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%I #15 Apr 27 2019 05:23:06
%S 21,960540,16418498901144294337512360,
%T 436066841882071117095002459324085167366543342937477344818646196279385305441506861017701946929489111120
%N Solutions z of the Diophantine equation x^3 + y^3 = 6z^3.
%C A. Nitaj proved Erdős's conjecture (1975) and claimed that there exist infinitely many triples of 3-powerful numbers a,b,c with (a,b) = 1, such that a+b=c, because the equation x^3 + y^3 = 6z^3 admits an infinite number of solutions, and given by the recurrence equations (see formula). It is proved that a=x(k)^3, b=y(k)^3, and c=6c^3, and are 3-powerful numbers for each k >= 1.
%D J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 348.
%D Mordell, L. J. (1969). Diophantine equations. Academic Press. ISBN 0-12-506250-8
%H P. Erdős, C. Pomerance, and A. Sárközy, <a href="http://www.math.dartmouth.edu/~carlp/PDF/58.pdf">On locally repeated values of certain arithmetic functions, II</a>, Acta Math. Hungarica 49 (1987), pp. 251-259. [<a href="http://www.renyi.hu/~p_erdos/1987-14.pdf">alternate link</a>]
%H A. Nitaj, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.108.4355">On a conjecture of Erdős on 3-powerful numbers</a>, Bull. London Math. Soc. 27 (1995), no. 4, 317-318.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Diophantine_equation">Diophantine equation</a>
%F We generate the solutions (x(k),y(k),z(k)) from the initial solution x(0) = 37, y(0)=17, z(0)=21 x(k+1) = x(k)*(x(k)^3 + 2*y(k)^3) y(k+1) = -y(k)*(2*x(k)^3 + y(k)^3) z(k+1) = z(k)*(x(k)^3 - y(k)^3).
%e 37^3 + 17^3 = 6*21^3.
%p x0:=37:y0:=17:z0:=21: for p from 1 to 5 do: x1:=x0*(x0^3+ 2*y0^3):y1:=-y0*(2*x0^3+ y0^3):z1:=z0*(x0^3- y0^3): print(z1) : x0 :=x1 :y0 :=y1 :z0 :=z1 :od :
%Y Cf. A050240, A050241, A057521, A060859, A113839, A115645, A115651, A115676, A115686, A115687, A115689, A115691, A115693, A115695, A115697, A116064, A140172.
%K nonn
%O 1,1
%A _Michel Lagneau_, Feb 20 2010
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