A143936 Subsequence of A050791, "Fermat near misses", generated by iteration of a linear form derived from Ramanujan's parametric formula for equal sums of two pairs of cubes.

5262, 2262756, 972979926, 418379105532, 179902042398942, 77357459852439636

Offset: 1

Author

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

Keywords

• nonn

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.

Examples

			1 + 5262^3 = 4528^3 + 3753^3 = 145697644729
1 + 2262756^3 = 1947250^3 + 1613673^3 = 11585457155467377217
1 + 972979926^3 = 837313192^3 + 693875529^3 = 921110304262410135315034777
		

References

• Charles Edward Sandifer, The Early Mathematics of Leonhard Euler, 2007, pp. 102-103.

Links

• Wolfram Mathworld, Diophantine Equation 3rd Powers

Crossrefs

Cf. A050791, A141326.

Programs

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. */

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.