ON THE DIOPHANTINE EQUATION Pn a x4 = Pn a y4 i=1 i i j=1 j j FARZALI IZADI AND MEHDI BAGHALAGHDAM Abstract. In this paper, by using the elliptic curves theory, we study 7 the fourth power Diophantine equation Pni=1aix4i =Pnj=1ajyj4, where 1 a andn≥3arefixedarbitraryintegers.Wesolvetheequationforsome i 0 values of a and n = 3,4, and find nontrivial solutions for each case in i 2 naturalnumbers.Byourmethod,wemayfindinfinitelymanynontrivial n solutionsfortheaboveDiophantineequationandshow,amongtheother a things, that how some numbers can be written as sums of three, four, J or more biquadrates in two different ways. While our method can be 6 used for solving the equation for every a and n≥ 3, this paper will be i ] restrictedtothe exampleswheren=3,4.Inthe end,weexplainhowto T solve it in general cases without giving concrete examples. N . h t a m 1. Introduction [ Number theory is a vast and fascinating field of mathematics, sometimes 1 v called ”higher arithmetic”, consisting of the study of the properties of whole 5 numbers. In mathematics, a Diophantine equations is a polynomial equa- 0 6 tion, usually in two or more unknowns, such that only the integer solutions 2 0 are studied. The word Diophantine refers to the Hellenistic mathematician . 1 of the 3rd century, Diophantus of Alexanderia, who made a study of such 0 7 equations. While individual equation present a kind of puzzle and have 1 been considered throughout the history, the formulation of general theo- : v i ries of Diophantine equations (beyond the theory of quadratic forms) was X an achievement of the twentieth century. As a history work, Euler conjec- r a tured in 1969 that the Diophantine equation A4 +B4 +C4 = D4, or more generally AN + AN + ···+ AN = AN, (N ≥ 4), has no solution in pos- 1 2 N−1 N itive integers (see [1]). Nearly two centuries later, a computer search (see [3]) found the first counterexample to the general conjecture (for N = 5): 275 +845 +1105 +1335 = 1445. In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the K = 4 case (see [2]). His smallest counterexample was: 26824404+153656394+187967604 = 206156734. 2010 Mathematics Subject Classification. 11D45, 11D72, 11D25, 11G05 and 14H52. Key words and phrases. Diophantineequations.FourthpowerDiophantineequations, Elliptic curves. 1 2 F. IZADIAND M. BAGHALAGHDAM Inthis paper, we areinterested inthe study ofthe Diophantine equation: Pn a x4 = Pn a y4, i=1 i i j=1 j j where a , n ≥ 3 are fixed arbitrary integers. i Now we are going to solve the equation for the cases n = 3,4, and in the end we explain how to solve it in general cases (n ≥ 5). Our main results are the followings: Main Theorem 1.1. Consider the Diophantine equation : ax4+by4+cz4 = au4+bv4+cw4, where all the coefficients are fixed arbitrary integers. Let Y2 = X3 + fX2 + gX + h, be an elliptic curve in which the coefficients of f, g, and h are all functions of a, b, c, and two other rational parameters A and B, yet to be found later. If the elliptic curve has positive rank, depending on the values of A and B, the Diophantine equation has infinitely many integer solutions. Proof. Let: x = m+p, y = m−q, z = m−s, u = m−p, v = m+q and w = m+s, where all variables are rational numbers. By substituting these variables in the above Diophantine equation, and after some simplifications, we get: m2(ap−bq −cs) = −ap3 +bq3 +cs3. We may assume that ap−bq−cs = 1 and m2 = −ap3+bq3+cs3. Also let q = As+B, where A, B ∈ Q. By plugging q = As+B and p = bq+ cs+1, a a a into the equation m2 = −ap3+bq3+cs3, and after some simplifications, we obtain the elliptic curve: m2 = (−a(bA+c)3 +bA3 +c)s3 +(−3a(bA+c)2.(bB+1)+3A2Bb)s2 a a a +(−3a(bA+c).(bB+1)2 +3AB2b)s+(−a(bB+1)3 +bB3). a a a Multiplying the both sides of this elliptic curve by (−a(bA+c)3 +bA3 +c)2 and letting a bA+c (1.1) Y = (−a( )3 +bA3 +c)m, a and bA+c (1.2) X = (−a( )3 +bA3 +c)s, a we get the new elliptic curve Y2 = X3 +fX2 +gX +h, where bA+c bB +1 f = (−3a( )2.( )+3A2Bb), a a ON THE DIOPHANTINE EQUATION Pni=1aix4i =Pnj=1ajyj4 3 bA+c bB +1 bA+c g = (−3a( ).( )2 +3AB2b).(−a( )3 +bA3 +c), a a a bB +1 bA+c h = (−a( )3 +bB3).(−a( )3 +bA3 +c)2. a a If the above elliptic curve has positive rank (for every a, b, c, this is done by choosing appropriate values for A and B.), by calculating m, s, q, p, m ± p, m ± q, m ± s, from the relations (1.1), (1.2), q = As + B and p = bq + cs + 1 , and after some simplifications and canceling the a a a denominators of m±p, m±q, m±s, we may obtain infinitely many inte- gersolutionsfortheaboveDiophantineequation.Nowtheproofiscomplete. Now we are going to solve some couple of examples: Example 1. We wish to solve the Diophantine equation: x4 +2y4+3z4 = u4 +2v4 +3w4, We may assume that A = 4, B = 0. (we may choose the other appropriate values for A and B so that the corresponding elliptic curve has positive rank). Then we get the elliptic curve: Y2 = X3 −363X2+39600X −12002. The rank of this elliptic curve is 1 and its generator is the point P = (X,Y) = (3625, 46525). Because of this, the above elliptic curve has 16 64 infinitely many rational points and we may obtain infinitely many solutions for the Diophantine equation too. Since X = 3625 and Y = 46525, by calcu- 16 64 lating m, s, q, p, m±p, m±q, m±s, from the relations (1.1), (1.2) and q = As+B, p = bq + cs+ 1, and after some simplifications and canceling a a a the denominators of m±p, m±q, m±s, we get the identity: 51694+2.(459)4 +3.(1281)4 = 14474+2.(4181)4 +3.(2441)4. By choosing the other points on the elliptic curve such as 2P, 3P, ..., we obtain infinitely many solutions for the Diophantine equation. Example 2. Let a = 1, b = 1, c = 61, then by letting A = 2, B = 0, we get the elliptic curve 4 F. IZADIAND M. BAGHALAGHDAM Y2 = X3 −11907X2+47245842X −2499782. The rank of this elliptic curve is 1 and its generator is the point P = (X,Y) = (2613213887380271422, −35386313782867169078293498). 612348332222929 15152971591283964136217 By calculating theabove values and after some simplifications, we obtain a solution for the Diophantine equation: x4+y4+61z4 = u4+v4+61w4 as 1834886844435757755944694+1205840310799481812579854 +61.(73244546207202190584444)4 = 2352988071124881754162754 +687739084110357814361794+61.(21434423538289790762638)4. By choosing the other points on the elliptic curve such as 2P, 3P, ..., we obtain infinitely many solutions for the Diophantine equation. Example 3. Sums of three biquadrates in two different ways: In this case we have a = b = c = 1. Letting A = −10, B = 0, we get the elliptic curve Y2 = X3 −243X2−7290X −2702. The rank of this elliptic curve is 1 and its generator is the point P = (X,Y) = (450,6210). By calculating the above values and after some simplifications, we obtain a solution for the Diophantine equation: x4 +y4 +z4 = u4 +v4 +w4 as 194 +744 +1174 = 214 +644 +1194. Also we have: 2P = (X′,Y′) = (606357, −115780401) and 2116 97336 3P = (X′′,Y′′) = (255306774610, −118288360159623370). 164070481 2101578791129 By using these two new points, we obtain the two other solutions for the Diophantine equation respectively: 179480134+438560694+97653314 = 436769914+181270914+159636474, 88288913602203134+150990604919418274+115018135683643884 = 148287806717043614+85586115399828474+121558584635602864. Choosing the other points such as 4P, 5P, ..., give rise to infinitely many solutions for the Diophantine equation. ON THE DIOPHANTINE EQUATION Pni=1aix4i =Pnj=1ajyj4 5 Main Theorem 1.2. Consider the Diophantine equation: ax4 +by4 +cz4 +dt4 = au4 +bv4 +cw4 +dh4, where all the coefficients are fixed arbitrary integers. Let Y2 = X3+fX2+gX+h, be an elliptic curve in which the coefficients of f, g, and h are all functions of a, b, c, d, and four other rational param- eters A, B, D, F, yet to be found later. If the elliptic curve has positive rank, depending on the values of A, B, D, F, the Diophantine equation has infinitely many integer solutions. Proof. Let x = m + p, y = m − q, z = m − s, t = m+ r, u = m − p, v = m + q, w = m + s and h = m − r, where all variables are rational numbers. By substituting these variables in the above Diophantine equa- tion, and after some simplifications, we get: m2(ap−bq −cs+dr) = −ap3 +bq3 +cs3 −dr3. wemay assume that ap−bq−cs+dr = 1 andm2 = −ap3+bq3+cs3−dr3. Also let q = As+B and r = Ds+F, where A, B, D, F ∈ Q. By plugging q = As + B, r = Ds + F and p = bq + cs − dr + 1 into the equation a a a a m2 = −ap3 +bq3 +cs3−dr3, and after some simplifications, we obtain the elliptic curve: m2 = (−a(bA+c−dD)3+bA3 +c−dD3)s3 +(−3a(bA+c−dD)2.(bB−dF+1)+ a a a 3A2Bb − 3dD2F)s2 + (−3a(bA+c−dD).(bB−dF+1)2 + 3AB2b − 3DF2d)s + a a (−a(bB−dF+1)3 +bB3 −dF3). a Multiplying the both sides of this elliptic curve by (−a(bA+c−dD)3 +bA3 +c−dD3)2 and letting a bA+c−dD (1.3) Y = (−a( )3 +bA3 +c−dD3)m, a and bA+c−dD (1.4) X = (−a( )3 +bA3 +c−dD3)s, a we get the elliptic curve Y2 = X3 +fX2 +gX +h, where bA+c−dD bB −dF +1 f = (−3a( )2.( )+3A2Bb−3dD2F), a a 6 F. IZADIAND M. BAGHALAGHDAM bA+c−dD bB −dF +1 g = (−3a( ).( )2 +3AB2b−3DF2d) a a bA+c−dD (−a( )3 +bA3 +c−dD3), a bB −dF +1 bA+c−dD h = (−a( )3+bB3−dF3).(−a( )3+bA3+c−dD3)2. a a If the above elliptic curve has positive rank, then by calculating m, s, q, p,r,m±p, m±q,m±s, m±r, fromtherelations(1.3),(1.4)andq = As+B, r = Ds+F, p = bq+ cs− dr+ 1, after some simplifications and canceling a a a a the denominators of m±p, m±q, m±s, m±r, we may obtain infinitely many integer solutions for the Diophantine equation. The proof is complete. Now we are going to solve some couple of examples: Example 1. Let us to solve the Diophantine equation: x4 +y4 +z4 +19t4 = u4 +v4 +w4 +19h4, Suppose that A = 2, B = 0, D = 4, F = 0. Then we get the elliptic curve: Y2 = X3 −15987X2+84930390X −3878102. The rank of this elliptic curve is 1 and its generator is the point P = (X,Y) = (8832851584572306, −260518741182457285866354). 887637201025 836282950759698625 Because of this, the above elliptic curve has infinitely many rational pointsandthisgives riseinfinitely many solutions fortheDiophantine equa- tion too. By using the point P = (X,Y) and calculating m, s, q, p, r, m±p, m±q, m±s, m±r, from the above relations and after some simplifications, we get the identity: 29230818163820454535494+14901204037352206254794 +14453793985946460274344+19.(1221674372891773037209)4 = 1218050294739025947714+13111563831729222332994 +13558973883134968313444+19.(1579602414016369821569)4. By choosing the other points on the elliptic curve such as 2P, 3P, ..., we obtain infinitely many solutions for the Diophantine equation. ON THE DIOPHANTINE EQUATION Pni=1aix4i =Pnj=1ajyj4 7 Example 2. Let a = 1, b = c = d = 1000. By taking A = 2, B = 0, D = 3, F = 0, we get the elliptic curve Y2 = X3 −180002. The rank of this elliptic curve is 1 and its generator is the point P = (X,Y) = (1000,26000). By calculating the above values and after some simplification, we obtain a solution for the Diophantine equation as x4 +1000y4+1000z4 +1000t4 = u4 +1000v4 +1000w4+1000h4, 84+1000.(24)4+1000.(25)4+1000.(29)4 = 444+1000.(23)4+1000.(27)4+ 1000.(28)4. For the above solution it is interesting to see that 24 + 25 + 29 = 23 + 27 + 28. Again by choosing the other points such as 2P, 3P, ..., one can obtain infinitely many solutions for the Diophantine equation. Example 3. Sums of four biquadrates in two different ways. In this case for the coefficients of the diophantine equation we have a = b = c = d = 1. By letting A = 3, B = 0, D = 7, F = 0, we get the elliptic curve: Y2 = X3 −27X2 −2592X −2882. The rank of this elliptic curve is 1 and its generator is the point P = (X,Y) = (328,5608). By calculating the above values and after some simplifications, we obtain a solution for the Diophantine equation as x4 +y4 +z4 +t4 = u4 +v4 +w4 +h4, 2714 +2894 +3304+4944 = 2074 +3714 +4124 +4304. Also we have: 2P = (X′,Y′) = (48232180, 203244176836) and 491401 344472101 3P = (X′′,Y′′) = (61771836632004160, −3119953836085429109330528). 797318963764569 22513765616829041228253 By using these two new points, we obtain the other solutions for the Diophantine equation respectively: 3254921514+127264877874+211791773324+549899355124 = 504855520584+380845564224+296318668774+41788913034, 4636450681324307603372864+2610211775809693892830094+ 1520060974454362362053894+2840542230966963761050914 = 8 F. IZADIAND M. BAGHALAGHDAM 2686479533770914410041284+660240628256300699498514+ 429910173099030831277694+4790513378520356954382494. By choosing the other points on the elliptic curve such as 4P, 5P, ..., we obtain infinitely many solutions for the Diophantine equation. In the end, we prove the general result without without discussing any concrete examples. Main Theorem 1.3. Consider the Diophantine equation: Pn a x4 = Pn a y4, i=1 i i j=1 j j where all the coefficients are fixed arbitrary integers and n ≥ 5 . Let Y2 = X3+fX2+gX+h, be an elliptic curve in which the coefficients of f, g, and h are all functions of a , and the other rational parameters A , i i B , yet to be found later. If the elliptic curve has positive rank, depending on i the values of A , B , the Diophantine equation has infinitely many integer i i solutions. Proof. Let: x = m+p and y = m−p , (For 1 ≤ i ≤ n), where m and i i i i p ∈ Q. After substituting these variables in the Diophantine equation, we i obtain: m2(a p +···+a p ) = −a p3 −···−a p3. 1 1 n n 1 1 n n Wemay assume that (a p +···+a p ) = 1 andm2 = −a p3−···−a p3. 1 1 n n 1 1 n n Also we let p = A p +B (for 2 ≤ i ≤ n−1). i i n i By plugging p1 = −aa21p2−···− ana−11pn−1− aan1pn+ a11 = Gpn+H, where G = −a2A2−a3A3−···−an−1An−1−an, a1 and H = −a2B2−a3B3−···−an−1Bn−1+1, a1 and p = A p +B (for 2 ≤ i ≤ n−1), into the equation i i n i m2 = −a p3 −···−a p3, and after some simplifications, we obtain the 1 1 n n elliptic curve m2 = L p3 +L p2 +L p +L4, where 1 n 2 n 3 n L1 = (−a1G3 −a2A32 −a3A33 −···−an−1A3n−1 −an), ON THE DIOPHANTINE EQUATION Pni=1aix4i =Pnj=1ajyj4 9 L2 = (−3a1G2H −3a2A22B2 −3a3A23B3 −···−3an−1A2n−1Bn−1), L3 = (−3a1GH2 −3a2A2B22 −3a3A3B32 −···−3an−1An−1Bn2−1), L4 = (−a1H3 −a2B23 −a3B33 −···−an−1Bn3−1). Multiplying the both sides of this elliptic curve by L21 = (−a1G3 −a2A32 −a3A33 −···−an−1A3n−1 −an)2 and letting (1.5) Y = (−a1G3 −a2A32 −a3A33 −···−an−1A3n−1 −an)m, and (1.6) X = (−a1G3 −a2A32 −a3A33 −···−an−1A3n−1 −an)pn, we get the elliptic curve Y2 = X3 +fX2 +gX +h, where f = (−3a1G2H −3a2A22B2 −3a3A23B3 −···−3an−1A2n−1Bn−1), g = (−3a1GH2 −3a2A2B22 −3a3A3B32 −···−3an−1An−1Bn2−1) (−a1G3 −a2A32 −a3A33 −···−an−1A3n−1 −an), h = (−a1H3 −a2B23 −a3B33 −···−an−1Bn3−1) (−a1G3 −a2A32 −a3A33 −···−an−1A3n−1 −an)2. Then if the above elliptic curve has positive rank (this is done by choos- ing appropriate values for A and B .), by calculating m, p , m±p , from the i i i i relations (1.5), (1.6) and p = A p +B (for 2 ≤ i ≤ n−1), p = Gp +H, i i n i 1 n and after some simplifications and canceling the denominators of m±p , we i may obtain infinitely many integer solutions for the Diophantine equation. The proof is complete. Finally we mention that each point on the elliptic curve can be repre- sented in the form of ( r , t ), where r, s, t ∈ Z. s2 s3 Then for every generator P of the curve we have nP = (rn, tn). This gives s2 s3 n n rise to a parametric solution for each case of the Diophantine equations. Also by choosing the other appropriate values of A and B and getting i i the new elliptic curve of rank ≥ 1( and repeating the above process), we 10 F. IZADIAND M. BAGHALAGHDAM may obtain infinitely many nontrivial parametric solutions for each case of Diophantine equations. References [1] L.E.DICKSON,HistoryoftheTheoryofNumbers,Vol.II:DiophantineAnalysis, G. E. Stechertk Co., New York, 1934. [2] N. ELKIES, ”OnA4 + B4 + C4 = D4”. Mathematics of Computation. 1988. 51(184):825835. [3] L. J. LANDER and T. R. PARKIN, ”Counterexamples to Euler’s conjecture on sums of like powers, ”Bull. Amer. Math. Soc, Vol. 72,1966,p.1079. Farzali Izadi, Department of Mathematics, Faculty of Science, Urmia University, Urmia 165-57153, Iran E-mail address: [email protected] Mehdi Baghalaghdam, Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran E-mail address: [email protected]