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%I #15 Apr 13 2014 11:23:46
%S 252,342513171,3169663601
%N Numbers that are expressible as C(m,5) + C(n,5) in at least 2 ways.
%C Did not find any other couple (m, n) with n <= m <= 600. - _Michel Marcus_, Feb 03 2014
%C No additional solutions with n <= m <= 25000 exist. Conjecture: no additional solutions exist. - _Jon E. Schoenfield_, Apr 13 2014
%H J. Leech, <a href="http://dx.doi.org/10.1017/S0305004100032850">Some solutions of Diophantine equations</a>, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
%e C(9,5) + C(9,5) = C(10,5) + C(4,5).
%e C(118,5) + C(117,5) = C(133,5) + C(78,5).
%e C(197,5) + C(160,5) = C(209,5) + C(53,5).
%o (PARI) findother(s, ma) = {forstep (i = ma-1, 4, -1, my(ci = binomial(i, 5)); if (2*ci < s, return([])); forstep (j = i, 4, -1, my(cj = binomial(j, 5)); if (ci+cj == s, return ([i, j])););); return ([]);}
%o lista(ma, mb) = {for (m = ma, mb, my(cm = binomial(m, 5)); forstep (n = m, 4, -1, my(cn = binomial(n, 5)); my(s = cm+cn); vsol = findother(s, m); if (#vsol, print1(s, ", "); /*vsol = concat(vsol, [m,n]); print(vsol);*/);););} \\ _Michel Marcus_, Feb 03 2014
%Y Cf. A000389.
%K nonn,more,bref
%O 1,1
%A _N. J. A. Sloane_
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