1966:
1124:
1961:{\displaystyle g(k)={\begin{cases}2^{k}+\lfloor (3/2)^{k}\rfloor -2&{\text{if}}\quad 2^{k}\{(3/2)^{k}\}+\lfloor (3/2)^{k}\rfloor \leq 2^{k},\\2^{k}+\lfloor (3/2)^{k}\rfloor +\lfloor (4/3)^{k}\rfloor -2&{\text{if}}\quad 2^{k}\{(3/2)^{k}\}+\lfloor (3/2)^{k}\rfloor >2^{k}{\text{ and }}\lfloor (4/3)^{k}\rfloor \lfloor (3/2)^{k}\rfloor +\lfloor (4/3)^{k}\rfloor +\lfloor (3/2)^{k}\rfloor =2^{k},\\2^{k}+\lfloor (3/2)^{k}\rfloor +\lfloor (4/3)^{k}\rfloor -3&{\text{if}}\quad 2^{k}\{(3/2)^{k}\}+\lfloor (3/2)^{k}\rfloor >2^{k}{\text{ and }}\lfloor (4/3)^{k}\rfloor \lfloor (3/2)^{k}\rfloor +\lfloor (4/3)^{k}\rfloor +\lfloor (3/2)^{k}\rfloor >2^{k}.\end{cases}}}
104:
in 1770, the same year Waring made his conjecture. Waring sought to generalize this problem by trying to represent all positive integers as the sum of cubes, integers to the fourth power, and so forth, to show that any positive integer may be represented as the sum of other integers raised to a
3690:
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].
2436:
in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1986 and 1989 reduced the 14 biquadrates successively to 13 and 12). The exact value of
3662:
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].
2806:
and 10 required at most 16, and Kawada, Wooley and
Deshouillers extended Davenport's 1939 result to show that every number above 10 required at most 16). Numbers of the form 31·16 always require 16 fourth powers.
2092:
3046:
783:
2251:
3183:
3273:
1092:
3547:
910:
3472:
2939:
2170:
647:
2764:
is due to Linnik in 1943. (All nonnegative integers require at most 9 cubes, and the largest integers requiring 9, 8, 7, 6 and 5 cubes are conjectured to be 239, 454, 8042,
348:
2872:
313:
278:
3097:
970:
614:
575:
532:
489:
389:
243:
997:
937:
837:
810:
673:
3069:
form of the Hardy–Ramanujan–Littlewood–Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors,
2303:
1026:
443:
168:
4197:(2002). "Waring's Problem: A Survey". In Bennet, Michael A.; Berndt, Bruce C.; Boston, Nigel; Diamond, Harold G.; Hildebrand, Adolf J.; Philipp, Walter (eds.).
2271:
2136:
2116:
1989:
857:
701:
208:
188:
139:
105:
specific exponent, and that there was always a maximum number of integers raised to a certain exponent required to represent all positive integers in this way.
4126:
4620:
2388:(1) = 1. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that
2316:
4480:
4239:
4206:
60:. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by
1994:
245:. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth powers; these examples show that
392:
356:
of 1770 states that every natural number is the sum of at most four squares. Since three squares are not enough, this theorem establishes
93:
4256:
2950:
1107:
617:
4539:, Hilbert's proof of Waring's conjecture and the Hardy–Littlewood proof of the asymptotic formula for the number of ways to represent
3567:
73:
4625:
4569:
4520:
4334:
92:
greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of
Diophantus by
353:
101:
89:
210:
th powers of naturals needed to represent all positive integers. Every positive integer is the sum of one first power, itself, so
3399:
Remember we restrict ourselves to natural numbers. With general integers, it is not hard to write 23 as the sum of 4 cubes, e.g.
2799:
is the largest number to require 17 fourth powers (Deshouillers, Hennecart and
Landreau showed in 2000 that every number between
706:
2309:
1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, ... (sequence
4610:
3786:
3287:
4615:
4361:
3070:
2175:
412:
Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example,
4593:
4508:
3105:
535:
3301:
3201:
1031:
4588:
4393:
2876:
4110:
U. V. Linnik. "On the representation of large numbers as sums of seven cubes". Mat. Sb. N.S. 12(54), 218–224 (1943).
4326:
3477:
3328:
4367:
G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, "Theory of multiple trigonometric sums". Moscow: Nauka, (1987).
3310:, an algorithmic problem that can be used to find the shortest representation of a given number as a sum of powers
862:
3402:
4536:
3586:"Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt"
3353:"Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)"
3313:
4078:
Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard; I. Gusti Putu
Purnaba, Appendix by (2000).
2885:
2822:
is the last known number that requires 9 fifth powers (Integer sequence S001057, Tony D. Noe, Jul 04 2017),
2341:
450:
3768:
3559:
1103:
539:
2757:, and the authors give reasonable arguments there that this may be the largest possible. The upper bound
4427:
3962:
3357:
2141:
97:
626:
4496:
4295:
4155:
Deshouillers, Jean-Marc; Kawada, Koichi; Wooley, Trevor D. (2005). "On Sums of
Sixteen Biquadrates".
3971:
3876:
4583:
3624:
3585:
1148:
3318:
3866:
Kubina, Jeffrey M.; Wunderlich, Marvin C. (1990). "Extending Waring's conjecture to 471,600,000".
4468:
4447:
4364:, "Trigonometric sums in number theory and analysis". Berlin–New-York: Walter de Gruyter, (2004).
3987:
3892:
3803:
3747:
3644:
3605:
3382:
3307:
2373:
318:
283:
248:
3076:
2376:
integer (i.e. every integer greater than some constant) can be represented as a sum of at most
4565:
4516:
4476:
4330:
4235:
4202:
4172:
4048:
3939:
496:
942:
584:
545:
502:
4526:
4439:
4340:
4303:
4164:
4135:
4091:
4040:
4010:
3979:
3957:
3931:
3884:
3840:
3795:
3731:
3636:
3597:
3581:
3366:
2708:(3) is at least 4 (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3
2426:
1111:
492:
459:
413:
406:
359:
213:
46:
4216:
4121:
3904:
3852:
3815:
3743:
3704:
3676:
3378:
975:
915:
815:
788:
652:
4553:
4530:
4512:
4344:
4212:
3900:
3848:
3811:
3739:
3700:
3672:
3374:
2279:
1002:
680:
419:
144:
4418:
I. M. Vinogradov, A. A. Karatsuba, "The method of trigonometric sums in number theory",
4299:
3975:
3880:
2256:
2121:
2101:
1974:
1099:
842:
686:
193:
173:
124:
39:
3829:
Mahler, Kurt (1957). "On the fractional parts of the powers of a rational number II".
4604:
4557:
4383:
4370:
4194:
3781:
3751:
3648:
3609:
3386:
3348:
2865:
2861:
1115:
676:
69:
61:
31:
4307:
453:
showed that all sufficiently large numbers are the sum of at most 19 fourth powers.
1095:
578:
4096:
4079:
4422:, 168, 3–30 (1986); translation from Trudy Mat. Inst. Steklova, 168, 4–30 (1984).
3922:: IV. The singular series in Waring's Problem and the value of the number G(k)".
3868:
3831:
2337:
2095:
446:
401:
4014:
4234:. Translated by Roth, K.F.; Davenport, Anne. Mineola, NY: Dover Publications.
3844:
396:
85:
17:
4373:, "An elementary solution of the problem of Waring by Schnirelman's method".
4176:
4052:
3943:
3278:
Vaughan and Wooley's survey article from 2002 was comprehensive at the time.
4325:. Cambridge Tracts in Mathematics. Vol. 125 (2nd ed.). Cambridge:
2674:
In the absence of congruence restrictions, a density argument suggests that
4572:). Has a proof of the Lagrange theorem, accessible to high-school students.
4120:
Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard (2000).
350:. Waring conjectured that these lower bounds were in fact exact values.
4451:
4044:
3991:
3935:
3896:
3807:
3735:
3640:
3601:
3370:
4464:), a simplified version of Hilbert's proof and a wealth of references.
4168:
3331:, discusses whether every integer is the sum of four cubes of integers
4140:
4031:
Vaughan, R. C. (1989). "A new iterative method in Waring's problem".
3063:
2087:{\displaystyle 2^{k}\{(3/2)^{k}\}+\lfloor (3/2)^{k}\rfloor >2^{k}}
88:
had asked whether every positive integer could be represented as the
4443:
4005:
Vaughan, R. C. (1986). "On Waring's
Problem for Smaller Exponents".
3983:
3888:
3799:
3352:
3764:
3041:{\displaystyle G(k)\leq k(2\log k+2\log \log k+C\log \log \log k)}
2722:
is the last to require 6 cubes, and the number of numbers between
4396:, "The method of trigonometrical sums in the theory of numbers".
3304:, the problem of representing numbers as sums of powers of primes
64:, after whom it is named. Its affirmative answer, known as the
4122:"Waring's Problem for sixteen biquadrates – numerical results"
4201:. Vol. III. Natick, MA: A. K. Peters. pp. 301–340.
2741:; the largest number now known not to be a sum of 4 cubes is
4232:
2311:
2172:. Thus it is conjectured that this never happens, that is,
1954:
778:{\displaystyle c=2^{k}\lfloor (3/2)^{k}\rfloor -1<3^{k}}
3918:
Hardy, G. H.; Littlewood, J. E. (1922). "Some problems of
4026:
4024:
3188:
Further refinements were obtained by
Vaughan in 1989.
2246:{\displaystyle g(k)=2^{k}+\lfloor (3/2)^{k}\rfloor -2}
2098:
proved that there can only be a finite number of such
3480:
3405:
3204:
3108:
3079:
2953:
2888:
2282:
2259:
2178:
2144:
2124:
2118:, and Kubina and Wunderlich have shown that any such
2104:
1997:
1977:
1127:
1034:
1005:
978:
945:
918:
865:
845:
818:
791:
709:
689:
655:
629:
587:
548:
505:
462:
422:
362:
321:
286:
251:
216:
196:
176:
147:
127:
52:
such that every natural number is the sum of at most
3178:{\displaystyle G(k)\leq k(2\log k+2\log \log k+12).}
391:. Lagrange's four-square theorem was conjectured in
3794:(1). The Johns Hopkins University Press: 137–143.
3784:(1944). "An unsolved case of the Waring problem".
3693:Comptes Rendus de l'Académie des Sciences, Série I
3665:Comptes Rendus de l'Académie des Sciences, Série I
3541:
3466:
3290:, that every positive integer is a sum of at most
3267:
3177:
3091:
3040:
2933:
2297:
2265:
2245:
2164:
2130:
2110:
2086:
1983:
1960:
1086:
1020:
991:
964:
931:
904:
851:
831:
804:
777:
695:
667:
641:
608:
569:
526:
483:
437:
383:
342:
307:
272:
237:
202:
182:
162:
133:
4188:
4186:
3960:(1939). "On Waring's Problem for Fourth Powers".
3268:{\displaystyle G(k)\leq k(\log k+\log \log k+C).}
1087:{\displaystyle 2^{k}+\lfloor (3/2)^{k}\rfloor -2}
409:claimed to have a proof, but did not publish it.
4386:, "A new iterative method in Waring's problem".
2734:at sufficient speed to have people believe that
80:Relationship with Lagrange's four-square theorem
3191:Wooley then established that for some constant
859:; the most economical representation requires
4007:Proceedings of the London Mathematical Society
3321:, discusses what numbers are the sum of three
2368:) is defined to be the least positive integer
4157:Mémoires de la Société Mathématique de France
3542:{\displaystyle 29^{3}+17^{3}+8^{3}+(-31)^{3}}
8:
4487:Has an elementary proof of the existence of
2730:requiring 5 cubes drops off with increasing
2234:
2207:
2068:
2041:
2035:
2008:
1932:
1905:
1899:
1872:
1866:
1839:
1836:
1809:
1788:
1761:
1755:
1728:
1701:
1674:
1668:
1641:
1605:
1578:
1572:
1545:
1539:
1512:
1509:
1482:
1461:
1434:
1428:
1401:
1374:
1347:
1341:
1314:
1278:
1251:
1245:
1218:
1191:
1164:
1075:
1048:
893:
866:
753:
726:
662:
656:
636:
630:
4505:Additive Number Theory: The Classical Bases
4230:Vinogradov, Ivan Matveevich (1 Sep 2004) .
3718:Pillai, S. S. (1940). "On Waring's problem
905:{\displaystyle \lfloor (3/2)^{k}\rfloor -1}
4278:Karatsuba, A. A. (1985). "On the function
4127:Journal de théorie des nombres de Bordeaux
3467:{\displaystyle 2^{3}+2^{3}+2^{3}+(-1)^{3}}
2879:published numerous refinements leading to
76:, 11P05, "Waring's problem and variants".
4456:Survey, contains the precise formula for
4403:I. M. Vinogradov, "On an upper bound for
4139:
4095:
4065:
3533:
3511:
3498:
3485:
3479:
3458:
3436:
3423:
3410:
3404:
3203:
3107:
3078:
2952:
2887:
2281:
2258:
2228:
2216:
2198:
2177:
2158:
2154:
2143:
2123:
2103:
2078:
2062:
2050:
2029:
2017:
2002:
1996:
1976:
1942:
1926:
1914:
1893:
1881:
1860:
1848:
1830:
1818:
1804:
1798:
1782:
1770:
1749:
1737:
1722:
1712:
1695:
1683:
1662:
1650:
1632:
1615:
1599:
1587:
1566:
1554:
1533:
1521:
1503:
1491:
1477:
1471:
1455:
1443:
1422:
1410:
1395:
1385:
1368:
1356:
1335:
1323:
1305:
1288:
1272:
1260:
1239:
1227:
1212:
1202:
1185:
1173:
1155:
1143:
1126:
1069:
1057:
1039:
1033:
1004:
983:
977:
950:
944:
923:
917:
887:
875:
864:
844:
823:
817:
796:
790:
769:
747:
735:
720:
708:
688:
654:
628:
586:
547:
504:
461:
421:
361:
320:
285:
250:
215:
195:
175:
146:
126:
2464:
3340:
4535:Has proofs of Lagrange's theorem, the
3566:. Vol. II: Diophantine Analysis.
2934:{\displaystyle G(k)\leq k(3\log k+11)}
2836:10 that requires 10 fifth powers, and
84:Long before Waring posed his problem,
72:in 1909. Waring's problem has its own
3625:"Bemerkungen zum Waringschen Problem"
3073:obtained in 1985 a new estimate, for
491:was established from 1909 to 1912 by
27:Mathematical problem in number theory
7:
56:natural numbers raised to the power
2853:The upper bounds on the right with
79:
4621:Unsolved problems in number theory
4360:G. I. Arkhipov, V. N. Chubarikov,
2380:positive integers to the power of
2165:{\displaystyle k>471\,600\,000}
74:Mathematics Subject Classification
25:
2873:Hardy–Ramanujan–Littlewood method
642:{\displaystyle \lfloor x\rfloor }
94:Claude Gaspard Bachet de Méziriac
4257:"On an upper bound for $ G(n)$ "
4199:Number Theory for the Millennium
3568:Carnegie Institute of Washington
3564:History of the Theory of Numbers
2846:is the last number less than 1.3
2832:is the last number less than 1.3
1118:and many others has proved that
4308:10.1070/IM1986v027n02ABEH001176
3787:American Journal of Mathematics
3288:Fermat polygonal number theorem
1717:
1390:
1207:
1102:, in about 1772. Later work by
4413:Izv. Akad. Nauk SSSR Ser. Mat.
4261:Izv. Akad. Nauk SSSR Ser. Mat.
3530:
3520:
3455:
3445:
3259:
3223:
3214:
3208:
3169:
3127:
3118:
3112:
3071:Anatolii Alexeevitch Karatsuba
3035:
2972:
2963:
2957:
2928:
2907:
2898:
2892:
2647:is a prime greater than 2 and
2619:is a prime greater than 2 and
2582:) is greater than or equal to
2292:
2286:
2225:
2210:
2188:
2182:
2059:
2044:
2026:
2011:
1923:
1908:
1890:
1875:
1857:
1842:
1827:
1812:
1779:
1764:
1746:
1731:
1692:
1677:
1659:
1644:
1596:
1581:
1563:
1548:
1530:
1515:
1500:
1485:
1452:
1437:
1419:
1404:
1365:
1350:
1332:
1317:
1269:
1254:
1236:
1221:
1182:
1167:
1137:
1131:
1066:
1051:
1015:
1009:
884:
869:
744:
729:
597:
591:
558:
552:
515:
509:
472:
466:
432:
426:
372:
366:
354:Lagrange's four-square theorem
331:
325:
296:
290:
261:
255:
226:
220:
157:
151:
1:
4509:Graduate Texts in Mathematics
4503:Nathanson, Melvyn B. (1996).
4473:Three Pearls of Number Theory
4432:American Mathematical Monthly
4288:Izv. Akad. Nauk SSSR Ser. Mat
4097:10.1090/S0025-5718-99-01116-3
4562:The Enjoyment of Mathematics
3051:for an unspecified constant
4589:Encyclopedia of Mathematics
4323:The Hardy–Littlewood method
2445:) is unknown for any other
2253:for every positive integer
343:{\displaystyle g(4)\geq 19}
90:sum of four perfect squares
4642:
4398:Trav. Inst. Math. Stekloff
4327:Cambridge University Press
4255:Vinogradov, I. M. (1959).
4084:Mathematics of Computation
3329:Sums of four cubes problem
2449:, but there exist bounds.
683:of a positive real number
308:{\displaystyle g(3)\geq 9}
273:{\displaystyle g(2)\geq 4}
170:denote the minimum number
4420:Proc. Steklov Inst. Math.
3924:Mathematische Zeitschrift
3845:10.1112/s0025579300001170
3092:{\displaystyle k\geq 400}
2944:in 1947 and, ultimately,
839:can be used to represent
4626:Squares in number theory
4537:polygonal number theorem
4286:) in Waring's problem".
4015:10.1112/plms/s3-52.3.445
3623:Kempner, Aubrey (1912).
3562:(1920). "Chapter VIII".
3323:not necessarily positive
2276:The first few values of
1028:is at least as large as
675:respectively denote the
4426:Ellison, W. J. (1971).
4321:Vaughan, R. C. (1997).
3560:Dickson, Leonard Eugene
3302:Waring–Goldbach problem
3055:and sufficiently large
2344:, the related quantity
965:{\displaystyle 2^{k}-1}
609:{\displaystyle g(6)=73}
570:{\displaystyle g(5)=37}
527:{\displaystyle g(4)=19}
96:, and it was solved by
4611:Additive number theory
4475:. Mineola, NY: Dover.
4009:. s3-52 (3): 445–463.
3724:Proc. Indian Acad. Sci
3543:
3468:
3269:
3179:
3093:
3042:
2935:
2299:
2267:
2247:
2166:
2132:
2112:
2088:
1985:
1962:
1088:
1022:
993:
966:
933:
906:
853:
833:
806:
779:
697:
669:
643:
610:
571:
528:
485:
484:{\displaystyle g(3)=9}
439:
385:
384:{\displaystyle g(2)=4}
344:
309:
274:
239:
238:{\displaystyle g(1)=1}
204:
184:
164:
135:
66:Hilbert–Waring theorem
4616:Mathematical problems
4415:(23), 637–642 (1959).
4400:(23), 109 pp. (1947).
4380:(54), 225–230 (1943).
3963:Annals of Mathematics
3722:(6) = 73".
3629:Mathematische Annalen
3590:Mathematische Annalen
3544:
3469:
3358:Mathematische Annalen
3314:Pollock's conjectures
3270:
3180:
3094:
3043:
2936:
2877:I. M. Vinogradov
2850:10 that requires 11.
2300:
2268:
2248:
2167:
2133:
2113:
2089:
1986:
1963:
1089:
1023:
994:
992:{\displaystyle 1^{k}}
967:
934:
932:{\displaystyle 2^{k}}
907:
854:
834:
832:{\displaystyle 1^{k}}
807:
805:{\displaystyle 2^{k}}
780:
698:
670:
668:{\displaystyle \{x\}}
644:
611:
572:
529:
486:
440:
386:
345:
310:
275:
240:
205:
185:
165:
136:
98:Joseph-Louis Lagrange
4497:Schnirelmann density
3771:(1), 203–204 (1862).
3478:
3403:
3202:
3106:
3077:
2951:
2886:
2651:= p(p − 1)/2;
2298:{\displaystyle g(k)}
2280:
2257:
2176:
2142:
2122:
2102:
1995:
1975:
1125:
1094:. This was noted by
1032:
1021:{\displaystyle g(k)}
1003:
976:
943:
916:
863:
843:
816:
789:
707:
687:
653:
627:
585:
546:
503:
460:
438:{\displaystyle g(4)}
420:
360:
319:
284:
249:
214:
194:
174:
163:{\displaystyle g(k)}
145:
125:
4390:(162), 1–71 (1989).
4300:1986IzMat..27..239K
3976:1939AnMat..40..731D
3881:1990MaCom..55..815K
3319:Sums of three cubes
2871:Using his improved
2352:) was studied with
1991:is known for which
703:. Given the number
395:'s 1621 edition of
102:four-square theorem
4428:"Waring's problem"
4045:10.1007/BF02392834
3936:10.1007/BF01482074
3920:Partitio Numerorum
3736:10.1007/BF03170721
3641:10.1007/BF01456723
3602:10.1007/BF01450913
3539:
3464:
3371:10.1007/bf01450405
3308:Subset sum problem
3265:
3175:
3089:
3038:
2931:
2418:, this shows that
2374:sufficiently large
2295:
2263:
2243:
2162:
2128:
2108:
2084:
1981:
1958:
1953:
1084:
1018:
999:. It follows that
989:
962:
929:
902:
849:
829:
802:
775:
693:
665:
639:
606:
567:
540:J.-M. Deshouillers
536:R. Balasubramanian
524:
481:
435:
381:
340:
305:
270:
235:
200:
180:
160:
131:
68:, was provided by
45:has an associated
38:asks whether each
4511:. Vol. 164.
4482:978-0-486-40026-6
4375:Mat. Sb., N. Ser.
4241:978-0-486-43878-8
4208:978-1-56881-152-9
4169:10.24033/msmf.413
3582:Wieferich, Arthur
2790:, respectively.)
2693:Upper bounds for
2670:
2669:
2662:for all integers
2572:
2571:
2568:25 ≤ G(20) ≤ 142
2563:20 ≤ G(19) ≤ 134
2558:27 ≤ G(18) ≤ 125
2553:18 ≤ G(17) ≤ 117
2548:64 ≤ G(16) ≤ 109
2543:16 ≤ G(15) ≤ 100
2453:Lower bounds for
2336:From the work of
2266:{\displaystyle k}
2131:{\displaystyle k}
2111:{\displaystyle k}
1984:{\displaystyle k}
1807:
1715:
1480:
1388:
1205:
852:{\displaystyle c}
696:{\displaystyle x}
203:{\displaystyle k}
183:{\displaystyle s}
134:{\displaystyle k}
16:(Redirected from
4633:
4597:
4584:"Waring problem"
4534:
4486:
4469:Khinchin, A. Ya.
4455:
4394:I. M. Vinogradov
4388:Acta Mathematica
4349:
4348:
4318:
4312:
4311:
4275:
4269:
4268:
4252:
4246:
4245:
4227:
4221:
4220:
4193:Vaughan, R. C.;
4190:
4181:
4180:
4152:
4146:
4145:
4143:
4141:10.5802/jtnb.287
4117:
4111:
4108:
4102:
4101:
4099:
4090:(229): 421–439.
4075:
4069:
4063:
4057:
4056:
4033:Acta Mathematica
4028:
4019:
4018:
4002:
3996:
3995:
3954:
3948:
3947:
3915:
3909:
3908:
3875:(192): 815–820.
3863:
3857:
3856:
3826:
3820:
3819:
3778:
3772:
3769:"Opera posthuma"
3762:
3756:
3755:
3715:
3709:
3708:
3687:
3681:
3680:
3659:
3653:
3652:
3620:
3614:
3613:
3578:
3572:
3571:
3556:
3550:
3548:
3546:
3545:
3540:
3538:
3537:
3516:
3515:
3503:
3502:
3490:
3489:
3473:
3471:
3470:
3465:
3463:
3462:
3441:
3440:
3428:
3427:
3415:
3414:
3397:
3391:
3390:
3345:
3274:
3272:
3271:
3266:
3184:
3182:
3181:
3176:
3098:
3096:
3095:
3090:
3047:
3045:
3044:
3039:
2940:
2938:
2937:
2932:
2859:
2849:
2845:
2844:
2841:
2835:
2831:
2830:
2827:
2821:
2820:
2817:
2814:
2805:
2804:
2798:
2797:
2789:
2788:
2785:
2782:
2779:
2773:
2772:
2769:
2763:
2756:
2755:
2752:
2749:
2746:
2740:
2721:
2720:
2717:
2711:
2688:
2666:greater than 1.
2587:
2586:
2538:15 ≤ G(14) ≤ 92
2533:14 ≤ G(13) ≤ 84
2528:16 ≤ G(12) ≤ 76
2523:12 ≤ G(11) ≤ 67
2518:12 ≤ G(10) ≤ 59
2465:
2435:
2424:
2413:
2394:
2372:such that every
2314:
2304:
2302:
2301:
2296:
2272:
2270:
2269:
2264:
2252:
2250:
2249:
2244:
2233:
2232:
2220:
2203:
2202:
2171:
2169:
2168:
2163:
2137:
2135:
2134:
2129:
2117:
2115:
2114:
2109:
2093:
2091:
2090:
2085:
2083:
2082:
2067:
2066:
2054:
2034:
2033:
2021:
2007:
2006:
1990:
1988:
1987:
1982:
1967:
1965:
1964:
1959:
1957:
1956:
1947:
1946:
1931:
1930:
1918:
1898:
1897:
1885:
1865:
1864:
1852:
1835:
1834:
1822:
1808:
1805:
1803:
1802:
1787:
1786:
1774:
1754:
1753:
1741:
1727:
1726:
1716:
1713:
1700:
1699:
1687:
1667:
1666:
1654:
1637:
1636:
1620:
1619:
1604:
1603:
1591:
1571:
1570:
1558:
1538:
1537:
1525:
1508:
1507:
1495:
1481:
1478:
1476:
1475:
1460:
1459:
1447:
1427:
1426:
1414:
1400:
1399:
1389:
1386:
1373:
1372:
1360:
1340:
1339:
1327:
1310:
1309:
1293:
1292:
1277:
1276:
1264:
1244:
1243:
1231:
1217:
1216:
1206:
1203:
1190:
1189:
1177:
1160:
1159:
1096:J. A. Euler
1093:
1091:
1090:
1085:
1074:
1073:
1061:
1044:
1043:
1027:
1025:
1024:
1019:
998:
996:
995:
990:
988:
987:
971:
969:
968:
963:
955:
954:
938:
936:
935:
930:
928:
927:
911:
909:
908:
903:
892:
891:
879:
858:
856:
855:
850:
838:
836:
835:
830:
828:
827:
811:
809:
808:
803:
801:
800:
784:
782:
781:
776:
774:
773:
752:
751:
739:
725:
724:
702:
700:
699:
694:
674:
672:
671:
666:
648:
646:
645:
640:
615:
613:
612:
607:
576:
574:
573:
568:
538:, F. Dress, and
533:
531:
530:
525:
490:
488:
487:
482:
444:
442:
441:
436:
390:
388:
387:
382:
349:
347:
346:
341:
314:
312:
311:
306:
279:
277:
276:
271:
244:
242:
241:
236:
209:
207:
206:
201:
189:
187:
186:
181:
169:
167:
166:
161:
140:
138:
137:
132:
47:positive integer
36:Waring's problem
21:
4641:
4640:
4636:
4635:
4634:
4632:
4631:
4630:
4601:
4600:
4582:
4579:
4554:Hans Rademacher
4523:
4513:Springer-Verlag
4502:
4483:
4467:
4444:10.2307/2317482
4425:
4362:A. A. Karatsuba
4357:
4352:
4337:
4320:
4319:
4315:
4277:
4276:
4272:
4254:
4253:
4249:
4242:
4229:
4228:
4224:
4209:
4192:
4191:
4184:
4154:
4153:
4149:
4119:
4118:
4114:
4109:
4105:
4080:"7373170279850"
4077:
4076:
4072:
4066:Nathanson (1996
4064:
4060:
4030:
4029:
4022:
4004:
4003:
3999:
3984:10.2307/1968889
3956:
3955:
3951:
3917:
3916:
3912:
3889:10.2307/2008448
3865:
3864:
3860:
3828:
3827:
3823:
3800:10.2307/2371901
3780:
3779:
3775:
3763:
3759:
3717:
3716:
3712:
3689:
3688:
3684:
3661:
3660:
3656:
3622:
3621:
3617:
3580:
3579:
3575:
3558:
3557:
3553:
3529:
3507:
3494:
3481:
3476:
3475:
3454:
3432:
3419:
3406:
3401:
3400:
3398:
3394:
3347:
3346:
3342:
3338:
3284:
3200:
3199:
3104:
3103:
3075:
3074:
2949:
2948:
2884:
2883:
2858:= 5, 6, ..., 20
2854:
2847:
2842:
2839:
2837:
2833:
2828:
2825:
2823:
2818:
2815:
2812:
2810:
2802:
2800:
2795:
2793:
2786:
2783:
2780:
2777:
2775:
2770:
2767:
2765:
2758:
2753:
2750:
2747:
2744:
2742:
2735:
2718:
2715:
2713:
2709:
2703:
2683:
2682:) should equal
2513:13 ≤ G(9) ≤ 50
2508:32 ≤ G(8) ≤ 42
2488:16 = G(4) = 16
2463:
2430:
2419:
2396:
2389:
2334:
2310:
2278:
2277:
2255:
2254:
2224:
2194:
2174:
2173:
2140:
2139:
2120:
2119:
2100:
2099:
2074:
2058:
2025:
1998:
1993:
1992:
1973:
1972:
1952:
1951:
1938:
1922:
1889:
1856:
1826:
1806: and
1794:
1778:
1745:
1718:
1710:
1691:
1658:
1628:
1625:
1624:
1611:
1595:
1562:
1529:
1499:
1479: and
1467:
1451:
1418:
1391:
1383:
1364:
1331:
1301:
1298:
1297:
1284:
1268:
1235:
1208:
1200:
1181:
1151:
1144:
1123:
1122:
1065:
1035:
1030:
1029:
1001:
1000:
979:
974:
973:
946:
941:
940:
919:
914:
913:
883:
861:
860:
841:
840:
819:
814:
813:
792:
787:
786:
765:
743:
716:
705:
704:
685:
684:
681:fractional part
651:
650:
625:
624:
583:
582:
544:
543:
501:
500:
458:
457:
445:is at most 53.
418:
417:
358:
357:
317:
316:
282:
281:
247:
246:
212:
211:
192:
191:
172:
171:
143:
142:
123:
122:
119:
82:
28:
23:
22:
15:
12:
11:
5:
4639:
4637:
4629:
4628:
4623:
4618:
4613:
4603:
4602:
4599:
4598:
4578:
4577:External links
4575:
4574:
4573:
4551:
4543:as the sum of
4521:
4500:
4481:
4465:
4423:
4416:
4401:
4391:
4381:
4368:
4365:
4356:
4353:
4351:
4350:
4335:
4313:
4294:(4): 935–947.
4270:
4263:(in Russian).
4247:
4240:
4222:
4207:
4195:Wooley, Trevor
4182:
4147:
4134:(2): 411–422.
4112:
4103:
4070:
4068:, p. 71).
4058:
4020:
3997:
3970:(4): 731–747.
3949:
3930:(1): 161–188.
3910:
3858:
3839:(2): 122–124.
3821:
3782:Niven, Ivan M.
3773:
3757:
3710:
3699:(5): 161–163.
3682:
3654:
3635:(3): 387–399.
3615:
3573:
3551:
3536:
3532:
3528:
3525:
3522:
3519:
3514:
3510:
3506:
3501:
3497:
3493:
3488:
3484:
3461:
3457:
3453:
3450:
3447:
3444:
3439:
3435:
3431:
3426:
3422:
3418:
3413:
3409:
3392:
3365:(3): 281–300.
3349:Hilbert, David
3339:
3337:
3334:
3333:
3332:
3326:
3316:
3311:
3305:
3299:
3298:-gonal numbers
3283:
3280:
3276:
3275:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3186:
3185:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3088:
3085:
3082:
3049:
3048:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2942:
2941:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2702:
2691:
2672:
2671:
2668:
2667:
2660:
2653:
2652:
2641:
2633:
2632:
2613:
2607:
2606:
2601:≥ 2, or
2591:
2570:
2569:
2565:
2564:
2560:
2559:
2555:
2554:
2550:
2549:
2545:
2544:
2540:
2539:
2535:
2534:
2530:
2529:
2525:
2524:
2520:
2519:
2515:
2514:
2510:
2509:
2505:
2504:
2503:8 ≤ G(7) ≤ 33
2500:
2499:
2498:9 ≤ G(6) ≤ 24
2495:
2494:
2493:6 ≤ G(5) ≤ 17
2490:
2489:
2485:
2484:
2480:
2479:
2475:
2474:
2470:
2469:
2462:
2451:
2333:
2322:
2321:
2320:
2294:
2291:
2288:
2285:
2262:
2242:
2239:
2236:
2231:
2227:
2223:
2219:
2215:
2212:
2209:
2206:
2201:
2197:
2193:
2190:
2187:
2184:
2181:
2161:
2157:
2153:
2150:
2147:
2127:
2107:
2081:
2077:
2073:
2070:
2065:
2061:
2057:
2053:
2049:
2046:
2043:
2040:
2037:
2032:
2028:
2024:
2020:
2016:
2013:
2010:
2005:
2001:
1980:
1969:
1968:
1955:
1950:
1945:
1941:
1937:
1934:
1929:
1925:
1921:
1917:
1913:
1910:
1907:
1904:
1901:
1896:
1892:
1888:
1884:
1880:
1877:
1874:
1871:
1868:
1863:
1859:
1855:
1851:
1847:
1844:
1841:
1838:
1833:
1829:
1825:
1821:
1817:
1814:
1811:
1801:
1797:
1793:
1790:
1785:
1781:
1777:
1773:
1769:
1766:
1763:
1760:
1757:
1752:
1748:
1744:
1740:
1736:
1733:
1730:
1725:
1721:
1711:
1709:
1706:
1703:
1698:
1694:
1690:
1686:
1682:
1679:
1676:
1673:
1670:
1665:
1661:
1657:
1653:
1649:
1646:
1643:
1640:
1635:
1631:
1627:
1626:
1623:
1618:
1614:
1610:
1607:
1602:
1598:
1594:
1590:
1586:
1583:
1580:
1577:
1574:
1569:
1565:
1561:
1557:
1553:
1550:
1547:
1544:
1541:
1536:
1532:
1528:
1524:
1520:
1517:
1514:
1511:
1506:
1502:
1498:
1494:
1490:
1487:
1484:
1474:
1470:
1466:
1463:
1458:
1454:
1450:
1446:
1442:
1439:
1436:
1433:
1430:
1425:
1421:
1417:
1413:
1409:
1406:
1403:
1398:
1394:
1384:
1382:
1379:
1376:
1371:
1367:
1363:
1359:
1355:
1352:
1349:
1346:
1343:
1338:
1334:
1330:
1326:
1322:
1319:
1316:
1313:
1308:
1304:
1300:
1299:
1296:
1291:
1287:
1283:
1280:
1275:
1271:
1267:
1263:
1259:
1256:
1253:
1250:
1247:
1242:
1238:
1234:
1230:
1226:
1223:
1220:
1215:
1211:
1201:
1199:
1196:
1193:
1188:
1184:
1180:
1176:
1172:
1169:
1166:
1163:
1158:
1154:
1150:
1149:
1147:
1142:
1139:
1136:
1133:
1130:
1100:Leonhard Euler
1083:
1080:
1077:
1072:
1068:
1064:
1060:
1056:
1053:
1050:
1047:
1042:
1038:
1017:
1014:
1011:
1008:
986:
982:
961:
958:
953:
949:
926:
922:
901:
898:
895:
890:
886:
882:
878:
874:
871:
868:
848:
826:
822:
799:
795:
772:
768:
764:
761:
758:
755:
750:
746:
742:
738:
734:
731:
728:
723:
719:
715:
712:
692:
664:
661:
658:
638:
635:
632:
605:
602:
599:
596:
593:
590:
566:
563:
560:
557:
554:
551:
523:
520:
517:
514:
511:
508:
480:
477:
474:
471:
468:
465:
434:
431:
428:
425:
380:
377:
374:
371:
368:
365:
339:
336:
333:
330:
327:
324:
304:
301:
298:
295:
292:
289:
269:
266:
263:
260:
257:
254:
234:
231:
228:
225:
222:
219:
199:
179:
159:
156:
153:
150:
130:
118:
107:
81:
78:
40:natural number
26:
24:
18:Waring problem
14:
13:
10:
9:
6:
4:
3:
2:
4638:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4608:
4606:
4595:
4591:
4590:
4585:
4581:
4580:
4576:
4571:
4570:0-691-02351-4
4567:
4563:
4559:
4558:Otto Toeplitz
4555:
4552:
4549:
4546:
4542:
4538:
4532:
4528:
4524:
4522:0-387-94656-X
4518:
4514:
4510:
4506:
4501:
4498:
4494:
4490:
4484:
4478:
4474:
4470:
4466:
4463:
4459:
4453:
4449:
4445:
4441:
4437:
4433:
4429:
4424:
4421:
4417:
4414:
4410:
4406:
4402:
4399:
4395:
4392:
4389:
4385:
4384:R. C. Vaughan
4382:
4379:
4376:
4372:
4371:Yu. V. Linnik
4369:
4366:
4363:
4359:
4358:
4354:
4346:
4342:
4338:
4336:0-521-57347-5
4332:
4328:
4324:
4317:
4314:
4309:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4274:
4271:
4267:(5): 637–642.
4266:
4262:
4258:
4251:
4248:
4243:
4237:
4233:
4226:
4223:
4218:
4214:
4210:
4204:
4200:
4196:
4189:
4187:
4183:
4178:
4174:
4170:
4166:
4162:
4158:
4151:
4148:
4142:
4137:
4133:
4129:
4128:
4123:
4116:
4113:
4107:
4104:
4098:
4093:
4089:
4085:
4081:
4074:
4071:
4067:
4062:
4059:
4054:
4050:
4046:
4042:
4038:
4034:
4027:
4025:
4021:
4016:
4012:
4008:
4001:
3998:
3993:
3989:
3985:
3981:
3977:
3973:
3969:
3965:
3964:
3959:
3958:Davenport, H.
3953:
3950:
3945:
3941:
3937:
3933:
3929:
3925:
3921:
3914:
3911:
3906:
3902:
3898:
3894:
3890:
3886:
3882:
3878:
3874:
3871:
3870:
3862:
3859:
3854:
3850:
3846:
3842:
3838:
3834:
3833:
3825:
3822:
3817:
3813:
3809:
3805:
3801:
3797:
3793:
3789:
3788:
3783:
3777:
3774:
3770:
3766:
3761:
3758:
3753:
3749:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3714:
3711:
3706:
3702:
3698:
3695:(in French).
3694:
3686:
3683:
3678:
3674:
3670:
3667:(in French).
3666:
3658:
3655:
3650:
3646:
3642:
3638:
3634:
3631:(in German).
3630:
3626:
3619:
3616:
3611:
3607:
3603:
3599:
3596:(1): 95–101.
3595:
3592:(in German).
3591:
3587:
3583:
3577:
3574:
3569:
3565:
3561:
3555:
3552:
3534:
3526:
3523:
3517:
3512:
3508:
3504:
3499:
3495:
3491:
3486:
3482:
3459:
3451:
3448:
3442:
3437:
3433:
3429:
3424:
3420:
3416:
3411:
3407:
3396:
3393:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3361:(in German).
3360:
3359:
3354:
3350:
3344:
3341:
3335:
3330:
3327:
3324:
3320:
3317:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3293:
3289:
3286:
3285:
3281:
3279:
3262:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3220:
3217:
3211:
3205:
3198:
3197:
3196:
3194:
3189:
3172:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3124:
3121:
3115:
3109:
3102:
3101:
3100:
3086:
3083:
3080:
3072:
3068:
3066:
3062:Applying his
3060:
3058:
3054:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2969:
2966:
2960:
2954:
2947:
2946:
2945:
2925:
2922:
2919:
2916:
2913:
2910:
2904:
2901:
2895:
2889:
2882:
2881:
2880:
2878:
2874:
2869:
2867:
2863:
2857:
2851:
2808:
2791:
2761:
2738:
2733:
2729:
2725:
2707:
2700:
2696:
2692:
2690:
2686:
2681:
2677:
2665:
2661:
2658:
2655:
2654:
2650:
2646:
2642:
2640:− 1)/2
2639:
2635:
2634:
2630:
2626:
2622:
2618:
2614:
2612:
2609:
2608:
2604:
2600:
2596:
2592:
2589:
2588:
2585:
2584:
2583:
2581:
2577:
2567:
2566:
2562:
2561:
2557:
2556:
2552:
2551:
2547:
2546:
2542:
2541:
2537:
2536:
2532:
2531:
2527:
2526:
2522:
2521:
2517:
2516:
2512:
2511:
2507:
2506:
2502:
2501:
2497:
2496:
2492:
2491:
2487:
2486:
2483:4 ≤ G(3) ≤ 7
2482:
2481:
2478:4 = G(2) = 4
2477:
2476:
2473:1 = G(1) = 1
2472:
2471:
2467:
2466:
2460:
2456:
2452:
2450:
2448:
2444:
2440:
2433:
2428:
2422:
2417:
2411:
2407:
2403:
2399:
2392:
2387:
2383:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2351:
2347:
2343:
2339:
2331:
2327:
2323:
2318:
2313:
2308:
2307:
2306:
2289:
2283:
2274:
2260:
2240:
2237:
2229:
2221:
2217:
2213:
2204:
2199:
2195:
2191:
2185:
2179:
2159:
2155:
2151:
2148:
2145:
2138:must satisfy
2125:
2105:
2097:
2079:
2075:
2071:
2063:
2055:
2051:
2047:
2038:
2030:
2022:
2018:
2014:
2003:
1999:
1978:
1948:
1943:
1939:
1935:
1927:
1919:
1915:
1911:
1902:
1894:
1886:
1882:
1878:
1869:
1861:
1853:
1849:
1845:
1831:
1823:
1819:
1815:
1799:
1795:
1791:
1783:
1775:
1771:
1767:
1758:
1750:
1742:
1738:
1734:
1723:
1719:
1707:
1704:
1696:
1688:
1684:
1680:
1671:
1663:
1655:
1651:
1647:
1638:
1633:
1629:
1621:
1616:
1612:
1608:
1600:
1592:
1588:
1584:
1575:
1567:
1559:
1555:
1551:
1542:
1534:
1526:
1522:
1518:
1504:
1496:
1492:
1488:
1472:
1468:
1464:
1456:
1448:
1444:
1440:
1431:
1423:
1415:
1411:
1407:
1396:
1392:
1380:
1377:
1369:
1361:
1357:
1353:
1344:
1336:
1328:
1324:
1320:
1311:
1306:
1302:
1294:
1289:
1285:
1281:
1273:
1265:
1261:
1257:
1248:
1240:
1232:
1228:
1224:
1213:
1209:
1197:
1194:
1186:
1178:
1174:
1170:
1161:
1156:
1152:
1145:
1140:
1134:
1128:
1121:
1120:
1119:
1117:
1113:
1109:
1105:
1101:
1098:, the son of
1097:
1081:
1078:
1070:
1062:
1058:
1054:
1045:
1040:
1036:
1012:
1006:
984:
980:
959:
956:
951:
947:
924:
920:
899:
896:
888:
880:
876:
872:
846:
824:
820:
797:
793:
770:
766:
762:
759:
756:
748:
740:
736:
732:
721:
717:
713:
710:
690:
682:
678:
659:
633:
621:
619:
603:
600:
594:
588:
580:
564:
561:
555:
549:
541:
537:
521:
518:
512:
506:
498:
497:A. J. Kempner
494:
478:
475:
469:
463:
454:
452:
448:
429:
423:
415:
410:
408:
404:
403:
398:
394:
378:
375:
369:
363:
355:
351:
337:
334:
328:
322:
302:
299:
293:
287:
267:
264:
258:
252:
232:
229:
223:
217:
197:
177:
154:
148:
128:
116:
112:
108:
106:
103:
99:
95:
91:
87:
77:
75:
71:
67:
63:
62:Edward Waring
59:
55:
51:
48:
44:
41:
37:
33:
32:number theory
19:
4587:
4561:
4547:
4544:
4540:
4504:
4492:
4488:
4472:
4461:
4457:
4438:(1): 10–36.
4435:
4431:
4419:
4412:
4408:
4404:
4397:
4387:
4377:
4374:
4322:
4316:
4291:
4287:
4283:
4279:
4273:
4264:
4260:
4250:
4231:
4225:
4198:
4160:
4156:
4150:
4131:
4125:
4115:
4106:
4087:
4083:
4073:
4061:
4036:
4032:
4006:
4000:
3967:
3961:
3952:
3927:
3923:
3919:
3913:
3872:
3867:
3861:
3836:
3830:
3824:
3791:
3785:
3776:
3760:
3727:
3723:
3719:
3713:
3696:
3692:
3685:
3671:(4): 85–88.
3668:
3664:
3657:
3632:
3628:
3618:
3593:
3589:
3576:
3563:
3554:
3395:
3362:
3356:
3343:
3322:
3295:
3291:
3277:
3192:
3190:
3187:
3064:
3061:
3056:
3052:
3050:
2943:
2870:
2855:
2852:
2809:
2792:
2759:
2736:
2731:
2727:
2723:
2705:
2704:
2698:
2694:
2684:
2679:
2675:
2673:
2663:
2656:
2648:
2644:
2637:
2631:− 1);
2628:
2624:
2620:
2616:
2610:
2602:
2598:
2594:
2579:
2575:
2573:
2458:
2454:
2446:
2442:
2438:
2431:
2429:showed that
2420:
2415:
2409:
2405:
2401:
2397:
2390:
2385:
2381:
2377:
2369:
2365:
2361:
2357:
2353:
2349:
2345:
2335:
2329:
2325:
2275:
1971:No value of
1970:
622:
579:Chen Jingrun
455:
416:showed that
411:
400:
352:
120:
114:
110:
83:
65:
57:
53:
49:
42:
35:
29:
3869:Math. Comp.
3832:Mathematika
2860:are due to
2574:The number
2384:. Clearly,
2324:The number
616:in 1940 by
577:in 1964 by
534:in 1986 by
402:Arithmetica
109:The number
4605:Categories
4550:th powers.
4531:0859.11002
4355:References
4345:0868.11046
2342:Littlewood
1112:Rubugunday
451:Littlewood
397:Diophantus
121:For every
86:Diophantus
4594:EMS Press
4177:0249-633X
4163:: 1–120.
4053:0001-5962
3944:0025-5874
3752:185097940
3730:: 30–40.
3649:120101223
3610:121386035
3524:−
3449:−
3387:179177986
3248:
3242:
3230:
3218:≤
3158:
3152:
3137:
3122:≤
3084:≥
3059:in 1959.
3030:
3024:
3018:
3003:
2997:
2982:
2967:≤
2917:
2902:≤
2605:= 3 × 2;
2597:= 2 with
2427:Davenport
2238:−
2235:⌋
2208:⌊
2069:⌋
2042:⌊
1933:⌋
1906:⌊
1900:⌋
1873:⌊
1867:⌋
1840:⌊
1837:⌋
1810:⌊
1789:⌋
1762:⌊
1705:−
1702:⌋
1675:⌊
1669:⌋
1642:⌊
1606:⌋
1579:⌊
1573:⌋
1546:⌊
1540:⌋
1513:⌊
1510:⌋
1483:⌊
1462:⌋
1435:⌊
1378:−
1375:⌋
1348:⌊
1342:⌋
1315:⌊
1282:≤
1279:⌋
1252:⌊
1195:−
1192:⌋
1165:⌊
1079:−
1076:⌋
1049:⌊
972:terms of
957:−
912:terms of
897:−
894:⌋
867:⌊
757:−
754:⌋
727:⌊
637:⌋
631:⌊
493:Wieferich
414:Liouville
335:≥
300:≥
265:≥
4564:(1933) (
4495:) using
4471:(1998).
4039:: 1–71.
3765:L. Euler
3584:(1909).
3351:(1909).
3282:See also
2434:(4) = 16
2414:for all
2395:. Since
677:integral
4596:, 2001
4452:2317482
4296:Bibcode
4217:1956283
3992:1968889
3972:Bibcode
3905:1035936
3897:2008448
3877:Bibcode
3853:0093509
3816:0009386
3808:2371901
3744:0002993
3705:0854724
3677:0853592
3379:1511530
3294:of the
2862:Vaughan
2762:(3) ≤ 7
2739:(3) = 4
2468:Bounds
2423:(2) = 4
2393:(2) ≥ 4
2315:in the
2312:A002804
1104:Dickson
785:, only
100:in his
70:Hilbert
4568:
4529:
4519:
4479:
4450:
4343:
4333:
4238:
4215:
4205:
4175:
4051:
3990:
3942:
3903:
3895:
3851:
3814:
3806:
3750:
3742:
3703:
3675:
3647:
3608:
3385:
3377:
2866:Wooley
2096:Mahler
1108:Pillai
618:Pillai
581:, and
407:Fermat
393:Bachet
315:, and
141:, let
4448:JSTOR
3988:JSTOR
3893:JSTOR
3804:JSTOR
3748:S2CID
3645:S2CID
3606:S2CID
3383:S2CID
3336:Notes
3325:cubes
3067:-adic
2726:and 2
2338:Hardy
2305:are:
1116:Niven
456:That
447:Hardy
4566:ISBN
4556:and
4517:ISBN
4477:ISBN
4411:)".
4331:ISBN
4236:ISBN
4203:ISBN
4173:ISSN
4049:ISSN
3940:ISSN
2864:and
2774:and
2712:10,
2404:) ≤
2360:).
2340:and
2317:OEIS
2149:>
2072:>
1936:>
1792:>
1465:>
939:and
812:and
763:<
679:and
649:and
623:Let
495:and
449:and
4527:Zbl
4440:doi
4341:Zbl
4304:doi
4165:doi
4136:doi
4092:doi
4041:doi
4037:162
4011:doi
3980:doi
3932:doi
3885:doi
3841:doi
3796:doi
3732:doi
3697:303
3669:303
3637:doi
3598:doi
3474:or
3367:doi
3245:log
3239:log
3227:log
3155:log
3149:log
3134:log
3087:400
3027:log
3021:log
3015:log
3000:log
2994:log
2979:log
2914:log
2843:617
2840:033
2829:724
2826:597
2824:617
2819:422
2816:904
2813:578
2803:793
2796:792
2787:850
2784:279
2781:170
2778:373
2771:740
2768:290
2754:850
2751:279
2748:170
2745:373
2719:740
2716:290
2687:+ 1
2659:+ 1
2643:if
2615:if
2593:if
2160:000
2156:600
2152:471
399:'s
190:of
30:In
4607::
4592:,
4586:,
4560:,
4525:.
4515:.
4507:.
4446:.
4436:78
4434:.
4430:.
4378:12
4339:.
4329:.
4302:.
4292:27
4290:.
4265:23
4259:.
4213:MR
4211:.
4185:^
4171:.
4159:.
4132:12
4130:.
4124:.
4088:69
4086:.
4082:.
4047:.
4035:.
4023:^
3986:.
3978:.
3968:40
3966:.
3938:.
3928:12
3926:.
3901:MR
3899:.
3891:.
3883:.
3873:55
3849:MR
3847:.
3835:.
3812:MR
3810:.
3802:.
3792:66
3790:.
3767:,
3746:.
3740:MR
3738:.
3728:12
3726:.
3701:MR
3673:MR
3643:.
3633:72
3627:.
3604:.
3594:66
3588:.
3527:31
3496:17
3483:29
3381:.
3375:MR
3373:.
3363:67
3355:.
3195:,
3167:12
3099::
2926:11
2875:,
2868:.
2838:51
2811:68
2801:13
2794:13
2689:.
2623:=
2425:.
2319:).
2273:.
2094:.
1714:if
1387:if
1204:if
1114:,
1110:,
1106:,
620:.
604:73
565:37
542:,
522:19
499:,
405:;
338:19
280:,
34:,
4548:k
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