THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( 3) Zrinka Franušić and Ivan Soldo

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1 RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 8 = 59 (04): 5-5 THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) Zrinka Franušić and Ivan Soldo Abstract. We solve the problem of Diophantus for integers of the quadratic field Q( ) by finding a D(z)-quadruple in Z[( + )/] for each z that can be represented as a difference of two squares of integers in Q( ), up to finitely many possible exceptions.. Introduction and preliminaries Let R be a commutative ring with unity and n R. The set of nonzero and distinct elements {a, a, a 3, a 4 } in R such that a i a j +n is a perfect square in R for i < j 4 is called a Diophantine quadruple with the property D(n) in R or just a D(n)-quadruple. If n = then a quadruple with a given property is called a Diophantine quadruple. The problem of constructing such sets was first studied by Diophantus of Alexandria who found the rational quadruple { 6, 33 6, 7 4, 05 6 } with the property D(). Fermat found the first Diophantine quadruple in the ring of integers Z - the set {, 3, 8, 0}. The problem on the existence of D(n)-quadruples has been studied in different rings, but mainly in rings of integers of numbers fields. The following assertion is shown to be true in many cases: There exists a D(n)-quadruple if and only if n can be represented as a difference of two squares, up to finitely many exceptions. In the ring Z one part of the assertion is proved independently by several authors (Brown, Gupta, Singh, Mohanty, Ramamsamy, see [6, 5, 7]), and another by Dujella in [7]. The set of possible exceptions S = { 4,,, 3, 5, 8,, 0} is still an open problem studied by many authors. The conjecture is that for n S there does not exist a Diophantine quadruple with the property D(n). In the ring of integers Z well studied is the case of n =. There is a conjecture that D( )-quadruple does not exist in Z. That is known as the D( )-quadruple conjecture and it was presented explicitly in [] for the first time. While it is conjectured that D( )-quadruples do not exist in integers 00 Mathematics Subject Classification. D09, R. Key words and phrases. Diophantine quadruples, quadratic field. 5

2 6 Z. FRANUŠIĆ AND I. SOLDO (see []), it is known that no D( )-quintuple exists and that if {a, b, c, d} is a D( )-quadruple with a < b < c < d, then a = (see [5]). It is proved that some infinite families of D( )-triples cannot be extended to a D( )-quadruple. The non-extendibility of {, b, c} was confirmed for b = by Dujella in [0], for b = 5 partially by Abu Muriefah and Al Rashed in [], and completely by Filipin in [8]. The statement was also proved for b = 0 by Filipin in [8], and for b = 7, 6, 37, 50 by Fujita in [4]. Dujella, Filipin and Fuchs in [3] proved that there are at most finitely many D( )-quadruples, by giving an upper bound of for the number of D( )-quadruples. This bound was improved several times: to by Filipin and Fujita ([9]), to by Bonciocat, Cipu and Mignotte ([5]) and very recently to by Elsholtz, Filipin and Fujita ([7]). In the ring of Gaussian integers Z[i] the above assertion was proved in [9]. Namely, if a + bi is not representable as a difference of the squares of two elements in Z[i], and in contrary if a + bi is not of such form and a + bi {±, ± ± i, ±4i}, then D(a + bi)-quadruple exists. Franušić in [0 ] found that a similar statement is true for rings of integers of some real quadratic fields, i.e. it can be seen that there exist infinitely many D(n)-quadruples if and only if n can be represented as a difference of two squares of integers. To be more precise, assuming the solvability of certain Pellian equation (x dy = ± or x dy = 4 in odd numbers) we are able to obtain an effective characterization of integers that can be represented as a difference of two squares of integers in Q( d) and then apply some polynomial formulas for Diophantine quadruples in a combination with elements of a small norm. Also, in [3] the existence problem in the ring of integers of the pure cubic field Q( 3 ) has been completely solved. The case of complex quadratic fields is more demanding because the set of elements with a small norm is poor (while in the real case there exist infinitely many units). A group of authors ([, 6, 8]) worked on the problem of the existence of D(z)-quadruples in Z[ ] and contributed that the problem is almost completely solved. As a prominent case, there appear the case z =, which could not be solved by the standard method via polynomial formulas. In [9] and [30] Soldo studied D( )-triples of the form {, b, c} and the existence of D( )-quadruples of the form {, b, c, d} in the ring Z[ t], t > 0, for b =, 5, 0, 7, 6, 37 or 50. He proved a more general result i.e. if positive integer b is a prime, twice prime or twice prime squared such that {, b, c} is a D( )-triple in the ring Z[ t], t > 0, then c has to be an integer. As a consequence of this result, he showed that for t {, 4, 9, 6, 5, 36, 49} there does not exist a subset ofz[ t] of the form {, b, c, d} with the property that the product of any two of its distinct elements diminished by is a square of an element in Z[ t]. For those exceptional cases of t he showed that there exist infinitely many D( )-quadruples of the form {, b, c, d}, c, d > 0 in Z[ t ].

3 THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) 7 In this paper, we verify assertion on the existence of D(z)-quadruples in complex quadratic field Q( ), i.e. in the corresponding ring of integers Z[( + )/]. In other words, we show the following theorems. Theorem.. There exists a D(z)-quadruple in the ring of integers of the quadratic field Q( ) if and only if z can be represented as a difference of two squares of integers in Q( ), up to possible exceptions z {, 3,, + }. Theorem.. There exists a D(z)-quadruple in the ring Z[ ] if and only if z can be represented as a difference of two squares of elements in Z[ ], up to possible exceptions z { 4,, 3,, + }. Although we have mentioned that the case of complex quadratic fields is rather complicated, observe that the Pellian equation x dy = 4 is solvable for d = in Z (the only solution is + ). To begin with, we will list briefly all statements that we require for the proofs of Theorem. and Theorem.. Lemma.3 ([8, Theorem ]). Let R be a commutative ring with the unity and m, k R. The set (.) {m, m(3k + ) + k, m(3k + ) + k +, 9m(k + ) + 8k + 4} has the D(m(k + ) + )-property. The set (.) is a D(m(k + ) + )-quadruple if it contains no equal elements or elements equal to zero. Lemma.4. If u is an element of a commutative ring R with the unity and {w, w, w 3, w 4 } is a D(w)-quadruple in R, then {w u, w u, w 3 u, w 4 u} is a D(wu )-quadruple in R. Lemma.5 ([4, Theorem ]). An integer z Q( ) can be represented as a difference of two squares of elements in Z[ ] if and only if is one of the following forms m, n Z. m + + n, 4m + 4n, 4m + + (4n + ), Lemma.6 ([4, Theorem ]). An integer z Q( ) can be represented as a difference of two squares of elements in Z[( + )/] if and only if is one of the following forms m + + n, m + (n + ), 4m + 4n, 4m + + (4n + ), m, n Z. m + + n +,

4 8 Z. FRANUŠIĆ AND I. SOLDO Lemma.7 ([, Lemma 5]). For each M, N Z, there exist k Z[( + )/] such that. M + + N = m(k + ) +, where m = +,. 4M (4N + ) = m(k + ) +, where m = +, 3. M + (N + ) = m(k + ) +, where m = +, 4. M + + (N + ) = m(k + ) +, where m = +, M N+ = m (k + ) +, where m = +. By using Lemmas.3,.4 and.7, we effectively construct Diophantine quadruples for integers of the forms given in Lemmas.5 and.6. The following assertion gives the nonexistence of a D(z)-quadruple in Z[ + ] if z cannot be represented as a difference of two squares in Z[( + )/], i.e. if and only if z is of the form 4m + + 4n, 4m + (4n + ), m + + (n + ). Lemma.8 ([, Theorem ]). If z has one of the forms 4m + + 4n, 4m + (4n + ), m + + (n + ), where m, n Z, then a D(z)-quadruple in Z[( + )/] does not exist. The nonexistence of a D(z)-quadruple in Z[ ] if z cannot be represented as a difference of two squares in Z[ ] follows partially from Lemma.8 (if z = 4m++4n or z = 4m+(4n+) ) and from the following assertion. Lemma.9. Let d Z is not a perfect square. Then there is no D(m + (n + ) d)-quadruple in the ring Z[ d]. Proof. The proof of Proposition in [] given for d = can be immediately rewritten for an arbitrary d. Let us denote the set. D(z)-quadruples in Z[ + ] D 4 = {mu, (m(3k+) +k)u, (m(3k+) +k+)u, (9m(k+) +8k+4)u}. According to Lemmas.3 and.4, D 4 is D((m(k + ) + )u )-quadruple if it contains no equal elements or elements equal to zero. This polynomial formula combining with specific values of m and u solves our problem, up to finitely many cases. Our results are listed in the tables of the following subsections... D(m + + n )-quadruples. In this subsection, for integers A and B, we will separate the cases of z = 4A (4B + ) and z = 4A + + 4B to corresponding subcases.

5 THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) 9 z k m u D 4 in exceptions of z 4A+3+4B A + B Z[ ], 3 4A++(4B+) +A + +B + Z[ ] 8A+3+(8B+) A+3B + A+B + Z[ ] 3 + 8A+7+(8B+6) A+3B+ + A+B + Z[ ] 8A+7+(8B+) AB + A+B+ Z[ ] + 8A+3+(8B+6) AB : + A+B+ Z[ ] 3 8A+5+8B A + B Z[ ] 5, 8A++(8B+4) A + B+ Z[ ] 8A++8B A + B 4 Z[ ], 9, 7 8A+5+(8B+4) 4A B 4 + B+ 4 4 Z[ ] Table It is easy to check that for those exceptions of z in Table, the polynomial formula D 4 gives the set with two equal elements, or some element is equal to zero. Therefore, in those exceptions of z (and all further exceptions), we used the method for the first time described in [8] (but only for quadruples in Z), to construct D(z)-quadruples with all distinct elements, of the form {u, v, u+v +r, u+4v +4r}, for some u, v, r Z[ ], or u, v, r Z[ + ], respectively. Except in cases of z =, 3, we found the following D(z)- quadruples in Z[ ]: {3 +,,, 5 3 } is the D(3 + )-quadruple, {3, +,, } is the D(3 )-quadruple, { + 3, +,, } is the D( )-quadruple, { 3,,, + } is the D( + )-quadruple, {8, +,, 4} is the D(5)-quadruple, {, 3, 8, 0 } is the D()-quadruple, {, 3, 8, 0} is the D()-quadruple, {6,, +, 4} is the D(9)-quadruple, { +, +,, } is the D( 7)-quadruple... D(m + (n + ) )-quadruples. z k m u D 4 in exceptions of z 4A + (4B + 3) A+3B+ + A+B+ + Z[ + ] 4A + + (4B + ) A+3B + A+B + Z[ + ] + 4A + (4B + ) AB + A+B Z[ + ] 4A + + (4B + 3) AB + A+B+ Z[ + ] Table

6 0 Z. FRANUŠIĆ AND I. SOLDO For the exceptions noted in Table, we found the following D(z)- quadruples in Z[ + ]: { 3,,, 3 + } is the D( )-quadruple, { +, 7 +,, 3 + } is the D( + )- quadruple, { + 3,, +, 3 } is the D( )-quadruple, {, 7,, 3 } is the D( )- quadruple..3. D(4m + 4n )-quadruples. z k m u D 4 in exceptions of z 8A+8B A+3B A+B + Z[ + ] 0 8A+4+(8B+4) A+3B A+B+ + Z[ + ] Table 3 The set {,, 5, 3 4 } is the D( )-quadruple in Z[ ] and it is easy to see that there exits infinitely many D(0)-quadruples. We obtain a D(8A B )-quadruple by multiplying elements of a D(m + + n )-quadruple by u = except for z = 4,, but {, 7 + 7,, 3} is the D( 4)-quadruple, and {, 7 +, 7, 30} is the D()-quadruple. Also, a D(8A + (8B + 4) )-quadruple is obtained by multiplying elements of a D(m + (n + ) )-quadruple by u =. Obviously, the resulting sets are subsets of Z[ ] (except for z = 4)..4. D(4m + + (4n + ) )-quadruples. z k m u D 4 in exceptions of z 8A++(8B+) A B Z[ + ] 6+, + 8A+6+(8B+6) A+3 B Z[ + ] 8A++(8B+6) (A+) (B+) 4 Z[ + ] 6, 8A+6+(8B+) A+3 B+ 4 Z[ + ] Table 4

7 THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) While the polynomial formula D 4 gives sets with two equal elements, for those exceptions of z of Table 4, we found the following D(z)-quadruples in Z[ + ]: { +, 9, 5 )-quadruple, { +, )-quadruple, {, 9+, 5 + )-quadruple, {, 5 3 )-quadruple., 85 } is the D( 6+, +, } is the D( +, 85 + } is the D( 6,, 3 3 } is the D(.5. D( m+ + n+ )-quadruples. We derive D( m+ + n+ )-quadruples from D(m + + n ) and D(m+(n+) ) -quadruples by multiplying them by + and. Multiplying the elements of a D(m + + n )-quadruple by u = + we obtain a D((m + + n )u )-quadruple except for z =, The number (m + + n )u is of the form A+ + B+ and for given A, B Z the equation (.) (m + + n )u = A + + B + has an integer solution (m, n Z) if and only if A + 3B (mod 4) and A+B 3(mod 4), i.e. (A, B) mod 4 {(0, 3), (, ), (, ), (3, 0)}. Concerning exceptions, the set { + 5, 3 5,, 5 } represents the D( )-quadruple, while we could not find the D( )-quadruple. Multiplying the elements of a D(m + + n )-quadruple by u = we obtain a D((m + + n )u )-quadruple except for z = +, 3 3. For given A, B Z the equation (.) has an integer solution if and only if A + 3B 0 (mod 4) and A B 0 (mod 4), i.e. (A, B) mod 4 {(0, 0), (, ), (, ), (3, 3)}. The set { 5, + 3 5, +, + 5 } is the D( 3 3 )-quadruple and we have not detected a D( )-quadruple. +

8 Z. FRANUŠIĆ AND I. SOLDO Multiplying the elements of a D(m + (n + ) )-quadruple by u = + we obtain a D( A+ + B+ )-quadruple. For given A, B Z the equation (.) (m + (n + ) )u = A + + B + has an integer solution if and only if A+3B 3 (mod 4) and A+B (mod 4), i.e. (A, B) mod 4 {(0, ), (, 0), (3, ), (, 3)}. Multiplying the elements of a D(m + (n + ) )-quadruple by u = we obtain a D( A+ + B+ )-quadruple. For given A, B Z the equation (.) has an integer solution if and only if A + 3B (mod 4) and A + B (mod 4), i.e. (A, B) mod 4 {(0, ), (, 0), (, 3), (3, )}. 3. D(z) quadruples in Z[ ] In the previous section we see that some D(z)-quadruples that have been constructed already lie in Z[ ] but some of them do not although z can be represented as a difference of squares in Z[ ]. Here we show that this can be improved. 3.. D(m + + n )-quadruples. z k m u D 4 in exceptions of z 4A + + (4B + ) A B+ + A+ /3 Z[ ], + Table 5 The set {, +,, } is a D( ), while the set {,,, 8 3 } is a D( + )-quadruple in Z[ ]. 3.. D(4m + + (4n + ) )-quadruples. Since there exist a D( m+ + n+ )-quadruple in Z[( + )/], by multiplying by the elements of this quadruple we obtain a D(4m++(4n+ ) )-quadruple in Z[ ], up to z =, +.

9 THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) D(4m + 4n )-quadruples. We have shown in.3. that D(8m+(8n+4) ) and D(8m+4+8n )- quadruples in Z[ ] are obtained by multiplying by the elements of D(m+(n+) ) and D(m++n )-quadruples in Z[(+ )/] up to the the D( 4)-quadruple whose elements are not in Z[ ]. The set {, 9k 8k, 9k k +, 36k 0k + } is D(8k)-quadruple ([7]) if k 0,, so there exists a D(8m + 8n )- quadruple in Z[ ]. z k m u D 4 in exceptions of z 8A (8B + 4) 3A B+4 + A+ 6 Z[ ] Table 6 It is easy to check that for those exceptions of z in Table 6, the polynomial formula D 4 gives the set with two equal elements. Therefore for certain z, we found the following D(z)-quadruples in Z[ ]: { +,, 3, 6 } is the D( + 4 )- quadruple, {, +, + 3, 6 + } is the D( 4 )- quadruple, {+, +, +, 0+5 } is the D(8)-quadruple. Remark 3.. Concerning the list of possible exceptions given in Theorem. and Theorem., we can easily observe that 3 = ( ), 4 =, ± = ( ) and ± = ( ). So, we are not surprised that the key point lies in an investigation on the existence of D( )-quadruples in rings Z[( + )/] and Z[ ]. In an analogy to D( )-quadruple conjecture in the ring of integers and the problem of existence of D( )-quadruples in Z[ t], t > 0 studied in [9] and [30], we might expect that for such z there does not exists a D(z)-quadruple. Acknowledgements. This work has been supported by Croatian Science Foundation under the project no. 64. References [] F. S. Abu Muriefah and A. Al-Rashed, Some Diophantine quadruples in the ring Z[ ], Math. Commun. 9 (004), 8.

10 4 Z. FRANUŠIĆ AND I. SOLDO [] F. S. Abu Muriefah and A. Al-Rashed, On the extendibility of the Diophantine triples {, 5, c}, Int. J. Math. Math. Sci. 33 (004), [3] S. Alaca and K. S. Williams, Introductory Algebraic Number Theory, Cambridge University Press, Cambridge, 004. [4] A. Baker and H. Davenport, The equations 3x = y and 8x 7 = z, Quart. J. Math. Oxford Ser. () 0 (969), [5] N. C. Bonciocat, M. Cipu and M. Mignotte, On D( )-quadruples, Publ. Mat. 56 (0), [6] E. Brown, Sets in which xy + k is always a square, Math. Comp. 45 (985), [7] A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (993), 5 7. [8] A. Dujella, Some polynomial formulas for Diophantine quadruples, Grazer Math. Ber. 38 (996), [9] A. Dujella, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 3 (997), 0. [0] A. Dujella, Complete solution of a family of simultanous Pellian equations, Acta Math. Inform. Univ. Ostraivensis 6 (998), [] A. Dujella, On the exceptional set in the problem of Diophantus and Davenport, Applications of Fibonacci Numbers 7 (998), [] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (004), [3] A. Dujella, A. Filipin and C. Fuchs, Effective solution of the D( )-quadruple conjecture, Acta Arith. 8 (007), [4] A. Dujella and Z. Franušić, On differences of two squares in some quadratic fields, Rocky Mountain J. Math. 37 (007), [5] A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 7 (005), [6] A. Dujella and I. Soldo, Diophantine quadruples in Z[ ], An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 8 (00), [7] C. Elsholtz, A. Filipin and Y. Fujita, On Diophantine quintuples and D( )- quadruples, Monatsh. Math. 75 (04), [8] A. Filipin, Nonextendibility of D( )-triples of the form {, 0, c}, Int. J. Math. Math. Sci. 4 (005), 7 6. [9] A. Filipin and Y. Fujita, The number of D( )-quadruples, Math. Commun. 5 (00), [0] Z. Franušić, Diophantine quadruples in the ring Z[ ], Math. Commun. 9(004), [] Z. Franušić, Diophantine quadruples in Z[ 4k + 3], Ramanujan J. 7 (008), [] Z. Franušić, A Diophantine problem in Z[( + d)/], Stud. Sci. Math. Hung. 46 (009), 03. [3] Z. Franušić, Diophantine quadruples in the ring of integers of Q( 3 ), Miskolc Math. Notes 4 (03), [4] Y. Fujita, The extensibility of D( )-triples {, b, c}, Pub. Math. Debrecen 70 (007), [5] H. Gupta and K. Singh, On k-triad sequences, Internat. J. Math. Math. Sci. 8 (985), [6] B. He and A. Togbé, On the D( )-triple {, k +, k + k + } and its unique D( )-extension, J. Number Theory 3 (0), [7] S. P. Mohanty and A. M. S. Ramasamy, On P r,k sequences, Fibonacci Quart. 3 (985), [8] I. Soldo, On the existence of Diophantine quadruples in Z[ ], Miskolc Math. Notes 4 (03), 6 73.

11 THE PROBLEM OF DIOPHANTUS FOR INTEGERS OF Q( ) 5 [9] I. Soldo, On the extensibility of D( )-triples {, b, c} in the ring Z[ t], t > 0, Studia Sci. Math. Hungar. 50 (03), [30] I. Soldo, D( )-triples of the form {, b, c} in the ring Z[ t], t > 0, Bull. Malays. Math. Sci. Soc., to appear. Diofantov problem za cijele brojeve kvadratnog polja Q( ) Zrinka Franušić i Ivan Soldo Sažetak. Rješavamo Diofantov problem za cijele brojeve kvadratnog polja Q( ) konstruiranjem D(z)-četvorki u prstenu Z[ ] za svaki z koji se može prikazati kao razlika dva kvadrata u Q( ), do na konačno mnogo mogućih izuzetaka. Zrinka Franušić Department of Mathematics University of Zagreb Bijenička cesta 30, HR Zagreb Croatia fran@math.hr Ivan Soldo Department of Mathematics University of Osijek Trg Ljudevita Gaja 6, HR Osijek Croatia isoldo@mathos.hr Received:

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