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A190580 Value of y in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

%I A190580 #31 Aug 05 2025 10:02:43
%S A190580 1,17,-1,1,19,2,397,-1,-2,1,17299,-1,1,107,-65,523,-359,2,-3,-71,1,-2,
%T A190580 -11267,62641,-1819,-14653,-4,7,-1,1,1208,-472663,-10441,17,-126,
%U A190580 -11951,53,-4,323,-2404889,5,-907929611,36,-431,3,-3547,-15616184186396177,-5,-3,-349,3527,-140131,17,-71,-901,-2741617,-2,10183412861,-1,1,-6,33728183
%N A190580 Value of y in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).
%C A190580 A190356(n)^3 + a(n)^3 = A020898(n)*z^3. Unknown z corresponds to sequence A190581.
%C A190580 The 4 sequences A020898 [i.e. n], A190356 [i.e. x], A190580 [i.e. y] and A190581 [i.e. z] satisfy the equation A190356^3 + A190580^3 = A020898 * A190581^3
%H A190580 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a Generalized Fermat-Wiles Equation</a> [broken link]
%H A190580 Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On a Generalized Fermat-Wiles Equation</a> [From the Wayback Machine]
%H A190580 Nakao Hisayasu, <a href="https://www.kaynet.or.jp/~kay/misc/nna2.html">Rational Points on Elliptic Curves: x^3+y^3=n</a> (nna2.html up to nna22.html)
%H A190580 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/x3y3s.txt">Solutions of Diophantine equation x^3+y^3=A.z^3 ...</a>
%e A190580 a(18) = 2  because  A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.
%t A190580 Table[ y /. First[ Solve[ A190356[[n]]^3 + y^3 == A020898[[n]] * A190581[[n]]^3 ] ], {n, 62}] (* _Jean-François Alcover_, Jan 04 2012 *)
%Y A190580 Cf. A020898, A060838, A190356, A190581.
%K A190580 sign
%O A190580 1,2
%A A190580 _Jean-François Alcover_, May 13 2011