3. Bremner's method : Using Elliptic Curve

For furthur computation, we may try,

• Extend the search range of (x,y,z) from 1000 to 10000
• Fix x in the small range (under 20), and extend y and z

But these simple modification cannot conquer the solution for n=564,

x =   122 44200 50100 02877 81163 51171 17995 21361 35134 91867 (48 digits)
y = -3460 69586 84255 04865 64589 22621 88752 08971 30654 24460 (49 digits)
z = 74807 19101 53025 27837 94583 60171 46464 94820 59055 28060 (50 digits)

So, let's consult to the Bremner's paper.

Bremner's method is,

Given Diophantine equation n=(x+y+z)(1/x+1/y+1/z) is equivalent under the following birational map,

X = -4(yz+zx+xy)/z²
Y = 2(X-4n)y/z-(n-1)X

to the elliptic curve E,

E : Y² = X(X²+(n²-6n-3)X+16n)

The map from the elliptic curve to the diophantine equation is,

x,y/z = {±Y-(n-1)X}/2(4n-X)

So, searching the rational point on the elliptic curve E, transform with the above birational map, and find out the solution of n=(x+y+z)(1/x+1/y+1/z).

The computation results with Cremona's mwrank are below.

solutions for -500 ≤ n ≤ -1
solutions for 1 ≤ n ≤ 500

Now we get;

n = -100 : (x,y,z) = (4450012553, 219887106322, -663397965750)
n = 94 : (x,y,z) = (571064, -1799160, 79045681)

Parametrized solutions

n	x	y	z
-(k2-5) (k≥3)	k+1	k-1	-k(k2-1)/2
-(k2+3) (k≥3)	(k3+3k2+4k+4)/2	(k3-3k2+4k-4)/2	-k(k4+3k2+4)/4
-(k-1)(k+2) (k≥3)	1	k	-k(k+1)
v4k+8	u2k-1	u2k+1	u2k-1v2ku2k+1
u2k2	-u2k-1	u2k+1	u2k-1u2ku2k+1

u_k and v_k are Fibonacci number and Lucas number, respectively. The definitions are following;

Fibonacci number u_k : u₁=1, u₂=1, u_k+1=u_k+u_k-1
Lucas number      v_k : v₁=1, v₂=3, v_k+1=v_k+v_k-1

For two parameter diophantine equation n=(x+y)(1/x+1/y),
the solutions are only the cases

n=0, 4.

And for four parameter diophantine equation n=(x+y+z+w)(1/x+1/y+1/z+1/w),
there is a parametrized solution such that

x = m²+m+1
y = m(m+1)(n-1)
z = (m+1)(n-1)
w = -m(n-1)