Computers as Fresh Mathematical Reality. II. Waring Problem
Abstract
In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th powers, n=x_1^k+ldots+x_s^k. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. The XIX century problem is still unsolved even or cubes. However, even the solution of the original Waring problem was [almost] finalised only in 1984, with heavy use of computers. In the present paper we document the history of this classical problem itself and its solution, as also discuss possibilities of using this and surrounding material in education, and some further related aspects.
References
M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Moscow: Binom, Laboratoriya znanii, 2015 (in Russian).
W. Borho, “Befreundete Zahlen. Ein zweitausend Jahre altes Thema der elementaren Zahlentheorie,” in Lebendige Numbers, Moscow: Mir, 1985, pp.11–41 (in Russian).
B. M. Bredikhin and T. I. Grishina, “An elementary estimate of G(n) in Waring’s problem,” Matematicheskie Zametki, vol. 24, no. 1, pp. 7–18, 1978 (in Russian).
A. Bufetov, A. Ya. Kanel’, “The new elementary solution of the Waring problem,” Fundam. Prikl. Mat., vol. 3, no. 4, 1239–1252, 1997 (in Russian).
N. A. Vavilov, “Computers as novel mathematical reality. I. Personal Account,” Computer tools in education, no. 2, pp. 5–26, 2020 (in Russian); doi: 10.32603/2071-2340-2020-2-5–26
N. A. Vavilov, Konkretnaya teoriya grupp [Concrete group theory], vol. 1, 2020 (in Russian). [Online]. Available: http://www.add3d.ru/wp-content/uploads/2019/10/Vavilov-Groups.pdf
N. A. Vavilov and V. G. Khalin, Zadachi po kursu Matematika i Komp’yuter. Vyp. 1. Arifmetika i teoriya chisel [Exercises for the course "Mathematics and Computers". Issue 1, Arithmetics and Number Theory], St. Petersburg, Russia: OTsEiM, 2005 (in Russian).
N. A. Vavilov and V. G. Khalin, Dopolnitel’nye zadachi po kursu Matematika i Komp’yuter [Supplementary exercises for the course "Mathematics and Computers"], St. Petersburg, Russia: OTsEiM, 2006 (in Russian).
N. A. Vavilov, V. G. Khalin, and A. V. Yurkov, Mathematica dlya nematematika [Mathematica for nonmathematician], [In Print], St. Petersburg, Russia, 2020 (in Russian).
B. A. Venkov, Issledovaniya po teorii chisel. Izbrannye trudy, [Studies in number theory. Selected works], [With a supplement by B. B. Venkov and A. V. Malyshev], Leningrad, USSR: Nauka, 1981 (in Russian).
I. M. Vinogradov, “On Waring’s theorem,” Izv. Akad. Nauk SSSR, Otd. Fiz.-Mat. Nauk, no. 4, pp. 393–400, 1928 (in Russian).
I. M. Vinogradov, “A new solution of Waring’s problem,” Dokl. Akad. Nauk SSSR, vol. 2, no. 6, pp. 337–341, 1934 (in Russian).
I. M. Vinogradov, “A new evaluation of G(n) in Waring’s problem,” Dokl. Akad. Nauk SSSR, no. 4, no. 5–6, 249– 53, 1934 (in Russian).
I. M. Vinogradov, “On the upper bound of G(n) in Waring’s problem,“ Izv. Akad. Nauk SSSR, Otd. Mat. Estestv. Nauk, no. 10, pp. 1455–1469, 1934, (in Russian).
I. M. Vinogradov, “A new variant of the proof of Waring’s theorem,” Trudy Matem. Instituta im. V. A. Steklova,
vol. 9, 5–15, 1935 (in Russian).
I. M. Vinogradov, Izbrannye trudy [Selected works], Moscow: Izdat. Akad. Nauk SSSR, 1952 (in Russian).
M. A. Gelbcke, “A propos de g (k) dans le probl´eme de Waring,” Izv. Akad. Nauk SSSR, Otd. Fiz.-Mat. Nauk, no. 5, pp. 631–640, 1933 (in Russian).
A. O. Gel’fond and Yu. V. Linnik, Elementary methods in analytic number theory, L. J. Mordell ed., Chicago, Ill.: Rand McNally & Co., 1965.
A. I. Generalov, “A combinatorial proof of Euler—Fermat’s theorem on presentation of primes of the form p = 8k +3 by the quadratic form x^2 +2y^2,” Zap. Nauchn. Sem. POMI, vol. 330, pp. 155–157, 2006 (in Russian).
Diophantus of Alexandria, Arifmetika i Kniga o mnogougol’nykh chislakh [Arithmetica and the Book of Polygonal Numbers], I. G. Bashmkova, ed., Moscow: Nauka, 1974 (in Russian).
A. Dubickas, “A lower bound for the quantity ||(3/2)^k ||,” Uspekhi Mat. Nauk, vol. 45, no. 4, pp. 153–154, 1990 (in Russian).
O. A. Ivanov, Izbrannye glavy elementarnoi matematiki [], St. Petersburg, Russia: Izd-vo SPbGU, 1995 (in Russian).
A. A. Karatsuba, The Hilbert—Kamke problem in analytic number theory. , Moscow: Nauka, 1983 (in Russian).
A. A. Karatsuba, “The Hilbert—Kamke problem in analytic number theory,” Mat. Zametki, vol. 41, no. 2, pp. 272–284, 1987 (in Russian).
J. H. Conway and D. A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry, St. Petersburg, Russia: OTsEiM, 2009 (in Russian).
U. V. Linnik, “An elementary solution of the problem of Waring by Schnirelman’s method,” Rec. Math. [Mat.
Sbornik], vol. 12, no. 2, pp. 225–230, 1943 (in Russian).
Yu. V. Linnik, “Quaternions and Cayley numbers; some applications of the arithmetic of quaternions,” Uspekhi
Mat. Nauk, vol. 4, no. 5, pp. 49–98, 1949 (in Russian).
Yu. V. Linnik and A. V. Malyshev, “Application of the arithmetic of quaternions to the theory of ternary quadratic forms and to the decomposition of numbers into cubes”, Uspekhi Mat. Nauk, vol. 8, no. 5, pp. 3–71, 1953 (in Russian).
Yu. V. Nesterenko, “On Waring’s problem (elementary methods),” Zap. Nauchn. Sem. POMI, vol. 322, pp. 149–175, 2005 (in Russian).
A. Ya. Khinchin, Three Pearls of Number Theory, Moscow: Nauka, 1979 (in Russian).
L. Hua, Die Abschatzung von Exponentialsummen und ihre Anwendg in der Zahlentheorie ¨ , Moscow: Mir, 1964 (in Russian).
L. G. Shnirel’man, “On the additive properties of numbers,” Uspekhi Mat. Nauk, no. 6, pp. 9–25, 1939 (in Russian).
L. G. Shnirel’man, “On the additive properties of numbers,” Uspekhi Mat. Nauk, no. 7, pp. 7–46, 1940 (in Russian).
R. G Archibald, “Waring’s problem: squares,” Scripta Math., vol. 7, pp. 33–48, 1940; doi: 10.2307/3606948
F. C. Auluck, “On Waring’s problem for biquadrates,” in Proc. Indian Acad. Sci.- Section A, vol. 11, no. 5, 1940, 437–450; doi:10.1007/BF03046010
P. Bachmann, Niedere Zahlentheorie. Zweiter Teil: Additive Zahlentheorie, Leipzig, Germany: B. G. Teubner, 1910.
W. S. Baer, “Udie Zerlegung der ganzen Zahlen in sieben Kuben, ¨ ” Math. Ann., vol. 74, no. 4, 511–514, 1913; doi: 10.1007/BF01456910
W. S. Baer, Beitrage zum Waringschen Problem ¨ , G¨ottingen, Germany: Dietrich, 1913.
R. Balasubramanian, “On Waring’s problem: g (4) ≤ 21,” Hardy—Ramanujan J., no. 2, pp. 1–31, 1979.
R. Balasubramanian, “On Waring’s problem: g (4) ≤ 20,” Hardy—Ramanujan J., no. 8, pp. 1–40, 1985.
R. Balasubramanian, “Highly composite,” R. Bhatia et al., eds., in Proc. of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010, vol. I, River Edge, NJ: World Scientific, New Delhi: Hindustan Book Agency, 2011, pp. 176–209.
R. Balasubramanian, J.-M. Deshouillers, and F. Dress, “Probl/’eme de Waring pour les bicarr´es,” C. R. Acad. Sci. Paris S´er. I Math, vol. 303, no. 4, pp. 85–88, 1986 (in French).
P. M. Batchelder, “Waring’s Problem,” Amer. Math. Monthly, vol. 43 , no. 1, pp. 21–27, 1936.
E. Becker, “Summen n-ter Potenzen in K¨orpern,” J. Reine Angew. Math., no. 307–308, pp. 8–30, 1979; doi: 10.1515/crll.1979.307-308.8
M. A. Bennett, “Fractional parts of powers of rational numbers,” Math. Proc. Cambridge Philos. Soc., vol. 114, no. 2, pp. 191–201, 1993; doi: 10.1017/S0305004100071528
M. A. Bennett, “Fractional parts of powers and related topics,” Ph. D. Thesis, Univ. British Columbia, Canada, 1993; doi: 10.14288/1.0079903
M. A. Bennett “An ideal Waring problem with restricted summands,” Acta Arith., vol. 66, no. 2, pp. 125–132, 1994; doi: 10.4064/aa-66-2-125-132
A. C. Robin, “Pi: A source book (3rd edn.),” by L. Berggren, J. Borwein, and P. Borwein, The Mathematical Gazette, vol. 90, no. 518, pp. 375–376, 2006; doi: 10.1017/S002555720018009X
F. Bertault , O. Ramar´e, and P. Zimmermann, “On sums of seven cubes,” Math. Comp., vol. 68, no. 227, pp. 1303–1310, 1999; doi: 10.1090/S0025-5718-99-01071-6
F. Beukers, “Fractional Parts of Powers of Rational Numbers,” Math. Proc. Cambridge Philos. Soc., vol. 90, pp. 13–20, 1981.
Ch. Binder, “100 Jahre Mertenssche Vermutung,” Int. Math. Nachr. Wien, vol. 178, pp. 2–6, 1998.
J. Bohman and C. E. Fr¨oberg, “Numerical investigations of Waring’s problem for cubes,” BIT Numer. Math, vol. 21, pp. 118–122, 1981; doi: 10.1007/s10543-020-00827-y
K. D. Boklan and N. D. Elkies, “Every multiple of 4 except 212, 364, 420, and 428 is the sum of seven cubes,” in arXiv:0903.4503v1 [math.NT], 26 Mar 2009. [Online]. Available: https://arxiv.org/pdf/0903.4503.pdf
C. A. Bretschneider, “Tafeln fur die Zerlegung der Zahlen bis 4100 in Biquadrate, ¨ ” J. Reine Angew. Math., vol. 46, pp. 1–23, 1853.
J. W. S. Cassels and R. C. Vaughan, “Obituary: Ivan Matveevich Vinogradov,” Bull. London Math. Soc., vol. 17, no. 6, pp. 584–600, 1985.
J.-R. Chen, “Waring’s problem for g (5),” Sci. Record (N.S.), vol. 3, pp. 27–330, 1959.
J.-R. Chen, “Waring’s problem for g (5) = 37,” Chinese Math. Acta, vol. 6, pp. 105–127, 1965.
J.-R. Chen, “An estimate for g (4) in Waring’s problem,” Acta Math. Sinica, vol. 17, no. 2, pp. 131–142, 1974 (in Chinese).
Yu.-Yo. Cheng, “Explicit estimate on primes between consecutive cubes,” Rocky Mountain J. Math., vol. 40, no. 1, pp. 117–153, 2010; doi: 10.1216/RMJ-2010-40-1-117
S. Chowla, “A remark on g (n),” Proc. Indian Acad. Sci. A, vol. 9, pp. 20–21, 1939; doi: 10.1007/BF03045445
S. Chowla, “On g (k) in Waring’s problem,” in Proc. Lahore Philos. Soc., vol. 6, no. 2, 1944, pp. 16–17.
P. L. Clark, “Number theory: a contemporary introduction,” in Dep. of Mathematics, University of Georgia.[Online]. Available: http://math.uga.edu/ pete/4400FULL.pdf
R. J. Cook, “An effective seven cube theorem,” Bull. Austral. Math. Soc., vol. 30, no. 3, pp. 381–38, 1984; doi: 10.1017/S0004972700002094
Z. Dahse, “Der Kreis-Umfang fur den Durchmesser 1 auf 200 Decimalstellen berechnet, ¨ ” J. Reine Angew. Math., vol. 1844, no. 27, p. 198 (in German); doi: 184410.1515/crll.1844.27.198
R. D. von Sterneck, “Uber die kleinste Anzahl Kuben, aus welchen jede Zahl bis 40000 zusammengesetzt werden kann,” Akad. Wiss. Wien, Math.-Natur. Kl. Sitz., vol. 112, pp. 1627–1666, 1903.
H. Davenport, “On Waring’s problem for cubes,” Acta Math., vol. 71, pp. 123–143, 1939; doi: 10.1007/BF02547752
H. Davenport, “On Waring’s problem for fourth powers,” Ann. Math., vol. 40, no. 4, pp. 731–747, 1939; doi: 10.2307/1968889
H. Davenport, “On Waring’s problem for fifth and sixth powers,” Amer. J. Math., vol. 64, pp. 199–207, 1942; doi: 10.2307/2371678
H. Davenport, “Some aspects of Hardy’s mathematical work. V. Waring’s problem,” J. London Math. Soc., vol. 25, pp. 119–125, 1950; doi: 10.1112/jlms/s1-25.2.119
H. Davenport and H. Heilbronn, “On Waring’s problem for fourth powers,” Proc. London Math. Soc., vol. 41, no. 2, pp. 143–150, 1936; doi: 10.1112/plms/s2-41.2.143
J. H. Davenport Computer Algebra, Univ. Bath, 2917
F. Delmer and J.-M. Deshouillers, “On the computation of g (k) in Waring’s problem,” Math. Comp., vol. 54, no. 190, pp. 885–893, 1990.
J.-M. Deshouillers “Probl`eme de Waring pour les bicarr´es: le point en 1984,” . Th´eor, Analyt. Nombres Paris,
exp. 33, pp. 1–5, 1984–85.
J.-M. Deshouillers, “Probl`eme de Waring pour les bicarr´es,” in Seminar on number theory 1984–1985, no. 14, Univ. Bordeaux I, Talence, France, 1985, pp. 1–47 (in French).
J.-M. Deshouillers and PRC Mathematiques et Informatique, “Waring’s problem and the circle-method,” Number theory and applications, Dordrecht, Nederland: Kluwer Acad. Publ., 1989, pp. 37–44.
J.-M. Deshouillers, “La majoration de sommes de Weyl biquadratiques” [Upper bounds for biquadratic Weyl sums], Ann. Scuola Norm. Sup. Pisa - Cl. Sci., Serie 4, vol. 19, no. 2, pp. 291–304, 1992 (in French).
J.-M. Deshouillers and F. Dress, “Sommes de diviseurs et structure multiplicative des entiers,” Acta Arith., vol. 49, no. 4, pp. 341–375, 1988; doi: 10.4064/aa-49-4-341-375
J.-M. Deshouillers and F. Dress, “Sums of 19 biquadrates: on the representation of large integers,” Ann. Scuola Norm. Sup. Pisa - Cl. Sci., Serie 4, vol. 19, no. 1, pp. 113–153, 1992 (in French).
J.-M. Deshouillers and F. Dress, “Numerical results for sums of five and seven biquadrates and consequences for sums of 19 biquadrates,” Math. Comp., vol. 61, pp. 195–207, 1993; doi: 10.1090/S0025-5718-1993-1201766-8
J.-M. Deshouillers, F. Hennecart, and B. Landreau, “Sums of powers: an arithmetic refinement to the probabilistic model of Erd˝os and R´enyi,” Acta Arithmetica, vol. 85, pp. 13–33, 1998; doi: 10.4064/aa-85-1-13-33
J.-M. Deshouillers, F. Hennecart, and B. Landreau, “7373170279850 (With an appendix by I. Gusti Putu Purnaba),” Math. Comp., vol. 69, no. 229, pp. 421–439, 2000.
J.-M. Deshouillers, F. Hennecart, and B. Landreau, “Waring’s problem for sixteen biquadrates — numerical results,” J. Th´eor. Nombres Bordeaux, vol. 12, no. 2, pp. 411–422, 2000; doi: 10.5802/jtnb.287
J.-M. Deshouillers, K. Kawada, and T. D Wooley, “On sums of sixteen biquadrates,” M´em. Soc. Math. Fr., no. 100, pp. a-120, 2005.
L. E. Dickson, “Generalization of Waring’s theorem on fourth, sixth, and eighth powers,” Amer. J. Math., vol. 49, no. 2, pp. 241–250, 1927; doi: 10.2307/2370754
L. E. Dickson, “Extensions of Waring’s theorem on nine cubes,” Amer. Math. Monthly, vol. 34, no. 4, pp. 177–83, 1927; doi: 10.1080/00029890.1927.11986678
L. E. Dickson, “Extensions of Waring’s theorem on fourth powers,” Bull. Amer. Math. Soc., vol. 33, pp. 319–327, 1927; doi: 10.1090/S0002-9904-1927-04365-8
L. E. Dickson, “A generalization of Waring’s theorem on nine cubes,” Bull. Amer. Mat. Soc., vol. 33, pp. 299–300, 1927; doi: 10.1090/S0002-9904-1927-04357-9
L. E. Dickson, “Simpler proofs of Waring’s theorem on cubes with various generalizations,” Trans. Amer. Math. Soc., vol. 30, no. 1, pp. 1–18, 1928; doi: 10.2307/1989262
L. E. Dickson, “Proof of a Waring theorem on fifth powers,” Bull. Amer. Math. Soc., vol. 37, pp. 549–553, 1931; doi: 10.1090/S0002-9904-1931-05198-3
L. E. Dickson, “Minimum decomposition into n-th powers,” Amer. J. Math., vol. 55, no. 1, pp. 593–602, 1933; doi: 10.2307/2371152
L. E. Dickson, Minimum decompositions into fifth powers. (Math. tables. 3), London: Office of the Brit. Assoc. f. the Advancement of Science, 1933.
L. E. Dickson, “Recent progress on Waring’s theorem and its generalizations,” Bull. Amer. Math. Soc., vol. 39, pp. 701–727, 1933; doi: 10.1090/S0002-9904-1933-05719-1
L. E. Dickson, “A new method for universal Waring theorems with details for seventh powers,” Amer. Math. Monthly, vol. 41, pp. 547–555, 1934; doi: 10.1080/00029890.1934.11987646
L. E. Dickson, “Waring’s problem for cubic functions,” Trans. Amer. Math. Soc., vol. 36, no. 1, pp. 1–12, 1934;
doi: 10.1090/S0002-9947-1934-1501731-6
L. E. Dickson, “Waring’s problem for ninth powers,” Bull. Amer. Math. Soc., vol. 40, pp. 487–493, 1934; doi: 10.1090/S0002-9904-1934-05905-6
L. E. Dickson, “Universal Waring theorem for eleventh powers,” J. Lond. Math. Soc., vol. 9, pp. 201–206, 1934; doi: 10.1112/jlms/s1-9.3.201
L. E. Dickson, “A new method for universal Waring theorems with details for seventh powers,” Amer. Math. Monthly, vol. 41, pp. 547–555, 1934; doi: 10.1080/00029890.1934.11987646
L. E. Dickson, “A new method for Waring theorems with polynomial summands,” Trans. Amer. Math. Soc., I: vol. 36, no. 4, pp. 731–748, 1934; II: vol. 39, no. 2, pp. 205–208, 1936; doi: 10.1090/S0002-9947-1936-1501842-7
L. E. Dickson, “Cyclotomy, higher congruences, and Waring’s problem,” Amer. J. Math., I. vol. no. 2, 57, pp. 391–424, 1935; II. vol. 57, no. 3, pp. 463–474, 1935; doi: 10.2307/2371217
L. E. Dickson, Researches on Waring’s problem, Washington, USA: Carnegie Inst. of Washington Publ, no. 464, 1935.
L. E. Dickson, “Universal Waring theorems with cubic summands,” Acta Arith., vol. 1, no. 1, pp. 184–196, 1935; doi: 10.4064/aa-1-2-184-196
L. E. Dickson, “On Waring’s problem and its generalization,” Ann. Math., vol. 37, no. 2, pp. 293–316, 1936; doi: 10.2307/1968443
L. E. Dickson, “Universal Waring theorems,” Monatsh. Math. Phys, vol. 43, pp. 391–400, 1936; doi: 10.1007/BF01707618
L. E. Dickson, “The ideal Waring theorem for twelfth powers,” Duke Math. J., vol. 2, no. 2, pp. 192–204, 1936; doi: 10.1215/S0012-7094-36-00218-1
L. E. Dickson, “Proof of the ideal Waring theorem for exponents,” Amer. J. Math., vol. 58, no. 3, pp. 521–529, 1936; doi: doi.org/10.2307/2370969
L. E. Dickson, “Solution of Waring’s problem,” Amer. J. Math., vol. 58, no. 3, pp. 530–535, 1936; doi: 10.2307/2370970
L. E. Dickson, “A generalization of Waring’s problem,” Bull. Amer. Math. Soc., vol. 42, pp. 525–529, 1936; doi: 10.1090/S0002-9904-1936-06348-2
L. E. Dickson, “The Waring problem and its generalizations,” Bull. Amer. Math. Soc., vol. 42, pp. 833–842, 1936; doi: 10.1090/S0002-9904-1936-06432-3
L. E. Dickson, “Universal forms S(ai xi^n) and Waring’s problem,” Acta Arith., vol. 2, no. 2, pp. 177–196, 1937.
L. E. Dickson, “All integers except 23 and 239 are sums of eight cubes.,” Bull. Amer. Math. Soc., vol. 45, pp. 588–591, 1939; doi: 10.1090/S0002-9904-1939-07041-9
L. E. Dickson, The collected mathematical papers of Leonard Eugene Dickson, Adrian Albert ed., New York: Chelsea Publishing Company, 1975.
L. E. Dickson, “Vol. I: Divisibility and primality,” in History of the theory of numbers. Mineola, NY: Dover Publications, 2005.
L. E. Dickson, “Vol. II: Diophantine analysis,” in History of the theory of numbers. Mineola, NY: Dover Publications, 2005.
L. E. Dickson, “Vol. III: Quadratic and higher forms,” in History of the theory of numbers, Mineola, NY: Dover Publications, 2005.
F. Dress, “Am´elioration de la majoration de g (4) dans le probl`eme de Waring: g (4) 6 34,” in S´eminaire Delange—Pisot—Poitou, Th´eorie des nombres, 11.1, 1969–1970, Paris, 1970, pp. 1–23 (in French).
F. Dress, “Sur le probl`eme de Waring pour les puissances quatri`emes,” Sci. Paris S´er. A–B, vol. 272, pp. A457–A459, 1971 (in French).
F. Dress, “M´ethodes ´el´ementaires dans le probl`eme de Waring pour les entiers,” Journ´ees Arithm´etiques Fran¸caises, 1971 (in French).
F. Dress, “Th´eorie additive des nombres et probl`eme de Waring,” in S´eminaire de Th´eorie des Nombres, Univ. Bordeaux I, Talence, 1970–1971, vol. 28, 1971, pp. 1–9 (in French).
F. Dress, “Th´eorie additive des nombres, probl`eme de Waring et th´eor`eme de Hilbert,” Enseign. Math. 2, vol. 18, pp. 175–190, 1972 [erratum, ibid. vol. 18, pp. 301–302, 1972.] (in French).
F. Dress, “Am´elioration de la majoration de g (4) dans le probl`eme de Waring: g (4) 6 30,” Acta Arith., vol. 22, pp. 137–147, 1973 (in French).
D. Dumbaugh and A. Shell-Gellasch, “The "wide influence” of Leonard Eugene Dickson,” Notices Amer. Math. Soc., vol. 64, no. 7, pp. 772–776, 2017.
H. Ehlich, “Zur Pillaischen Vermutung,” Arch. Math., vol. 16, pp. 223–226, 1965; doi: 10.1007/BF01220025
N. D. Elkies, “Every even number greater than 454 is the sum of seven cubes,” in arXiv:1009.3983 [math.NT], [v1], Tue, 21 Sep 2010, pp. 1–9.
W. J. Ellison, “Waring’s problem,” Amer. Math. Monthly, vol. 78, no. 1, pp. 10–36, 1971; doi: 10.1080/00029890.1971.11992689
T. Estermann, “Proof that every large integer is a sum of seventeen biquadrates,” Proc. London Math. Soc., vol. s2–41, no. 1, pp. 126–142, 1936.
T. Estermann, “On Waring’s problem for fourth and higher powers,” Acta Arith., vol. 2, pp. 197–211, 1936; doi: 10.4064/aa-2-2-197-211
L. Euler, “De numeris amicabilibus (1747),” in Euler Archive — All Works by Enestr¨om Number. E100, 2018. [Online]. Available: https://scholarlycommons.pacific.edu/euler-works/100
L. Euler, “De numeris amicabilibus (1750),” in Euler Archive — All Works by Enestr¨om Number. E152, 2018. [Online]. Available: https://scholarlycommons.pacific.edu/euler-works/152
L. Euler, “Opera postuma. Mathematica et physica. Anno 1844 detecta quae Academiae Scientiarum Petropolitanae. Tomus I,” Petropoli, 1862. [Online]. Available: https://ia800204.us.archive.org/18/items/ operapostumamath01euleuoft/operapostumamath01euleuoft.pdf
L. J. Everet, “Reviews: Innumeracy: Mathematical Illiteracy and its Consequences,” Amer. Math. Monthly, vol. 97, no. 1, pp. 88–91, 1990; doi: 10.1080/00029890.1990.11995554
A. Fleck, “Uber die Darstellung ganzer Zahlen als Summen von positiven Kuben und von Biquadraten ganzer ¨Zahlen,” Sitzungsber. Berl. Math. Ges., vol. 5, pp. 2–9, 1906.
A. Fleck, “Uber die Darstellung ganzer Zahlen als Summen von sechsten Potenzen ganzer Zahlen, ¨ ” Math. Ann., vol. 64, no. 4, pp. 561–566, 1907; doi: 10.1007/BF01450063
G. Frobenius, “Uber den Stridsbergschen Beweis des Waringschen Satzes, ¨ ” Berl. Ber., pp. 666–670, 1912.
J. von zur Gathen and J. Gerhard, Modern computer algebra, 3rd ed., Cambridge, UK: Cambridge University Press, 2013.
M. Gelbcke, “Zum Waringschen Problem,” Math. Ann., vol. 105, pp. 637–652, 1931; doi:10.1007/BF01455835
R. K. Guy, “The strong law of small numbers,” Amer. Math. Monthly, vol. 95, no. 8, pp. 697–712, 1988; doi:10.1080/00029890.1988.11972074
R. K. Guy, “The second strong law of small numbers,” Math. Mag, vol. 63, no. 1, pp. 3–20, 1990; doi:10.1080/0025570X.1990.11977475
L. Habsieger, “Explicit lower bounds for ||(3/2)k ||,” Acta Arith.,vol. 106, no. 3, pp. 299–309, 2003; doi:10.4064/aa106-3-7
H. Halberstam and K. F. Roth, Sequences, Clarendon Press, Oxford, 1966; doi: 10.1007/978-1-4613-8227-0
G. H. Hardy, “Some famous problems of the theory of numbers and in particular Waring’s problem. An inaugural lecture delivered before the University of Oxford,” Nature, vol. 106, pp. 239-240, 1920; doi:10.1038/106239c0
G. H. Hardy, A mathematician’s apology. With a foreword by C. P. Snow, Cambridge Univ. Press, Cambridge, 1967.
G. H. Hardy and J. E. Littlewood, “A new solution of Waring’s problem,” Quart. J. Math., vol. 48, pp. 272–293, 1919.
G. H. Hardy and J. E. Littlewood, “Some Problems of «partitio numerorum». I: A new solution of Waring’s problem,” Nachr. Akad. Wiss. G¨ottingen Math.–Phys. Kl, pp. 33–54, 1920.
G. H. Hardy and J. E. Littlewood, “Some problems of «partitio numerorum». II: Proof that every large number is the sum of at most 21 biquadrates,” Math. Z., vol. 9, pp. 14–27, 1921; doi:10.1007/BF01378332
G. H. Hardy and J. E. Littlewood, “Some problems of «partitio numerorum». IV: The singular series in Waring’s problem and the value of the number G(k),” Math. Z., vol. 12, pp. 161–168, 1922.
G. H. Hardy and J. E. Littlewood, “Some problems of «partitio numerorum». VI: Further researches in Waring’s problem,” Math. Z., vol. 23, pp. 1–37, 1925.
G. H. Hardy and J. E. Littlewood, The collected works of G. H. Hardy, vol. 1, New York: Oxford University Press, 1966.
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Oxford, UK: Oxford University Press, 1979.
A. Hurwitz, “Uber die Erzeugung der Invarianten durch Integration, ¨ ” Nachr. Ges. Wiss. G¨ottingen, pp. 71–90, 1897.
F. Hausdorff, “Zur Hilbertschen Losung des Waringschen Problems,” Math. Ann., vol. 67, pp. 301–305, 1909; doi: 10.1007/BF01450406
H. Heilbronn, “Uber das Waringsche Problem, ¨ ” Acta Arith., vol. 1, no. 2, pp. 212–221, 1935 (in German).
D. Hilbert, “Beweis fur die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl ¨ n-ter Potenzen (Waringsches problem). Dem Andenken an Hermann Minkowski gewidmet,” G¨ott. Nachr., pp. 17–36, 1909 [Abdruck mit Ver¨anderungen und Zus¨atzen: Math. Ann, vol. 67, pp. 281–300, 1909] (in German); doi: 10.1007/BF01450405
L.-K. Hua, “Hua, L. K.: On Waring’s problem,” Quart. J. Math., vol. 9, no. 1, pp. 199—202, 1938; doi: 10.1093/qmath/os-9.1.199
L.-K. Hua, “On Waring’s problem for fifth powers,” Proc. London Math. Soc., vol. 45, no. 1, pp. 144–160, 1939; doi: 10.1112/plms/s2-45.1.144
L.-K. Hua, Introduction to number theory, P. Shiu transl., Berlin-New York: Springer-Verlag, 1982; doi:10.1007/978-3-642-68130-1
L.-K. Hua, “Selected papers. Edited, with a preface and a biographical note of Hua by Heini Hadberstam,” [Reprint of the 1983 edition.], New York: Springer, 2015.
A. Hurwitz, “Uber die Darstellung der ganzen Zahlen als Summen von ¨n-ten Potenzen ganzer Zahlen,” Math.
Ann., vol. 65, pp. 424–427, 1908 (in German); doi: 10.1007/BF01456421
A. Hurwitz, “Uber de ¨ finite Polynome,” Math. Ann., vol. 73, pp. 173–176, 1912; doi: 10.1007/BF01456665
H. Iwaniec, “The sixtieth birthday of Jean-Marc Deshouillers,” Functiones et Approximatio Commentarii Mathematici, vol. 37, n. 1, pp. 7–16, 2007; doi: 10.7169/facm/1229618737
C. G. J. Jacobi, “Uber die Zusammensetzung der Zahlen aus ganzen positiven Cuben; nebst einer Tabelle fur die ¨kleinste Cubenanzahl, aus welcher jede Zahl bis 12000 zusammengesetzt werden kann,” J. Reine Angew. Math., vol. 42, pp. 41–69, 1851 [reprinted in C. G J. Jacobi, Gesammelte Werke. Bd. 6, Berlin, Verlag von Georg Reimer, pp. 322–354, 1891.] (in German); doi: 10.1515/crll.1851.42.41
R. D. James, “The value of the number g (k) in Waring’s problem,” Trans. Amer. Math. Soc., vol. 36, no. 2, pp. 395–444, 1934; doi: 10.2307/1989846
R. D. James, “On Waring’s problem for odd powers,” Proc. London Math. Soc. (2), vol. 37, no. 1, pp. 257–291, 1934; doi: 10.1112/plms/s2-37.1.257
R. D. James, “The constants in Waring’s problem for odd powers,” Bull. Amer. Math. Soc., vol. 41, pp. 689–694, 1935; doi: 10.1090/S0002-9904-1935-06176-2
R. D. James, “Zuckerman H. S. New results for the number g (n) in Waring’s problem,”Bull. Amer. Math. Soc., vol. 41, pp. 197–198, 1935.
H. Kadiri, “Short effective intervals containing primes in arithmetic progressions and the seven cubes problem,” Math. Comput., vol. 77, pp. 1733–1748, 2008; doi:10.1090/S0025-5718-08-02084-X
E. Kamke, “Verallgemeinerungen des Waring—Hilbertschen Satzes,” Math. Ann., vol. 83, pp. 85–112, 1921; doi:10.1007/BF01464230
E. Kamke, “Bemerkung zum allgemeinen Waringschen Problem,” Math. Zeitschr, vol. 15, pp. 188–194, 1922; doi:10.1007/BF01494392
E. Kamke, “Zum Waringschen Problem fur rationale Zahlen und Polynome, ¨ ” Math. Ann., vol. 87, pp. 238–245, 2009; doi:10.1007/BF01459066
A. Kempner, “Bemerkungen zum Waringschen Problem,” Math. Ann., vol. 72, pp. 387–399, 1912; doi:10.1007/BF01456723
A. J. Kempner, Uber das Waringsche Problem und einige Verallgemeinerungen ¨ , Diss. G¨ottingen., 1912 (in German).
A. J. Kempner, “The development of “partitio numerorum”, with particular reference to the work of Messrs. Hardy, Littlewood and Ramanujan,” Amer. Math. Monthly, vol. 30, pp. 354–369, 1923; doi: 10.1080/00029890.1923.11986272
M. Kreuzer and L. Robbiano, Computational commutative algebra, Springer-Verlag, Berlin, vol. 1, 2000, vol. 2, 2005; doi:10.1007/978-3-540-70628-1
J. M. Kubina and M. C. Wunderlich, “Extending Waring’s conjecture to 471,600,000,” Math. Comp., vol. 55, no. 192, pp. 815–820, 1990; doi:10.1090/S0025-5718-1990-1035936-6
J. Kursch ¨ ´ak, “Uber die Liouvillesche Identit ¨ ¨at,” Arch. Math. Phys., vol. 18, pp. 242–243, 1911 (in German).
E. Landau, “Uber die Darstellung einer ganzen Zahl als Summe von Biquadraten, ¨ ” Rend. Circ. Matem. Palermo, vol. 23, pp. 91–96, 1907 (in German); doi: 10.1007/BF03013509
E. Landau, “Uber eine Anwendung der Primzahltheorie auf das Waringsche Problem in der elementaren Zahlentheorie,” Math. Ann., vol. 66, no. 1, pp. 102–105, 1908 (in German); doi: 10.1007/BF01450914
E. Landau, “Zur Hardy—Littlewoodschen L¨osung des Waringschen Problems,” G¨ott. Nachr., pp. 88–92, 1921 (in German).
E. Landau, “Zum Waringschen Problem,” Math. Zeitschr, I: vol. 12, pp. 219–247, 1922; II: vol. 31, pp. 149–150, 1929; III: vol. 32, pp. 699–702, 1930 (in German).
E. Landau, “Die Winogradowsche Methode zum Beweis des Waring—Hilbert—Kamkeschen Satzes, Acta Math., vol. 48, pp. 217–253, 1926 (in German).
E. Landau, “Zum Waringschen Problem,” Proceedings L. M. S., vol. s2-25, pp. 484–486, 1926 (in German).
E. Landau, Vorlesungen uber Zahlentheorie. Bd. I ¨ –III, Hirzel, Leipzig, 1927 (in German).
E. Landau, “Uber die neue Winogradoffsche Behandlung des Waringschen Problems, ¨ ” Math. Z., vol. 31, pp. 319–338, 1930 (in German); doi: 10.1007/BF01246414
B. Landreau, “Mod`ele probabiliste pour les sommes de s puissances s-i`emes,” Compositio Math., vol. 99, pp. 1–31, 1995 (in French).
V.-A. Lebesgue, Exercises d’analyse num´erique, Paris, 1859 (in French).
U. V. Linnik, “On the representation of large numbers as sums of seven cubes,” Matem. sb., vol. 12, no. 2, pp. 218–224, 1943 (in French).
E. Lucas, “Sur la d´ecomposition des nombres en bicarr´es,”N. C. M., vol. IV, pp. 323–325, 1878 (in French).
E. Lucas, “Sur un th´eor`eme de M. Liouville concernant la d´ecomposition des nombres en bicarr´es,” Nouvelles annales de math´ematiques, vol. 17, pp. 536–537, 1878 (in French).
K. Mahler, “On the fractional parts of the powers of a rational number,” II. Mathematika, vol. 4, pp. 122–124, 1957(in French).
E. Maillet, “Sur la d´ecomposition d’un nombre entier en une somme de cubes d’entiers positifs, Assoc. Franc,” Bordeaux Notes M´em., vol. 24, pp. 242–247, 1895 (in French).
E. Maillet, “Quelques extensions de th´eor`eme de Fermat sur les nombres polygones,” J. Math. Pures et App., vol. 2, pp. 363–380, 1896 (in French).
E. Maillet, “Sur la d´ecomposition d’un entier en une somme de puissances huiti`emes d’entiers. (Probleme de Waring),” Bull. Soc. Math. France, vol. 36, pp. 69–77, 1908 (in French).
A. Malter, D. Schleicher, and D. Zagier, “New looks at old number theory,” Amer. Math. Monthly, vol. 120, no. 3, pp. 243–264, 2013; doi: 10.4169/amer.math.monthly.120.03.243
K. S. McCurley, “Explicit estimates for functions of primes in arithmetic progressions,” Ph. D. Thesis, Univ. Illinois at Urbana-Champaign, 1981.
K. S. McCurley, “An effective seven cube theorem,” J. Number Theory, vol. 19, no. 2, pp. 176–183, 1984; doi: 10.1016/0022-314X(84)90100-8
W. Narkiewicz, Teoria liczb, 3rd ed., Warszawa: PWN, 2003.
W. Narkiewicz, Classical problems in number theory, Warszawa: PWN, 1986.
W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 3rd ed., Berlin: Springer, 2004.
W. Narkiewicz, Rational number theory in the 20th century. From PNT to FLT, Berlin: Springer, 2012; doi: 10.1007/978-0-85729-532-3
M. B. Nathanson, Additive number theory. The classical bases. Graduate Texts in Mathematics, New York: Springer-Verlag, 1996.
M. B. Nathanson, Elementary methods in number theory. Graduate Texts in Mathematics, New York, Springer-Verlag, 2000.
D. J. Newman, “A simplified proof of Waring’s conjecture,” Michigan Math. J., vol. 7, no. 3, pp. 291–295, 1960; doi:10.1307/mmj/1028998439
I. Niven, “An unsolved case of the Waring problem,” Amer. J. Math., vol. 66, no. 1, pp. 137–143, 1944; doi:10.2307/2371901
A. Oppenheim, “Hilbert’s proof of Waring’s theorem,” Messenger, vol. 58, pp. 153–158, 1929.
A. Ostrowski, “Bemerkung zur Hardy—Littlewoodschen L¨osung des Waringschen Problems,” Math. Zeitschr., vol. 9, pp. 28–34, 1921; doi:10.1007/BF01378333
G. Pall, “Quaternions and sums of three squares,” Amer. J. Math., vol. 64, no. 1, pp. 503–513, 1942; doi:10.2307/2371700
The PARI Group, Univ. Bordeaux, PARI/GP version 2.11.0, 2018. [Web-site]. Available: http://pari.math.u-bordeaux.fr/
S. S. Pillai, “On Waring’s problem,” I: J. Annamalai Univ., vol. 5, pp. 145–166, 1936; III: vol. 6, pp. 50–53, 1936; IV: vol. 6, pp. 54–64, 1936; VI: Polynomial summands, vol. 6, pp. 171–197, 1937.
S. S. Pillai, “On Waring’s problem,” J. Indian Math. Soc., vol. 2, pp. 16–44, 1936; [Errata. J. Indian math. Soc., vol. 2, 1936]; V: “On g (6),” no. 2, pp. 213–214, 1937; VIII: “With polynomial summands,” vol. 3, 205–220, 1939; IX: “On universal Waring’s problem with powers of primes,” vol. 3, pp. 221–225, 1939.
S. S. Pillai, “On Waring’s problem: g (6) = 73,” in Proc. Indian Acad. Sci., Sect. A, vol. 12, pp. 30–40, 1940; doi:10.1007/BF03170721
S. S. Pillai, Collected works of S. Sivasankaranarayana Pillai, R. Balasubramanian and R. Thangadurai, eds., Mysore, India: Ramanujan Mathematical Society, 2010.
H. Poincar´e, “Rapport sur le prix Bolyai,” Palermo Rend, vol. 31, pp. 109–132, 1911 (in French);
P. Pollack, “Not always buried deep. A second course in elementary number theory,” Amer. Math. Soc., Providence, RI, 2009.
P. Pollack, “Paul On Hilbert’s solution of Waring’s problem,” Cent. Eur. J. Math., vol. 9, no. 2, pp. 294–301, 2011; doi:10.2478/s11533-011-0009-z
P. Pollack and P. Schorn, “Dirichlet’s proof of the three-square theorem: an algorithmic perspective,” Math. Comp., vol. 88, pp. 1007–1019, 2019; doi:10.1090/mcom/3349
O. Ramar´e, “An explicit seven cube theorem,” Acta Arith., vol. 118, no. 4, pp. 375–382, 2005.
O. Ramar´e, “An explicit result of the sum of seven cubes,” Manuscripta Math., vol. 124, no. 1, pp. 59–75, 2007.
O. Ramar´e, “Etat des lieux, ´ ” pp. 1–19. [Online] (in French). Available: http://math.univ-lille1.fr/~ramare/Maths/ExplicitJNTB.pdf
S. R´ealis, “Note sur un th´eor`eme d’arithm´etique,” Nouv. Corresp. Math., vol. 4, pp. 209–210, 1878 (in French).
R. Remak, “Bemerkung zu Herrn Stridsbergs Beweis des Waringschen Theorems,” Math. Ann., vol. 72, pp. 153–156, 1912; doi: 10.1007/BF01667320
G. J. Rieger, “Zur Hilbertschen L¨osung des Waringschen Problems: Absch¨atzung von g (n),” Arch. Math., vol. 4, pp. 275–281, 1953 (in German); doi: 10.1007/BF01899890
G. J. Rieger, “Zu Linniks L¨osung des Waringschen Problems: Absch¨atzung von g (n),” Math. Z., vol. 60, pp. 213–234, 1954 (in German); doi: 10.1007/BF01187372
O. Ramar´e, “Zum Waringschen Problem fur algebraische Zahlen und Polynome, ¨ ” J. Reine Angew. Math., vol. 195, pp. 108–120, 1956 (in German); doi: 10.1515/crll.1955.195.108
F. Romani, “Computations concerning Waring’s problem for cubes,” Calcolo, vol. 19, pp. 415–431, 1982; doi:10.1007/BF02575769
R. Rubugunday “On g (k) in Waring’s problem,” J. Indian Math. Soc., vol. 6, pp. 192–198, 1942; doi:10.18311/jims/1942/17193
E. Schmidt, “Erhard Zum Hilbertschen Beweise des Waringschen Theorems,"Math. Ann., vol. 74, pp. 271–274,
(in German); doi: 10.1007/BF01456042
W. Sierpinski, ´ Elementary theory of numbers, Warszawa: PWN, 1964.
S. Siksek, “Every integer greater than 454 is the sum of at most seven positive cubes,” Algebra Number Theory, vol. 10, no. 10, pp. 2093–2119, 2016; doi: 10.2140/ant.2016.10.2093
C. Small, “Waring’s problem,” Math. Mag., vol. 50, no. 1, pp. 12–16, 1977; doi: 10.2307/2689743
R. M. Stemmler “The ideal Waring theorem for exponents 401–200,000,” Math. Comput., vol. 18, pp. 144–146, 1964.
E. Stridsberg, “Ofret Hilberts bevis f ¨ ¨or Warings sats. Not 1 och 2.,” in Arkiv f¨or Math., Astr. Fys., vol. 6, I: no. 32, 1910; II: no. 39, 1911 (in German).
H. E. Thomas, “A numerical approach to Waring’s problem for fourth powers,” Ph. D. Thesis, Univ. Michigan, 1973.
H. E. Thomas, “Waring’s problem for twenty-two biquadrates,” Trans. Amer. Math. Soc., vol. 193, pp. 427–430, 1974; doi:10.2307/1996923.
E. Trost, “Eine Bemerkung zum Waringschen Problem,” Elem. Math., vol. 13, pp. 73–75, 1958 (in German).
R. C. Vaughan, “On Waring’s problem for cubes,” I: J. Reine Angew. Math., vol. 363, pp 122–170, 1986; II: J. London Math. Soc., vol. 39, pp 205–218, 1989.
R. C. Vaughan, The Hardy—Littlewood Method, Second edition, Cambridge, UK: Cambridge University Press, 1997.
R. C. Vaughan and T. D. Wooley, “Waring’s problem: a survey,” in Number Theory for the Millennium, III (Urbana, IL, 2000), 301–340, 2002. [Online]. Available: https://www.researchgate.net/publication/2842101_WaringT2Atextquoterights_Problem_A_Survey
I. Vinogradov, “Une nouvelle variante de la d´emonstration du th´eor`eme de Waring,” Compt. Rend. Acad. Sci.
Paris, vol. 200, pp. 182–184, 1935 (in French).
I. Vinogradov, “An asymptotic formula for the number of representations in Waring’s problem,” Matem. sb., vol. 42, no. 5, pp. 531–534, 1935.
I. Vinogradov, “On Waring’s problem,” Ann. of Math., vol. 36, no. 2, pp. 395–405, 1935; doi: 10.2307/1968579
I. Vinogradov, “On asymptotic formula in Warings problem,” Matem. sb., vol. 43, no. 2, 169–174, 1936.
E. Waring, “Meditationes Algebraicae,” in Archive.org, 1772. [Online]. Available: https://archive.org/details/bub_gb_1MNbAAAAQAAJ/page/n11/mode/2up
G. L. Watson, “A proof of the seven cube theorem,” J. London Math. Soc., vol. 26, no. 2, pp. 153–156, 1951.
G. L. Watson, “A simple proof that all large integers are sums of at most eight cubes,” Math. Gaz., vol. 37, no. 321, pp. 209–211, 1953; doi: 10.2307/3608305
S. M. Watt, “Making Computer Algebra More Symbolic,” in Proc. Transgressive Computing 2006: A conference in honor or Jean Della Dora, (TC 2006), April 24–26 2006, Granada, Spain, 2006, pp. 43–49.
S. M. Watt, “Two Families of Algorithms for Symbolic Polynomials,” in Computer Algebra 2006: Latest Advances in Symbolic Algorithms, Proceedings of the Waterloo Workshop, World Scientific, 2007, pp. 193–210.
S. M. Watt, “Symbolic Polynomials with Sparse Exponents,” in Proc. Milestones in Computer Algebra: a Conference in Honour of Keith Geddes’ 60th Birthday, (MICA 2008), May 1–3 2008, Stonehaven Bay, Trinidad and Tobago, University of Western Ontario, 2008, pp. 91–97.
S. M. Watt, “Functional Decomposition of Symbolic Polynomials,” in Proc. International Conference on Computational Sciences and its Applications, (ICCSA 2008), June 30–July 3 2008, Perugia, Italy, IEEE Computer Society, 2008, pp. 353–362.
A. Weil, “Two lectures on number theory, past and present,” Enseign. Math., vol. 20, no. 2, pp. 87–110, 1974.
A. Weil, Number theory. An approach through history from Hammurapi to Legendre, [Reprint of the 1984 edition], Boston, MA: Birkh¨auser Boston Inc., 2007.
A. E. Western, “Computations concerning numbers representable by four or five cubes,” J. London Math. Soc., vol. s1-1, no. 4, pp. 244–250, 1926.
H. Weyl, “Bemerkung uber die Hardy ¨ —Littlewoodschen Untersuchungen zum Waringschen Problem,” G¨ott. Nachr., pp. 189–192, 1921 (in German); doi: 10.1112/jlms/s1-1.4.244
A. Wieferich, “Zur Darstellung der Zahlen als Summen von funften und siebenten Potenzen positiver ganzer
Zahlen,” Math. Ann., vol. 67, pp. 61–75, 1909 (in German); doi: 10.1007/BF01451870
A. Wieferich, “Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von h¨ochstens neun positiven Kuben darstellen l¨aßt,” Math. Ann., vol. 66, pp. 95–101, 1909 (in German).
A. Wieferich, “Uber die Darstellung der Zahlen als Summen von Biquadraten, ¨ ” Math. Ann., vol. 66, pp. 106–108, 1909 (in German).
J. M. Winogradow, “Sur un th´eor`eme g´en´eral de Waring,” Matem. sb., vol. 31, no. 3–4, pp. 490–507, 1924 (in French).
T. D. Wooley, “Breaking classical convexity in Waring’s problem: sums of cubes and quasi-diagonal behaviour,” Invent. Math., vol. 122, pp. 421–451, 1995.
D. Zagier, “A one-sentence proof that every prime p ≡ 1 mod 4 is a sum of two squares,” Amer. Math. Monthly, vol. 97, no. 2, p. 144, 1990.
A. Zornow, “De compositione numerorum e cubis integris positivis,” J. Reine Angew. Math., vol. 1835, no. 14, pp. 276–280, 1835; doi: 10.1515/crll.1835.14.276
H. S. Zuckerman, “New results for the number g (n) in Waring’s problem,” Amer. J. Math., vol. 58, pp. 545–552, 1936; doi: 10.2307/2370972
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