Fourth Power Diophantine Equations in Gaussian Integers

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1 Manuscript Noname manuscript No. (will be inserted by the editor) Fourth Power Diophantine Equations in Gaussian Integers Farzali Izadi Rasool Naghdali Forooshani Amaneh Amiryousefi Varnousfaderani. Received: date / Accepted: date Abstract In this paper we examine a class of fourth power Diophantine equations of the form x +kx y +y = z and ax +by = cz, in the Gaussian Integers, where a and b are prime integers. Keywords Quartic Diophantine equation Gaussian integers Elliptic curve rank torsion group Mathematics Subject Classification (0) MSC D MSC G0 1 Introduction Lebesque noted that x ± m y = z has integral solutions only when m = n±. Likewise, the Diophantine equation m x y = z has solution only when m = n + 1, but x ± y = m z is impossible in the integers []. L. Euler proved that x ±y = z has no integer solution for x y by means of the fact that x y are not squares[]. W. Mantel proved by descent that x + m y z unless n (mod ). The method of Yasutaka Suzuki [] determined all solutions of the equation a X r + b Y s = c Z t in nonzerointegersx, Y, Z,where a, b,carenon-negative integers, and r,s,t are or, and X, Y, Z are pairwise relatively primes. In [] Suzuki show that if the equation a X + b Y = c Z has an integer solution Farzali Izadi Department of Pure Mathematics, Faculty of Science, Urmia University, Urmia 1-, Iran. f.izadi@urmia.ac.ir Rasool Naghdali Forooshani Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz -, rnfmath@gmail.com Amaneh Amiryousefi Varnousfaderani Department of Mathematics, Isfahan University of technology, Isfahan, Iran. aa math@yahoo.com

2 F. Izadi et al. then a+1 = b+1 = c. We prove all of these cases by means of elliptic curve method. F. Najman [] showed that the equation x y = iz has only trivial solutions in the Gaussian integers. He also showed that the only nontrivial Gaussian solutions of the equation x +y = iz, are x,y {±i,±1} and z = ±i(1+i).using these results, we solve some new Diophantine equations of theformx +kx y +y = z where,k = ±.AlsoweexaminetheDiophantine equations x ±p y = iz and x ±p y = iz for some prime integers p. Elliptic curves method In this section we describe the method which we use for proving our results. Let E(Q) denote the group of rational points on elliptic curve with Weierstrass equation E : y = x +ax +bx. Let Q be the multiplicative groupofnon-zero rational numbers and Q denote the subgroup of squares of elements of Q. Define the group -descent homomorphismα from E(Q) to Q /Q as follows: 1 (mod Q ) if P =, α(p) = b (mod Q ) if P = (0,0), (1) x (mod Q ) if P = (x,y) with x 0. Similarly, take the isogenous curve Ê : y = x ax +(a b)x with group of rational points Ê(Q). The group -descent homomorphism α from Ê(Q) to Q /Q given by 1 (mod Q ) if P =, α( P) = a b (mod Q ) if P = (0,0), () x (mod Q ) if P = (x,y) with x 0. Proposition 1 By the above notations, we have the following equality for the rank r of E(Q) r = Im(α) Im( α). () Theorem 1 The group α(e(q)) is equal to the classes of 1, b and the positive and negative divisors b 1 of b modulo squares such that the quartic equation N = b 1 M +am e + b b 1 e has solution in integers with M,N and e pairwise coprime such that Me 0. If (M,N,e) is such a solution then the point P = ( b1m e, b1mn e ) is in E(Q) and α(p) = b 1. Remark 1 A similar theorem is true for α. For more details and the the proof of the above proposition and theorems see [1, Section..].

3 Fourth Power Diophantine Equations in Gaussian Integers Remark It is well-known (see e.g. []) that if an elliptic curve E is defined over Q, then the rank of E over Q(i) is given by rank(e(q(i))) = rank(e(q))+rank(e 1 (Q)) where E 1 is the ( 1)-twist of E over Q. We use it during the proofs of this article. In order to determine the torsion subgroup of E(Q(i)), we use the extended Lutz-Nagell theorem [, Chapter ], which is a generalization of the Lutz- Nagell theorem from E(Q) to E(Q(i)). Therefore throughout this article, the following extension of the Lutz-Nagell theorem is used to compute the torsion groups of elliptic curves. Theorem (Extended Lutz-Nagell Theorem) Let E : y = x +Ax+B with A,B Z[i]. If a point (x,y) E(Q(i)) has finite order, then 1. Both x,y Z[i], and. Either y = 0 or y (A +B ). Now we are ready to state our result about the elliptic curves. Theorem 1. Let F p : Y = X +p X, where p (mod ) is a prime integer. Then F p (Q(i)) = {,(0,0),(ip,0),( ip,0)}.. For prime integers p (mod ) let E p : Y = X p X. Then E p (Q(i)) = {,(0,0),( ip,0),(ip,0)}. Proof 1. The biquadratic equation of the homogeneous space of the elliptic curve F p is N = b 1 M + p b 1 e where b 1 {±1,±p±p } and 1 Im(α). Clearly, the equation has no solutions for negative b 1. Considering b 1 mod squares, we have to examine b 1 = p and hence we have pm +pe = N. Consideringthisequationmod,weseethatthelefthandsideisequivalent to, while the right hand side is equivalent to 0,1,. Therefore, we get Im(α) = {1}.Next,weconsidertheisogenouscurve F p : Ŷ = X p X. The biquadratic equation of the homogeneous space of this curve is where N = b 1 M p b 1 ê b 1 {±1,±,±,±p,±p,±p,±p,±p,±p }. We have 1, 1 Im α. Considering b 1 mod square, we have to examine the equation for b 1 = ±,±p,±p. For b 1 = we have M p ê = N M = N mod p but then is a square mod p so p ±1 mod which is false. Since Im( α) is a multiplicative group, / Im( α). For b 1 = p the equation is p M +pê = N. Since M is odd, the left hand is or mod

4 F. Izadi et al. while the right hand side is 0,1,. Also p / Im( α) since Im( α) is multiplicative. Therefore, Im( α). By proposition 1 the rankf p (Q) = 0; Using the Extended Lutz-Nagell theorem, Fp = p and so if (X,Y) is a torsion point, Y = 0 or ap k where a = ±1,±,±i,± and k = 0,,,. For k = 0, by comparing the power of p in Y = X +p X we have Y ap k for all a = ±1,±,±i,± and k =,,. For Y =, suppose that q is a prime divisor of x in Z[i]. Then q hence, q = ω = 1 + i. Comparing the powers of ω in both sides, we deduce that Y. In a similar way, we have Y ±1,±i. Only for Y = 0 do we have X = 0,ip which means that F p (Q(i)) Tor = {,(0,0),(ip,0),( ip,0)}.. It is similar to the part one. Main results In this section we study some quartic Diophantine equations in the Gaussian integers..1 On the Diophantine equations y ±p x = z and y ±x = pz Theorem 1. Let p (mod ). The Diophantine equations y p x = ±z and y +p x = ±iz has only trivial solutions in Z[i].. For p (mod ), Diophantine equations y +p x = ±z and y p x = ±iz has no nontrivial solution in Z[i]. Proof In the equations y ±x = ±pz, we divide both sides by x and put s = y x,t = z x. We have s ±1 = pt. Let r = s and multiplying two last terms we have r ±r = p(st), which leads to the elliptic curve Y = X ±p X using X = pr and Y = p st. By theorem the rank of theses curves is zero and the torsion point lead to trivial solution. The other cases are similar. Corollary 1 Let for n N {0}: 1. Let p (mod ) The Diophantine equations y p x = ± n z, y p x = n z, y +p x = ± n iz and y +p x = n iz have only trivial solutions in Z[i].. For p (mod ), the Diophantine equations y + p x = ± n z, y + p x = n z, y p x = ± n iz and y p x = n iz have only trivial solutions in Z(i).

5 Fourth Power Diophantine Equations in Gaussian Integers. On the Diophantine equations m x ± n y = z and m x ± n y = iz First we state results of Najman []: 1. x ±y = z has only trivial solutions in the Gaussian integers. (Hilbert). The equation x y = iz has only trivial solutions in the Gaussian integers.. TheonlynontrivialGaussianintegersolutionsofthe equationx +y = iz are (x,y,z), where x,y {±i,±1},z = ±iω and gcd(x,y,z) = 1. It is easy to change the Diophantine equations x ±y = ± m z, x ±y = ± m iz, x ± m y = z, x ± m y = iz, x ± m y = n iz, to one of the above equations and study their solvability in the Gaussian integers. Note that we have ω = i and ω =. For instance, Diophantine equation x +y = z transforms to X +Y = iz by X = x,y = y,z = iωz. So, the nontrivial solutions satisfying gcd(x, y, z) = 1 in the Gaussian integers of the equation x +y = z are (x,y,z), where x,y {±i,±1},z = ±i. In [] some quartic Diophantine equations of the form x + kx y + y = z, with trivial integer solutions were studied. Some of these equations have solutions in the Gaussian integers. For instance, x + x y + y = z is impossible in the integers, while (x,y,z) = (1,i,1) is a nontrivial Gaussian integer solution. Corollary 1. The only nontrivial Gaussian integer solutions of the Diophantine equation x +x y +y = z are (x,y,z) where gcd(x,y) = 1, x,y {±ω,±ω} and z {±}.. Triples (x,y,z) where x,y {±ω} and z {±}, are the only nontrivial solutions of Diophantine equation x x y + y = z in the Gaussian integers.. The equations x ± x y + y = z have no nontrivial solutions in the Gaussian integers.. The Diophantine equations x ±x y +y = iz have only trivial solutions in the Gaussian integers. Proof 1. Thisequationimpliesthat(x+y) +(x y) = z.byourdiscussion before the corollary, we have (x + y),(x y) {±i,±1} and z {±1}. An elementary calculation rise to eight nontrivial solutions in Q(i) and therefore eight solutions in Z[i].. Similar to the first part (ix+y) +(ix y) = z. So, it is sufficient to change the x coordinate of the above solutions to ix.. If (x,y,z) is a solution of x +x y +y = z then (x+y) +(x y) = (z). This equation has no solution in Z[i].. From x +x y +y = iz we have the Diophantine equation (x+y) + (x y) = (ωz) with only trivial solution. Note that by the mapping x ωx and y ωy, we can obtain many Diophantine equations from the above equations and discuss about their solutions. Now consider the equation x ± m y = z. Without loss of generality we suppose that 0 m. The cases m = 0, have been considered above. The

6 F. Izadi et al. Diophantine equation x y = z and x +y = z have Gaussian integer solutions such as (i,,)and (i,i,), respectively. The equation x y = z can be written in the form x + (ωy) = z. So, it is sufficient to consider x +y = z. Theorem The solutions of the Diophantine equation x + y = z are trivial in Z[i]. Proof Let (x,y,z) be a nontrivial solution of x + y = z. Dividing the equation by y and considering the change of variables s = x y and t = z y, we have s + = t for s,t Q(i). Let X = s X + = t. Multiplying both sides of these equations together and letting Y = st, we have the elliptic curve Y = X +X. The ( 1) twist of this curve is isomorphic to itself. Using sage and remark, we found that the rank of this curve is zero over Q(i). The only torsion point (0,0) on this curve leads to the trivial solution for the original equation. Corollary The Diophantine equations x +y = iz and x y = iz have only trivial solutions in Z[i]. Proof The first one is obvious. For the second one we have: x y = iz (y) x = (iωz) (y) +(ωx) = (iωz). The last equation has only trivial Gaussian integer solutions. Acknowledgements We are indebted to an anonymous reviewer of an earlier paper for providing insightful comments and providing directions for additional work which has resulted in this paper. References 1. H. Cohen, Number Theory, Volume I, Tools and Diophantine Equations, Graduate Texts in Math. Springer, New york, (00).. L. E. Dickson, History of The Theory of Number, Volume II,Diophantine Analysis, Chelsea Publishing Company, New york, (11).. Mordell, L.J., Diophantine equations, volume 0, Academic Press Inc., (London)LTD, England, (1).. F. Najman, The Diophantine equation x ± y = iz in the Gaussian integers. Amer. Math. Monthly, 1, -, (0).. Adam Parker, Who solved the Bernoulli equation and how did they do it?, Coll. Math. J.,,-, (01).. Sage software, Version..,

7 Fourth Power Diophantine Equations in Gaussian Integers. U. Schneiders and H.G. Zimmer, The rank of elliptic curves upon quadratic extensions, Computational Number Theory (A. Petho, H.C. Williams,H.G. Zimmer, eds.), de Gruyter, -, Berlin, (11).. Y. Suzuki, On the Diophantine Equation a X + b Y = c Z, Proc. Japan Acad.,, Ser A, (1).. Y. Suzuki, All solutions of Diophantine equation a X r + b Y s = c Z t where r,s and t are or. Nihoika Math.J.,.. T. Thongjunthug, Elliptic curves over Q(i), Honours thesis, (00).