Waring's problem (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Waring's problem" in English language version.

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Hilbert, David (1909). "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)". Mathematische Annalen (in German). 67 (3): 281–300. doi:10.1007/bf01450405. MR 1511530. S2CID 179177986.
Balasubramanian, Ramachandran; Deshouillers, Jean-Marc; Dress, François (1986). "Problème de Waring pour les bicarrés. I. Schéma de la solution" [Waring's problem for biquadrates. I. Sketch of the solution]. Comptes Rendus de l'Académie des Sciences, Série I (in French). 303 (4): 85–88. MR 0853592.
Balasubramanian, Ramachandran; Deshouillers, Jean-Marc; Dress, François (1986). "Problème de Waring pour les bicarrés. II. Résultats auxiliaires pour le théorème asymptotique" [Waring's problem for biquadrates. II. Auxiliary results for the asymptotic theorem]. Comptes Rendus de l'Académie des Sciences, Série I (in French). 303 (5): 161–163. MR 0854724.
Pillai, S. S. (1940). "On Waring's problem g(6) = 73". Proc. Indian Acad. Sci. 12: 30–40. doi:10.1007/BF03170721. MR 0002993. S2CID 185097940.
Niven, Ivan M. (1944). "An unsolved case of the Waring problem". American Journal of Mathematics. 66 (1). The Johns Hopkins University Press: 137–143. doi:10.2307/2371901. JSTOR 2371901. MR 0009386.
Mahler, Kurt (1957). "On the fractional parts of the powers of a rational number II". Mathematika. 4 (2): 122–124. doi:10.1112/s0025579300001170. MR 0093509.
Kubina, Jeffrey M.; Wunderlich, Marvin C. (1990). "Extending Waring's conjecture to 471,600,000". Math. Comp. 55 (192): 815–820. Bibcode:1990MaCom..55..815K. doi:10.2307/2008448. JSTOR 2008448. MR 1035936.
Vaughan, R. C.; Wooley, Trevor (2002). "Waring's Problem: A Survey". In Bennet, Michael A.; Berndt, Bruce C.; Boston, Nigel; Diamond, Harold G.; Hildebrand, Adolf J.; Philipp, Walter (eds.). Number Theory for the Millennium. Vol. III. Natick, MA: A. K. Peters. pp. 301–340. ISBN 978-1-56881-152-9. MR 1956283.

archive.org

L. Euler, "Opera posthuma" (1), 203–204 (1862).

doi.org

Hilbert, David (1909). "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)". Mathematische Annalen (in German). 67 (3): 281–300. doi:10.1007/bf01450405. MR 1511530. S2CID 179177986.
Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen (in German). 66 (1): 95–101. doi:10.1007/BF01450913. S2CID 121386035.
Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen (in German). 72 (3): 387–399. doi:10.1007/BF01456723. S2CID 120101223.
Pillai, S. S. (1940). "On Waring's problem g(6) = 73". Proc. Indian Acad. Sci. 12: 30–40. doi:10.1007/BF03170721. MR 0002993. S2CID 185097940.
Niven, Ivan M. (1944). "An unsolved case of the Waring problem". American Journal of Mathematics. 66 (1). The Johns Hopkins University Press: 137–143. doi:10.2307/2371901. JSTOR 2371901. MR 0009386.
Mahler, Kurt (1957). "On the fractional parts of the powers of a rational number II". Mathematika. 4 (2): 122–124. doi:10.1112/s0025579300001170. MR 0093509.
Kubina, Jeffrey M.; Wunderlich, Marvin C. (1990). "Extending Waring's conjecture to 471,600,000". Math. Comp. 55 (192): 815–820. Bibcode:1990MaCom..55..815K. doi:10.2307/2008448. JSTOR 2008448. MR 1035936.
Hardy, G. H.; Littlewood, J. E. (1922). "Some problems of Partitio Numerorum: IV. The singular series in Waring's Problem and the value of the number G(k)". Mathematische Zeitschrift. 12 (1): 161–188. doi:10.1007/BF01482074. ISSN 0025-5874.
Davenport, H. (1939). "On Waring's Problem for Fourth Powers". Annals of Mathematics. 40 (4): 731–747. Bibcode:1939AnMat..40..731D. doi:10.2307/1968889. JSTOR 1968889.
Vaughan, R. C. (1986). "On Waring's Problem for Smaller Exponents". Proceedings of the London Mathematical Society. s3-52 (3): 445–463. doi:10.1112/plms/s3-52.3.445.
Vaughan, R. C. (1989). "A new iterative method in Waring's problem". Acta Mathematica. 162: 1–71. doi:10.1007/BF02392834. ISSN 0001-5962.
Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard; I. Gusti Putu Purnaba, Appendix by (2000). "7373170279850". Mathematics of Computation. 69 (229): 421–439. doi:10.1090/S0025-5718-99-01116-3.
Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard (2000). "Waring's Problem for sixteen biquadrates – numerical results". Journal de théorie des nombres de Bordeaux. 12 (2): 411–422. doi:10.5802/jtnb.287.
Deshouillers, Jean-Marc; Kawada, Koichi; Wooley, Trevor D. (2005). "On Sums of Sixteen Biquadrates". Mémoires de la Société Mathématique de France. 1: 1–120. doi:10.24033/msmf.413. ISSN 0249-633X.
Karatsuba, A. A. (1985). "On the function G(n) in Waring's problem". Izv. Akad. Nauk SSSR Ser. Mat. 27 (4): 935–947. Bibcode:1986IzMat..27..239K. doi:10.1070/IM1986v027n02ABEH001176.

ethz.ch

math.ethz.ch

Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard (2000). "Waring's Problem for sixteen biquadrates – numerical results". Journal de théorie des nombres de Bordeaux. 12 (2): 411–422. doi:10.5802/jtnb.287.

harvard.edu

ui.adsabs.harvard.edu

Kubina, Jeffrey M.; Wunderlich, Marvin C. (1990). "Extending Waring's conjecture to 471,600,000". Math. Comp. 55 (192): 815–820. Bibcode:1990MaCom..55..815K. doi:10.2307/2008448. JSTOR 2008448. MR 1035936.
Davenport, H. (1939). "On Waring's Problem for Fourth Powers". Annals of Mathematics. 40 (4): 731–747. Bibcode:1939AnMat..40..731D. doi:10.2307/1968889. JSTOR 1968889.
Karatsuba, A. A. (1985). "On the function G(n) in Waring's problem". Izv. Akad. Nauk SSSR Ser. Mat. 27 (4): 935–947. Bibcode:1986IzMat..27..239K. doi:10.1070/IM1986v027n02ABEH001176.

jstor.org

Niven, Ivan M. (1944). "An unsolved case of the Waring problem". American Journal of Mathematics. 66 (1). The Johns Hopkins University Press: 137–143. doi:10.2307/2371901. JSTOR 2371901. MR 0009386.
Kubina, Jeffrey M.; Wunderlich, Marvin C. (1990). "Extending Waring's conjecture to 471,600,000". Math. Comp. 55 (192): 815–820. Bibcode:1990MaCom..55..815K. doi:10.2307/2008448. JSTOR 2008448. MR 1035936.
Davenport, H. (1939). "On Waring's Problem for Fourth Powers". Annals of Mathematics. 40 (4): 731–747. Bibcode:1939AnMat..40..731D. doi:10.2307/1968889. JSTOR 1968889.

mathnet.ru

Vinogradov, I. M. (1959). "On an upper bound for $G(n)$". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 23 (5): 637–642.

semanticscholar.org

api.semanticscholar.org

Hilbert, David (1909). "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)". Mathematische Annalen (in German). 67 (3): 281–300. doi:10.1007/bf01450405. MR 1511530. S2CID 179177986.
Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen (in German). 66 (1): 95–101. doi:10.1007/BF01450913. S2CID 121386035.
Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen (in German). 72 (3): 387–399. doi:10.1007/BF01456723. S2CID 120101223.
Pillai, S. S. (1940). "On Waring's problem g(6) = 73". Proc. Indian Acad. Sci. 12: 30–40. doi:10.1007/BF03170721. MR 0002993. S2CID 185097940.

uni-goettingen.de

dz-srv1.sub.uni-goettingen.de

Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen (in German). 66 (1): 95–101. doi:10.1007/BF01450913. S2CID 121386035.
Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen (in German). 72 (3): 387–399. doi:10.1007/BF01456723. S2CID 120101223.

worldcat.org

search.worldcat.org

Hardy, G. H.; Littlewood, J. E. (1922). "Some problems of Partitio Numerorum: IV. The singular series in Waring's Problem and the value of the number G(k)". Mathematische Zeitschrift. 12 (1): 161–188. doi:10.1007/BF01482074. ISSN 0025-5874.
Vaughan, R. C. (1989). "A new iterative method in Waring's problem". Acta Mathematica. 162: 1–71. doi:10.1007/BF02392834. ISSN 0001-5962.
Deshouillers, Jean-Marc; Kawada, Koichi; Wooley, Trevor D. (2005). "On Sums of Sixteen Biquadrates". Mémoires de la Société Mathématique de France. 1: 1–120. doi:10.24033/msmf.413. ISSN 0249-633X.

zbmath.org

Nathanson (1996, p. 71). Nathanson, Melvyn B. (1996). Additive Number Theory: The Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. ISBN 0-387-94656-X. Zbl 0859.11002. Has proofs of Lagrange's theorem, the polygonal number theorem, Hilbert's proof of Waring's conjecture and the Hardy–Littlewood proof of the asymptotic formula for the number of ways to represent N as the sum of s kth powers.
Vaughan, R. C. (1997). The Hardy–Littlewood method. Cambridge Tracts in Mathematics. Vol. 125 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-57347-5. Zbl 0868.11046.

zenodo.org

Hilbert, David (1909). "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)". Mathematische Annalen (in German). 67 (3): 281–300. doi:10.1007/bf01450405. MR 1511530. S2CID 179177986.