1,1

The formulae 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.

The Early Mathematics of Leonhard Euler By Charles Edward Sandifer, pp. 102-103

Wolfram Mathworld, Diophantine Equation 3rd Powers

The Early Mathematics of Leonhard Euler p. 103, on Google Books

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.

1 + 5262^3 = 4528^3 + 3753^3 = 145697644729

1 + 2262756^3 = 1947250^3 + 1613673^3 = 11585457155467377217

1 + 972979926^3 = 837313192^3 + 693875529^3 = 921110304262410135315034777

(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. */

A050791, A141326

Sequence in context: A067224 A093182 A124658 this_sequence A053397 A133152 A031661

Adjacent sequences: A143933 A143934 A143935 this_sequence A143937 A143938 A143939

nonn

Lewis Mammel (l_mammel(AT)att.net), Sep 05 2008