A010330 Numbers k such that C(k,3) = C(x,3) + C(y,3) is solvable.

6, 17, 57, 60, 76, 111, 112, 121, 142, 177, 247, 296, 420, 437, 454, 476, 494, 530, 537, 552, 564, 590, 646, 690, 704, 716, 742, 749, 755, 820, 870, 910, 920, 1100, 1160, 1222, 1243, 1430, 1436, 1446, 1452, 1647, 1710, 1740, 1788, 1870, 2172, 2185, 2222, 2258

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Bombieri's Napkin Problem: Bombieri said that "the equation C(x,n)+C(y,n)=C(z,n) has no trivial solutions for n >= 3" (the joke being that he said "trivial" rather than "nontrivial"!).

Examples

			C(10,3) + C(16,3) = C(17,3) = 680, so 17 is a term.
		

References

• J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
• Van der Poorten, Notes on Fermat's Last Theorem, Wiley, p. 122.

Links

• T. D. Noe, Table of n, a(n) for n = 1..463 (n < 10^6)

Crossrefs

Cf. A034404.

Cf. A000292.

Programs

Haskell

a010330 = (+ 2) . a002311  -- Reinhard Zumkeller, May 02 2014

Mathematica

f[n_]:=Reduce[1 < x <= y < n && n(n-1)(n-2) == x(x-1)(x-2) + y(y-1)(y-2), {x,y}, Integers]; Select[Range[2260], (f[#] =!= False)&] (* Jean-François Alcover, Mar 30 2011 *)

Formula

a(n) = A002311(n) + 2. - Reinhard Zumkeller, May 02 2014

Extensions

More terms from David W. Wilson