers. We will use Matveev s result ([18], Theorem 2.1), whih is quoted below with some restritions and simplifiations. Lemma 14 ([18]) Let Λ be a linear form in logarithms of l multipliatively independent totally real algebrai numbers α 1,..., α l with rational integer oeffiients b 1,..., b l (b l 0). Let h(α j ) denotes the absolute logarithmi height of α j, 1 j l. Define the numbers D, A j, 1 j l, and B by D = [Q(α 1,..., α l ) : Q], A j = max{d h(α j ), log α j }, B = max{1, max{ b j A j A l : 1 j l}}. Then (61) log Λ > C(l)C 0 W 0 D 2 Ω, where C(l) = 8 (l 1)! (l + 2)(2l + 3)(4e(l + 1))l+1, C 0 = log(e 4.4l+7 l 5.5 D 2 log(ed)), W 0 = log(1.5ebd log(ed)), Ω = A 1 A l. Proposition 3 Assume that > max{b 5/3, 10 100 }. If v m = w n, then (62) m log(31.3(m + 1)) < 3.826 1012 log 2. Proof. We apply Lemma 14 to the form (60). We have l = 3, D = 4, α 1 = s + a, α 2 = t + b, b(x0 + z0 a) α 3 =. a(y1 + z1 b) Furthermore, A 1 = 2 log α 1 < 1.608 log, A 2 = 2 log α 2 < 1.608 log. The onjugates of α 3 are b(z0 a ± x0 ) a(z1 b ± y1 ), and the leading oeffiient of the minimal polynomial of α 3 is a 0 = a 2 ( b) 2. We proeed with the following estimates: b(x0 + z0 a) a(y1 + z1 b) < b 2x0 a y1 < 1.415 4 b 2 a, b(x0 + z0 a) b a(y1 z1 b) < 2x0 2y1 < 2.001 b 4 ab 2 a( b) 0.999 a b(x0 z0 a) a(y1 + z1 b) < b( a) a x0 y1 < b a, < 2.004 4 b 3 2 a,

There are only finitely many Diophantine quintuples 23 b(x0 z0 a) a(y1 z1 b) < b( a) 2y1 a( b) x0 < 1.415 4 b 3 a 2. Therefore, A 3 = 4h(α 3 ) < log(4.013 a 1/2 b 5/2 3 ) < 4.804 log. We also have A 3 log(a 2 ( b) 2 ) 1.9999 log. Sine max{m, n} {m, m + 1} (by Lemma 3), we onlude that B < 0.8041(m + 1). We may assume that m > 10 12. Therefore we have log 8 3 a(s + a) 2m < 0.9999 m log. It is lear that α 1, α 2 and α 3 are multipliatively independent and totally real. Hene, we may apply Lemma 14. Putting all the above estimates in (61), we obtain 0.9999 m log < 3.8255 10 12 log 3 log(31.3(m + 1)) and m log(31.3(m + 1)) < 3.826 1012 log 2. Note that the assumption > max{b 5/3, 10 100 } is not essential. It has an effet only on the onstant on the right hand side of (62) (see [9], Setion 10). Baker s method an be applied without any gap assumption. However, gap assumptions are neessary for obtaining lower bounds for solutions using the ongruene method of Setion 7. We now ompare the lower bounds from Setion 7 with the upper bound from Proposition 3 to prove Conjeture 1 for all standard triples with large enough. Proposition 4 Let {a, b, } be a standard Diophantine triple and 10 2171. If {a, b,, d} is a Diophantine quadruple and d >, then d = d +. Proof. Let ad + 1 = x 2, bd + 1 = y 2, d + 1 = z 2. Then there exist integers m, n 0 suh that z = v m = w n, where the sequenes (v m ) and (w n ) are defined by (12) and (13). Assume that d d +. Then from Lemma 5 it follows that m 3 and n 3. Hene we may apply Lemmas 10 13. We get that in all ases n > 0.01. Now, by Lemma 3, m + 1 n > 0.01. If we put this in (62), we obtain m log(31.3(m + 1)) log 2 (m + 1) < 3.826 1016, whih implies m < 5.108 10 21, and finally < (m + 1) 100 < 10 2171. Corollary 2 Let {a, b, } be a standard Diophantine triple suh that < 2.695 b 3.5 and 10 2171. If {a, b,, d} is a Diophantine quadruple, then d = d or d = d +.

There are only finitely many Diophantine quintuples 24 Proof. By Proposition 4, we may assume that d <. Sine < 2.695b 3.5, Proposition 1 implies that {a, b, d, } is a regular quadruple. From the proof of Lemma 5 it follows that (m, n) {(0, 0), (0, 1), (1, 0), (1, 1)}. Assume that (m, n) = (0, 0). By Lemma 8 and the regularity of {a, b, d, }, we onlude that z 0 = 1 or z 0 = r st. Sine d > 0, we have z 0 = r st and d = (z 2 0 1)/ = d. If (m, n) = (0, 1) then Lemma 8 implies z 0 = r st and d = d. Analogously, if (m, n) = (1, 0) then z 1 = r st and d = d. Assume finally that (m, n) = (1, 1). Then z 0, z 1 < 0 and Lemma 8 implies z 0 = t, z 1 = s. We have v 1 = r st and d = (v 2 1 1)/ = d. Corollary 3 Let {a, b, } be a Diophantine triple of the third or fourth kind and 10 2171. If {a, b,, d} is a Diophantine quadruple, then d = d or d = d +. Proof. The statement follows diretly from Corollary 2 sine 6ab 2 < 2.695b 3.5 for any Diophantine pair {a, b}. 9 Proof of Theorem 1 Let {a, b,, d, e} be a Diophantine quintuple and a < b < < d < e. Consider the quadruple {a, b,, d}. By Proposition 2, it ontains a standard triple, say {A, B, C}, A < B < C. If d 10 2171 then we may apply Proposition 4 on the triple {A, B, C}. We onlude that e = A + B + C + 2ABC + 2 (AB + 1)(AC + 1)(BC + 1) < 4d(b + 1) < d 3. On the other hand, by Corollary 1, we have e > 2.695d 3.5 b 2.5 > d 3, a ontradition. Hene d < 10 2171. Consider now the quadruple {A, B, C, e}. We have C d < 10 2171. Let (V m ) and (W n ) be the sequenes defined in the same manner as (v m ) and (w n ), using A, B, C instead of a, b,. Let e C + 1 = V 2 m. By Proposition 3, and m < 5.109 10 21. From (16) we obtain Therefore and Hene e < 10 1026. m < 9.561 1019 log(31.3(m + 1)) V m < 1.7C 4 AC(2 AC + 1) m 1 < 2 m d m+0.5. log 10 V m < m log 10 2 + (m + 0.5) 2171 < 1.1094 10 25 log 10 e < 2 log 10 V m < 2.2188 10 25.

There are only finitely many Diophantine quintuples 25 Corollary 4 If {a, b,, d, e} is a Diophantine quintuple and a < b < < d < e, then d < 10 2171 and e < 10 1026. Remark 1 We an use the theorem of Baker and Wüstholz [4] instead of the theorem of Matveev [18] in the proof of Theorem 1. In that way we obtain slightly larger onstants in Corollary 4, namely, d < 10 2411 and e < 10 1028. 10 There does not exist a Diophantine sextuple Sine the bound for the size of elements of a Diophantine quintuple from Corollary 4 is huge, it is omputationally infeasible to hek whether there exist any Diophantine quintuple. However, using a theorem of Bennett [5], Theorem 3.2, on simultaneous approximations of algebrai numbers, instead of the theorem of Matveev (or Baker and Wüstholz), we are able to prove that there is no Diophantine sextuple. Lemma 15 ([5]) If a i, p i, q and N are integers for 0 i 2, with a 0 < a 1 < a 2, a j = 0 for some 0 j 2, q nonzero and N > M 9, where M = max 0 i 2 { a i }, then we have { max 1 + a i 0 i 2 N p } i > (130Nγ) 1 q λ q where and log(33n γ) λ = 1 + log (1.7N 2 ) 0 i<j 2 (a i a j ) 2 γ = { (a2 a 0 ) 2 (a 2 a 1 ) 2 2a 2 a 0 a 1 if a 2 a 1 a 1 a 0, (a 2 a 0 ) 2 (a 1 a 0 ) 2 a 1 +a 2 2a 0 if a 2 a 1 < a 1 a 0. We will apply Lemma 15 to the numbers θ 1 = a s a and θ 2 = t b (a + 1)a θ 1 = a 2 = 1 + 1a = 1 + b ab, θ 2 = 1 + 1b = 1 + a ab. By [9], Lemma 12, it holds (63) ( max θ 1 sbx abz, θ 2 tay ) abz < 2a z 2. b. We have In order to apply Lemma 15, we have to assume that there is a big gap between b and. In the following two lemmas we will show that, if we assume that b and satisfy some strong gap onditions, the upper and lower bounds obtained in Setions 7 and 8 an be signifiantly improved. We will show in the proof of Theorem 2 that this strong gap onditions are satisfied by the seond and the fifth element of a Diophantine sextuple.

There are only finitely many Diophantine quintuples 26 Lemma 16 Let {a, b,, d}, a < b < < d, be a Diophantine quadruple. 1) If > 344.9 b 9.5 a 3.5, then d < 27.62. 2) If > 296.4 b 11.6 a 1.4, then d < 21.47. Proof. Let ad + 1 = x 2, bd + 1 = y 2, d + 1 = z 2. We apply Lemma 15 with a 0 = 0, a 1 = a, a 2 = b, N = ab, M = b, q = abz, p 1 = sbx, p 2 = tay. As in the proof of [9], Corollary 1, we obtain (64) log z < log (32.5a2 b 6 2 ) log (1.7 2 (b a) 2 ) log( 1.7 16.5ab 4 (b a) 2 ). Assume that > 344.9b 9.5 a 3.5. We have 32.5a 2 b 6 2 < 2+ 6 9.5, 1.7 2 (b a) 2 < 2, 1.7 16.5ab 4 (b a) 2 > 1 6 9.5. Inserting these estimates in (64), we obtain log z < 2 2.632 log2 0.368 log < 14.31 log. Hene, (65) z < 14.31 and d = z2 1 < 27.62. Assume now that > 296.4b 11.6 a 1.4. Then we have 32.5a 2 b 6 2 < 32.5a 0.9 b 7.1 2 7.1 2+ < 11.6 < 2.613, 1.7 2 (b a) 2 < 2, 1.7 16.5ab 6 > 1.7 6.3 16.5a 0.7 > 1 11.6 b6.3 > 0.456, and from (64) we obtain (66) z < 11.47 and d < 21.47. Lemma 17 Let {a, b, } be a Diophantine triple. Assume that v m = w n and n > 2. 1) If > max{b 11.6, 2.97 10 16 }, then n > 0.0815. 2) If b > 4a and > max{b 9.5, 1.5 10 13 }, then n > 0.11. Proof. 1) The proof is ompletely analogous to the proof of Lemma 10. 2) The proof is ompletely analogous to the proof of Lemma 11. Proposition 5 If {a, b,, d} is a Diophantine quadruple suh that > max{296.4 b 11.6 a 1.4, 2.97 10 16 } or b > 4a and > max{334.9 b 9.5 a 3.5, 1.5 10 13 }, and d >, then d = d +.

There are only finitely many Diophantine quintuples 27 Proof. Assume that d d +. Then n 3 and we may apply Lemma 17. If > max{296.4b 11.6 a 1.4, 2.97 10 16 }, then n > 0.0815. On the other hand, from (17) we have z = w n > 3.132 4 b (1.999 b) n 1 > n 2 + 1 4. From (66) we onlude that n 22 and it implies that < 12.97 10 16, a ontradition. If b > 4a and > max{334.9b 9.5 a 3.5, 1.5 10 13 }, then n > 0.11. From (65) it follows that n 28 and < 1.5 10 13, a ontradition. Proof of Theorem 2: Let {a, b,, d, e, f}, a < b < < d < e < f, be a Diophantine sextuple. By Corollary 1, we have e > 2.695d 3.5 b 2.5 > 2.695(4ab) 3.5 b 2.5 > 344.9b 9.5 a 3.5. If b < 4a then a + b + 2r > 9 4b, and we obtain e > 2.695 2.25 3.5 b 9.5 (4a) 1.4 b 2.1 > 296.4b 11.6 a 1.4. (67) Assume now that e > 2.97 10 16 or ( ) b < 4a and e > 1.5 10 13. Then we may apply Proposition 5. We onlude that the quadruple {a, b, e, f} is regular. It implies that f 4e(ab+1) < e 3. On the other hand, from Corollary 1 we have f > e 3.5, a ontradition. It remains to onsider the ase when the onditions (67) are not satisfied. If e 2.97 10 16 then d < 4 10 4, and it is easy to find all quadruples satisfying 2.695d 3.5 b 2.5 2.97 10 16 or b < 4a and 2.695d 3.5 b 2.5 1.5 10 13. There are exatly 10 suh quadruples: {1, 3, 8, 120}, {1, 3, 120, 1680}, {1, 8, 15, 528}, {2, 4, 12, 420}, {3, 5, 16, 1008}, {3, 8, 21, 2080}, {4, 6, 20, 1980}, {4, 12, 30, 5852}, {5, 7, 24, 3432} and {6, 8, 28, 5460}. However, we have already mentioned that Baker and Davenport [3] proved that {1, 3, 8} annot be extended to a quintuple. The same result was proved for the triples {1, 8, 15}, {1, 3, 120} in [16], and for all triples of the forms {k 1, k+1, 4k} and {F 2k, F 2k+2, F 2k+4 } (F n denotes the n th Fibonai number) in [7, 8]. Therefore, it suffies to show that {4, 12, 30, 5852} annot be extended to a Diophantine sextuple. But it is easy to prove, using the original Baker-Davenport method, that if d is a positive integer suh that {4, 12, 30, d} is a Diophantine quadruple, then d = 5852. First of all, in this ase we have z 0 = z 1 = ±1. If we apply Lemma 14 on the form (60) with (a, b, ) = (4, 12, 30), we obtain m < 2 10 15. Now we may apply the Baker-Davenport redution method [3] (see also [10], Lemma 5). In the first step of the redution we obtain m 7. The seond step gives m 1 if z 0 = 1, and m 2 if z 0 = 1, whih proves that d = 5852. Namely, d = 5852 orresponds to z 0 = 1 and m = n = 2. Aknowledgements. I would like to thank Professor Attila Pethő for inspirational disussions and his onstant enouragement during my work on this problem. I am grateful to Clemens Fuhs and the anonymous referee for valuable omments and suggestions that improved the previous version of the paper.

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