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A050787 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (0 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of z.

%I A050787 #55 Feb 16 2025 08:32:40
%S A050787 9,144,172,505,577,729,904,1010,1210,2304,3097,3753,5625,6081,6756,
%T A050787 8703,11664,12884,16849,18649,21609,24987,29737,36864,37513,38134,
%U A050787 38239,41545,49461,51762,59049,66465,68010,69709,71852,73627,78529
%N A050787 Consider the Diophantine equation x^3 + y^3 = z^3 - 1 (0 < x < y < z) or 'Fermat near misses'. Arrange solutions by increasing values of z. Sequence gives values of z.
%C A050787 n^3 - 1 is expressible as the sum of two distinct positive cubes. [corrected by _Altug Alkan_, Apr 11 2016]
%C A050787 The subsequence of primes in the sequence begins: 577, 38239, 69709. - _Jonathan Vos Post_, May 13 2010
%C A050787 Sequence is infinite. One subsequence is b (m) = 9 m^4 = {9, 144, 729, 2304, 5625, 11664, 21609, 36864, 59049, ...} = a (1, 2, 6, 10, 13, 17, 21, 24, 31, ...). - _Zak Seidov_, Sep 16 2013
%D A050787 Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
%D A050787 David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.
%H A050787 Jean-François Alcover and Charles R Greathouse IV, <a href="/A050787/b050787.txt">Table of n, a(n) for n = 1..104</a> (first 60 terms from Alcover)
%H A050787 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">Diophantine Equation - 3rd Powers</a>
%e A050787 2304 is in the sequence because 575^3 + 2292^3 = 2304^3 - 1.
%p A050787 N:= 10000: # to get all entries <= N
%p A050787 P:= proc(r)
%p A050787   local dcands, xs;
%p A050787   dcands:= select(d -> issqr(-3*d^4+12*d*r), numtheory[divisors](r));
%p A050787   xs:= map(d -> [solve(d^2-3*d*x+3*x^2-r/d,x)], dcands);
%p A050787   select(p -> p[1]<>p[2], select(type,xs,list(posint)));
%p A050787 end proc:
%p A050787 select(z -> nops(P(z^3-1))>0, [$1..N]); # _Robert Israel_, Jun 09 2014
%t A050787 r[z_] := Reduce[1 < x < y < z && x^3 + y^3 == z^3 - 1, {x, y}, Integers]; Reap[z = 4; While[z < 10^5, rz = r[z]; If[rz =!= False, Print[xyz = {x, y, z} /. ToRules[rz]]; Sow[xyz[[3]]]]; z++]][[2, 1]] (* _Jean-François Alcover_, Dec 27 2011, updated Feb 11 2014 *)
%o A050787 (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 A050787 (PARI) T=thueinit('x^3+1);
%o A050787 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 A050787 Cf. A050788, A050789, A050790, A050791.
%K A050787 nonn,nice
%O A050787 1,1
%A A050787 _Patrick De Geest_, Sep 15 1999
%E A050787 More terms from _Jud McCranie_, Dec 25 2000
%E A050787 More terms from _Don Reble_, Nov 29 2001
%E A050787 Definition corrected by _Robert Israel_, Jun 09 2014