OFFSET
1,1
COMMENTS
The formulas give an approximately geometric progression of values, z, such that 1 + z^3 = x^3 + y^3, along with the values for x and y. Iteration yields large values of x,y and z presumably unobtainable by exhaustive search.
REFERENCES
Charles Edward Sandifer, The Early Mathematics of Leonhard Euler, 2007, pp. 102-103.
LINKS
Wolfram Mathworld, Diophantine Equation 3rd Powers
FORMULA
In Ramanujan's parametric formula:
(a*x+y)^3 + (b+x^2*y)^3 = (b*x+y)^3 + (a+x^2*y)^3
with
a^2 + a*b + b^2 = x*y^2,
we set x=3, ax+y=1 and obtain a quadratic equation for b in terms of a
( Since 'a' is always negative we write it explicitly as '-a' and solve for positive 'a' )
The surd of the quadratic formula then becomes:
sqrt(321*a^2 + 216*a + 36)
and we require that this be an integer. After finding an initial value of 'a' which satisfies this condition by inspection of the sequence A050791, we use Euler's method to find the bilinear recursion: ( with s_i == sqrt(321*a_i^2 + 216*a_i + 36) )
a_i+1 = 215*a_i + 12*s_i + 72
s_i+1 = 215*s_i + 3852*a_i + 1296
and these yield the values of x,y and z from Ramanujan's formula.
EXAMPLE
1 + 5262^3 = 4528^3 + 3753^3 = 145697644729
1 + 2262756^3 = 1947250^3 + 1613673^3 = 11585457155467377217
1 + 972979926^3 = 837313192^3 + 693875529^3 = 921110304262410135315034777
PROG
(Other) /*
File: form.bc
Usage: bc form.bc
( In UNIX shell, e.g. bash on Cygwin )
*/
define a(x){ return( 321*x^2 + 216*x + 36 ); }
define b(x){ return( sqrt(a(x)) ); }
define n(z){ auto a, x; x=3; a = 215*z+12*b(z)+72 ;
a; b(a); return(v(a)); }
define v(z){ auto a, b, x, y, i, j, k, l;
a = z; b = ( a + b(a) )/2;
a = -a; x=3; y = 1-a*x;
i=a*x+y; j=b+x^2*y; k=b*x+y; l=a+x^2*y;
-a; b; i; j; k; l; i^3+j^3; k^3+l^3;
return ( -a ); }
z=144; v(z) ; z=n(z); z=n(z); z=n(z); /* ... etc. */
CROSSREFS
KEYWORD
nonn
AUTHOR
Lewis Mammel (l_mammel(AT)att.net), Sep 05 2008
STATUS
approved