login
This site is supported by donations to The OEIS Foundation.
Logo

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 0
5262, 2262756, 972979926, 418379105532, 179902042398942, 77357459852439636 (list; graph; refs; listen; history; internal format)
OFFSET

1,1

COMMENTS

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.

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

Cf. A050791, A141326.

Sequence in context: A124658 A206165 A206379 * A204281 A178027 A053397

Adjacent sequences:  A143933 A143934 A143935 * A143937 A143938 A143939

KEYWORD

nonn,changed

AUTHOR

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified February 14 17:06 EST 2012. Contains 205644 sequences.