A161884 Smallest k such that n^4 = a_1^4+...+a_k^4 and all a_i are positive integers less than n.

16, 6, 16, 5, 6, 6, 16, 6, 5, 7, 6, 6, 6, 5, 16, 6, 6, 6, 5, 6, 7, 6, 6, 5, 6, 6, 6, 6, 5, 5, 16, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 7, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 5, 6, 16, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6, 6, 6, 6, 5, 6

Offset: 2

Author

Dmitry Kamenetsky, Jun 21 2009

Keywords

• nonn

Comments

It follows from Balasubramanian, Deshouillers, & Dress' result g(4) = 19 that a(n) <= 20. Deshouillers, Hennecart, & Landreau and Deshouillers, Kawada, & Wooley together give an effective proof that G(4) = 16, from which it can be determined by checking the 96 exceptions that a(n) <= 17. Probably a(n) <= 16. [Charles R Greathouse IV, Jul 31 2011]

References

• J.-M. Deshouillers, K. Kawada, and T. D. Wooley, "On sums of sixteen biquadrates", Mem. Soc. Math. Fr. 100 (2005), 120 pp.

Links

• Giovanni Resta, Table of n, a(n) for n = 2..250
• R. Balasubramanian, J.-M. Deshouillers, and F. Dress, Problème de Waring pour les bicarrés I, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 303 (1986), pp. 85-88.
• R. Balasubramanian, J.-M. Deshouillers, and F. Dress, Problème de Waring pour les bicarrés II, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 303 (1986), pp. 161-163.
• J.-M. Deshouillers, F. Hennecart and B. Landreau, Waring's Problem for sixteen biquadrates - numerical results, Journal de Théorie des Nombres de Bordeaux 12 (2000), pp. 411-422.
• Jean-Charles Meyrignac, Computing minimal equal sums of like powers
• Manfred Scheucher, Sage Script
• Eric W. Weisstein, Diophantine Equation 4th Powers

Crossrefs

Cf. A161882, A161883, A161885, A099591.

Programs

PARI

a(n, verbose=0, m=4)={N=n^m; for(k=3, 99, forvec(v=vector(k-1, i, [1, n\sqrtn((k+1-i)*0.99999, m)]), ispower(N-sum(i=1, k-1, v[i]^m), m, &K)&&K>0&&!if(verbose,print1("/*"n" "v"*/"))&&return(k), 1))} \\ M. F. Hasler, Dec 17 2014

Extensions

a(51)-a(63) from M. F. Hasler, Dec 17 2014

a(64)-a(86) from Giovanni Resta, Aug 17 2015