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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.

%I A143936 #7 Feb 16 2025 08:33:08
%S A143936 5262,2262756,972979926,418379105532,179902042398942,77357459852439636
%N 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.
%C A143936 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.
%D A143936 Charles Edward Sandifer, The Early Mathematics of Leonhard Euler, 2007, pp. 102-103.
%H A143936 Wolfram Mathworld, <a href="https://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">Diophantine Equation 3rd Powers</a>
%F A143936 In Ramanujan's parametric formula:
%F A143936 (a*x+y)^3 + (b+x^2*y)^3 = (b*x+y)^3 + (a+x^2*y)^3
%F A143936 with
%F A143936 a^2 + a*b + b^2 = x*y^2,
%F A143936 we set x=3, ax+y=1 and obtain a quadratic equation for b in terms of a
%F A143936 ( Since 'a' is always negative we write it explicitly as '-a' and solve for positive 'a' )
%F A143936 The surd of the quadratic formula then becomes:
%F A143936 sqrt(321*a^2 + 216*a + 36)
%F A143936 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) )
%F A143936 a_i+1 = 215*a_i + 12*s_i + 72
%F A143936 s_i+1 = 215*s_i + 3852*a_i + 1296
%F A143936 and these yield the values of x,y and z from Ramanujan's formula.
%e A143936 1 + 5262^3 = 4528^3 + 3753^3 = 145697644729
%e A143936 1 + 2262756^3 = 1947250^3 + 1613673^3 = 11585457155467377217
%e A143936 1 + 972979926^3 = 837313192^3 + 693875529^3 = 921110304262410135315034777
%o A143936 (Other) /*
%o A143936 File: form.bc
%o A143936 Usage: bc form.bc
%o A143936 ( In UNIX shell, e.g. bash on Cygwin )
%o A143936 */
%o A143936 define a(x){ return( 321*x^2 + 216*x + 36 ); }
%o A143936 define b(x){ return( sqrt(a(x)) ); }
%o A143936 define n(z){ auto a,x; x=3; a = 215*z+12*b(z)+72 ;
%o A143936 a;b(a); return(v(a)); }
%o A143936 define v(z){ auto a,b,x,y,i,j,k,l;
%o A143936 a = z; b = ( a + b(a) )/2;
%o A143936 a = -a; x=3; y = 1-a*x;
%o A143936 i=a*x+y; j=b+x^2*y; k=b*x+y; l=a+x^2*y;
%o A143936 -a; b; i;j;k;l; i^3+j^3; k^3+l^3;
%o A143936 return ( -a ); }
%o A143936 z=144; v(z) ; z=n(z); z=n(z); z=n(z); /* ... etc. */
%Y A143936 Cf. A050791, A141326.
%K A143936 nonn
%O A143936 1,1
%A A143936 Lewis Mammel (l_mammel(AT)att.net), Sep 05 2008