We treat diophantine equation (x+2a+2b)(1/x+2/a+2/b) = n which is special case of (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n. We give a condition for positive rational solution of (x+2a+2b)(1/x+2/a+2/b) = n. Furthermore, we show that there are infinitely many parametric solutions. It seems that there are infinitely many positive integers n for (x+2a+2b)(1/x+2/a+2/b) = n.1.IntroductionDiophantine equation (x+2a+2b)(1/x+2/a+2/b) = n has a positive rational solution (x,a,b) if quartic curve (3) has a positive rational point P(U,V) satisfying the condition -7/8+1/8n-1/4sqrt(n)-1/8sqrt(-15-10n+28sqrt(n)+n^2-4n^(3/2)) < U > -7/8+1/8n-1/4sqrt(n)+1/8sqrt(-15-10n+28sqrt(n)+n^2-4n^(3/2)). a, b, and n>25 are positive integers. Proof. The final goal is to find a condition that x is a positive rational solution. a,b,n are positive integers. We assume n>25 and we use positive square root V^2( V>=0). (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n Let y=a, z=a, w=b, s=b. (x+2a+2b)(1/x+2/a+2/b) = n...........................................(1) (2ab^2+2a^2b)x^2+(9a^2b^2+4ab^3+4a^3b-a^2b^2n)x+2a^2b^3+2a^3b^2=0....(2) Discriminant for x must be perfect square number and let U=a/b, then we obtain V^2 = 16U^4+(56-8n)U^3+(81+n^2-18n)U^2+(56-8n)U+16...................(3) If quartic curve (3) has a positive rational point P(U,V), at least right hand side of equation (3) should be positive. 16U^4+(56-8n)U^3+(-18n+81+n^2)U^2+(56-8n)U+16>0......................(4) Let t=U+1/U, we obtain below 16t^2+49+(56-8n)t-18n+n^2>0..........................................(5) Hence we obtain t < -7/4+1/4n-1/2sqrt(n) or t>-7/4+1/4n+1/2sqrt(n). Since the second condition is excluded by the condition of x > 0, we use the first condition. Solving about U+1/U < -7/4+1/4n-1/2sqrt(n) for U. Thus the rational solutions exist within the range of -7/8+1/8n-1/4sqrt(n)-1/8sqrt(-15-10n+28sqrt(n)+n^2-4n^(3/2))< U <-7/8+1/8n-1/4sqrt(n)+1/8sqrt(-15-10n+28sqrt(n)+n^2-4n^(3/2)). This condition gives the positive rational solutions to U. From equation (2), we obtain x = 1/2(-9ab-4b^2-4a^2+abn+Vb^2)/(2b+2a). Since x should be positive, we obtain the condition -9ab-4b^2-4a^2+abn >0. This condition is included in above condition, so x becomes to be positive rational number. Thus diophantine equation (x+2a+2b)(1/x+2/a+2/b) = n has a positive rational solution (x,a,b). Certainly, rational solutions exist within range as follows. [n, U, lower bound, upper bound] [138, 22.55882353, 0.03725911, 26.83907083] [143, 12.60000000, 0.03573326, 27.98513638] [159, 7.129032258, 0.03158195, 31.66365795] [194, 24.86486486, 0.02515049, 39.76065537] [308, 64.44444444, 0.01504664, 66.45998896] [323, 65, 0.01428580, 69.99961382] [372, 72.50000000, 0.01225578, 81.59409346] [467, 104, 0.00959828, 104.1853103] Q.E.D.2.TheoremSince quartic equation (3) has a rational solution Q(U,V)=(0,4), this quartic equation (3) is birationally equivalent to an elliptic curve below. Y^2+(14-2n)YX+(448-64n)Y = X^3+(-4n+32)X^2-1024X+4096n-32768. U = (8X-32n+256)/Y. V = (4X^3-48X^2n+384X^2+128Xn^2-2048Xn+224nY-32n^2Y+12288X-16384n+131072)/(Y^2). The point corresponding to point Q is P(X,Y)=(4n-32, -56n+8n^2). This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points. Hence we can obtain infinitely many parametric solutions for equation (1). Case : m=2 x = 2(n-4)(-7+n) a = 84-12n b = 3n-48 Case : m=3 x = -6(5n-44)(n-4)(-7+n)(4n^2-41n+64) a = 4(-7+n)(-15n^2+81n-256+n^3)(n+2)(n-10) b = 3(5n-44)(n-4)(n-16)(n+2)(n-10)3.Parametric solutionsExample for n=110. A rational solution [110, 4446/7, 72, 72, 19, 19] leads to integer solution (4446+504+504+133+133)(1/4446+1/504+1/504+1/133+1/133)=110. 25 < n < 1000 [n x y z w s] [26, 3, 3, 3, 2, 2] [27, 2, 1, 1, 1, 1] [28, 2, 2, 2, 1, 1] [30, 3, 2, 2, 1, 1] [33, 3, 3, 3, 1, 1] [34, 4, 1, 1, 1, 1] [35, 4, 3, 3, 1, 1] [36, 2, 4, 4, 1, 1] [38, 6, 2, 2, 1, 1] [41, 3, 5, 5, 1, 1] [42, 15/4, 5, 5, 1, 1] [50, 6, 6, 6, 1, 1] [52, 10, 4, 4, 1, 1] [55, 12, 3, 3, 1, 1] [58, 6, 8, 8, 1, 1] [59, 18/5, 9, 9, 1, 1] [62, 65/3, 13, 13, 2, 2] [68, 30, 3, 3, 2, 2] [70, 12, 8, 8, 1, 1] [82, 5, 14, 14, 1, 1] [83, 18, 8, 8, 1, 1] [87, 44/3, 11, 11, 1, 1] [98, 68/9, 17, 17, 1, 1] [107, 230, 15, 15, 8, 8] [108, 14/3, 20, 20, 1, 1] [110, 4446/7, 72, 72, 19, 19] [122, 12, 21, 21, 1, 1] [123, 36, 8, 8, 1, 1] [126, 36, 9, 9, 1, 1] [131, 10, 24, 24, 1, 1] [132, 42, 6, 6, 1, 1] [138, 5251/9, 767, 767, 34, 34] [140, 70/3, 20, 20, 1, 1] [143, 765/4, 63, 63, 5, 5] [146, 315, 18, 18, 7, 7] [150, 156/7, 88, 88, 3, 3] [156, 42, 14, 14, 1, 1] [159, 14756/9, 221, 221, 31, 31] [174, 60, 5, 5, 1, 1] [175, 77/2, 21, 21, 1, 1] [187, 260, 8, 8, 5, 5] [188, 114/13, 38, 38, 1, 1] [194, 49358/33, 920, 920, 37, 37] [198, 170/7, 34, 34, 1, 1] [203, 1326/5, 247, 247, 8, 8] [206, 1105/3, 34, 34, 5, 5] [228, 161, 21, 21, 2, 2] [231, 60, 24, 24, 1, 1] [234, 1786/3, 133, 133, 8, 8] [238, 112716/13, 707, 707, 99, 99] [258, 27550/7, 75, 75, 58, 58] [267, 12, 56, 56, 1, 1] [270, 348/5, 29, 29, 1, 1] [290, 260, 216, 216, 5, 5] [299, 2010/7, 67, 67, 3, 3] [308, 2790/19, 580, 580, 9, 9] [314, 100, 24, 24, 1, 1] [322, 93, 30, 30, 1, 1] [323, 143/6, 65, 65, 1, 1] [330, 92/5, 69, 69, 1, 1] [345, 399, 35, 35, 3, 3] [348, 3531, 77, 77, 30, 30] [370, 91/4, 77, 77, 1, 1] [371, 430, 40, 40, 3, 3] [372, 203/3, 145, 145, 2, 2] [388, 154, 297, 297, 4, 4] [398, 1260, 55, 55, 8, 8] [406, 23161, 247, 247, 230, 230] [410, 5782/11, 531, 531, 8, 8] [418, 150, 24, 24, 1, 1] [425, 115, 45, 45, 1, 1] [442, 275, 75, 75, 2, 2] [459, 161/4, 91, 91, 1, 1] [467, 260/21, 104, 104, 1, 1] [468, 594/5, 54, 54, 1, 1] [470, 78/5, 104, 104, 1, 1] [548, 1241/13, 219, 219, 2, 2] [551, 475/4, 75, 75, 1, 1] [558, 6586/17, 623, 623, 6, 6] [566, 265/4, 105, 105, 1, 1] [567, 34151/4, 247, 247, 37, 37] [574, 1274/19, 245, 245, 2, 2] [606, 35, 273, 273, 2, 2] [609, 79577/3, 221, 221, 151, 151] [628, 42, 132, 132, 1, 1] [644, 437/5, 114, 114, 1, 1] [660, 2170, 21, 21, 10, 10] [698, 36, 152, 152, 1, 1] [731, 138047/12, 455, 455, 37, 37] [742, 15, 170, 170, 1, 1] [755, 60, 155, 155, 1, 1] [770, 310, 30, 30, 1, 1] [775, 266, 56, 56, 1, 1] [782, 13846/3, 115, 115, 14, 14] [812, 341/17, 186, 186, 1, 1] [858, 9238/29, 894, 894, 5, 5] [866, 1860, 88, 88, 5, 5] [867, 1092/5, 104, 104, 1, 1] [892, 770, 33, 33, 2, 2] [962, 1334/19, 435, 435, 2, 2] [972, 19710/11, 292, 292, 5, 5] [986, 22022/15, 968, 968, 7, 7]4.Results

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