A287286 a(n) = smallest integer s such that every element of the ring of integers mod t for any t can be written as a sum of s n-th powers.

1, 4, 4, 15, 5, 9, 4, 32, 13, 12, 11, 16, 6, 14, 15, 64, 6, 27, 4, 25, 24, 23, 23, 32, 10, 26, 40, 29, 29, 31, 5, 128, 33, 10, 35, 37, 9, 9, 39, 41, 41, 49, 12, 44, 15, 47, 10, 64, 13, 62, 51, 53, 53, 81, 60, 56, 14, 59, 5, 61, 11, 12, 63, 256, 65, 67, 12, 68, 69, 71, 6, 73, 16, 74, 75, 16, 14, 84

Offset

1,2

Comments

One needs only check a finite number of values (depending on the power).

See Small's paper in references for precise quantitive information.

a(2) <= 4 follows from Lagrange's four squares theorem.

Differs from A040004 only at k=4. - Andrey Zabolotskiy, Jun 03 2017

References

G. H. Hardy and J. E. Littlewood, Some Problems of "Partitio Numerorum" (VIII): The Number Gamma(k) in Waring's Problem, Proc London Math Soc. 28 (1928), 518--542. [G. H. Hardy, Collected Papers. Vols. 1-, Oxford Univ. Press, 1966-; see vol. 1, pp. 406-530.]

Wladyslaw Narkiewicz, Rational Number Theory in the 20th Century: From PNT to FLT, Springer Science & Business Media, 2011, pages 154-155.

Links

Table of n, a(n) for n=1..78.

H. Sekigawa and K. Koyama, Nonexistence Conditions of a Solution for the congruence x_1^k + ... + x_s^k = N (mod p^n), Math. Comp. 68 (1999), 1283--1297.

C. Small, Waring's problem mod n, Amer. Math. Monthly 84 (1977), no. 1, 12--25.

R. C. Vaughan and T. D. Wooley, Waring’s problem: a survey, Number Theory for the Millennium, III (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 301-340.

Example

a(3) <= 4 states that every element of every ring of integers mod m can be written as a sum of 4 (or fewer) cubes. a(3) >= 4, since in Z/9Z, the cubes are {0,1,8} so that 4 is not the sum of any three cubes in Z/9Z. Hence a(3) = 4.

Crossrefs

Cf. A079611, A174406, A040004.

Sequence in context: A331609 A174406 A270844 * A271546 A325655 A117187

Adjacent sequences: A287283 A287284 A287285 * A287287 A287288 A287289

Keyword

nonn

Author

David Covert, May 22 2017

Extensions

Edited by Andrey Zabolotskiy, Jun 10 2017

Status

approved