A050791 - OEIS

%I #94 Nov 30 2019 09:05:20

%S 12,103,150,249,495,738,1544,1852,1988,2316,4184,5262,5640,8657,9791,

%T 9953,11682,14258,21279,21630,31615,36620,36888,38599,38823,40362,

%U 41485,47584,57978,59076,63086,73967,79273,83711,83802,86166,90030

%N 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.

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

%C Values of z associated with A050794.

%C 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

%C Numbers n such that n^3+1 is a member of A001235. - _Altug Alkan_, May 09 2016

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

%H Lewis Mammel, <a href="/A050791/b050791.txt">Table of n, a(n) for n = 1..368</a>

%H Noam Elkies, <a href="http://arxiv.org/abs/math/0005139">Rational points near curves and small nonzero |x^3-y^2| via lattice reduction</a>, arXiv:math/0005139 [math.NT], 2000.

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q681.htm">Question 681</a>, J. Ind. Math. Soc.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">Diophantine Equation - 3rd Powers</a>

%e 12 is a term because 10^3 + 9^3 = 12^3 + 1 (= 1729).

%e 2316 is in the sequence because 577^3 + 2304^3 = 2316^3 + 1.

%t 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 *)

%o (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

%o (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

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

%K nonn,nice

%O 1,1

%A _Patrick De Geest_, Sep 15 1999

%E More terms from _Michel ten Voorde_

%E Extended through 47584 by _Jud McCranie_, Dec 25 2000

%E More terms from _Don Reble_, Nov 29 2001

%E Edited by _N. J. A. Sloane_, May 08 2007