A050791 Consider the Diophantine equation x^3 + y^3 = z^3 + 1 (1 < x < y < z) or 'Fermat near misses'. Sequence gives values of z in monotonic increasing order.

12, 103, 150, 249, 495, 738, 1544, 1852, 1988, 2316, 4184, 5262, 5640, 8657, 9791, 9953, 11682, 14258, 21279, 21630, 31615, 36620, 36888, 38599, 38823, 40362, 41485, 47584, 57978, 59076, 63086, 73967, 79273, 83711, 83802, 86166, 90030

Offset

1,1

Comments

Numbers n such that n^3+1 is expressible as the sum of two nonzero cubes (both greater than 1).

Values of z associated with A050794.

Sequence is infinite. One subsequence is (from x = 1 + 9 m^3, y = 9 m^4, z = 3*m*(3*m^3 + 1), x^3 + y^3 = z^3 + 1): z(m) = 3*m*(3*m^3 + 1) = {12, 150, 738, 2316, 5640, 11682, 21630, 36888, 59076, 90030, ...} = a (1, 3, 6, 10, 13, 17, 20, 23, 30, 37, ...). - Zak Seidov, Sep 16 2013

Numbers n such that n^3+1 is a member of A001235. - Altug Alkan, May 09 2016

References

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Links

Lewis Mammel, Table of n, a(n) for n = 1..368

Noam Elkies, Rational points near curves and small nonzero |x^3-y^2| via lattice reduction, arXiv:math/0005139 [math.NT], 2000.

S. Ramanujan, Question 681, J. Ind. Math. Soc.

Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers

Example

12 is a term because 10^3 + 9^3 = 12^3 + 1 (= 1729).
2316 is in the sequence because 577^3 + 2304^3 = 2316^3 + 1.

Mathematica

r[z_] := Reduce[ 1 < x < y < z && x^3 + y^3 == z^3 + 1, {x, y}, Integers]; z = 4; A050791 = {}; While[z < 10^4, If[r[z] =!= False, Print[z]; AppendTo[A050791, z]]; z++]; A050791 (* Jean-François Alcover, Dec 27 2011 *)

Prog

(PARI) is(n)=if(n<2, return(0)); my(c3=n^3); for(a=2, sqrtnint(c3-5, 3), if(ispower(c3-1-a^3, 3), return(1))); 0 \\ Charles R Greathouse IV, Oct 26 2014
(PARI) T=thueinit('x^3+1); is(n)=n>8&&#select(v->min(v[1], v[2])>1, thue(T, n^3+1))>0 \\ Charles R Greathouse IV, Oct 26 2014

Crossrefs

Cf. A050792, A050793, A050794, A050787, A229383.

Sequence in context: A133384 A052067 A307821 * A005771 A016228 A016276

Adjacent sequences: A050788 A050789 A050790 * A050792 A050793 A050794

Keyword

nonn,nice

Author

Patrick De Geest, Sep 15 1999

Extensions

More terms from Michel ten Voorde

Extended through 47584 by Jud McCranie, Dec 25 2000

More terms from Don Reble, Nov 29 2001

Edited by N. J. A. Sloane, May 08 2007

Status

approved