1.Introduction

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.


2.Theorem

Diophantine 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.



3.Parametric solutions

Since 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)



4.Results

Example 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]