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

Author

Patrick De Geest, Sep 15 1999

Keywords

• nonn
• nice

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

Examples

			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.
		

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

Crossrefs

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

Programs

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 *)

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

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