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Diophantine Equation--3rd Powers

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As a part of the study ofWaring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (g(3)=9), that every "sufficiently large" integer is a sum of no more than 7 positive cubes (Linnik 1943;G(3)<=7; although it is not known if 7 can be reduced), and that every integer is a sum of at most 5 signed cubes (eg(3)<=5; although it is not known if 5 can be reduced to 4).

Elkies (2010) settled the second case for even numbers by proving that everyeven number greater than 454 is the sum of at most seven positive cubes. Siksek (2015) subsequently proved thatall integers greater than 454 are the sum of at most seven positive cubes. The complete set of exceptional numbers requiring more than 7 positive cubes are 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, and 454 (OEISA018888), as conjectured by Jacobi (1851).

It is known that everyn can be written in the form

n=A^2+B^2-C^3.
(1)

Anelliptic curve of the formy^2=x^3+n forn an integer is known as aMordell curve.

The 3.1.2 equation

A^3=B^3+C^3
(2)

is a case ofFermat's last theorem withn=3. In fact, this particular case was known not to have any solutions long before the general validity ofFermat's last theorem was established. Thue showed that a Diophantine equationof the form

AX^3-BY^3=l
(3)

forA,B, andl integers, has only finite many solutions (Hardy 1999, pp. 78-79).

Miller and Woollett (1955) and Gardineret al.(1964) investigated integersolutions of

A^3+B^3+C^3=D,
(4)

i.e., numbers representable as the sum of three (positive or negative)cubicnumbers.

The general rational solution to the 3.1.3 equation

A^3=B^3+C^3+D^3
(5)

was found by Euler and Vieta (Hardy 1999, pp. 20-21; Dickson 2005, pp. 550-554). Hardy and Wright (1979, pp. 199-201) give a solution which can be based on the identities

a^3(a^3+b^3)^3=b^3(a^3+b^3)^3+a^3(a^3-2b^3)^3+b^3(2a^3-b^3)^3
(6)
a^3(a^3+2b^3)^3=a^3(a^3-b^3)^3+b^3(a^3-b^3)^3+b^3(2a^3+b^3)^3.
(7)

This is equivalent to the general 3.2.2 solution found by Ramanujan (Berndt 1994, pp. 54 and 107; Hardy 1999, p. 11, 68, and 237; Dickson 2005, pp. 500 and 554). A partial quadratic form identity was also given by Ramanujan (Berndt 1994, p. 56)

(3x^2+5xy-5y^2)^3+(4x^2-4xy+6y^2)^3+(5x^2-5xy-3y^2)^3 =(6x^2-4xy+4y^2)^3,
(8)

the first instance of which gives the nice equation3^3+4^3+5^3=6^3=216, which is one ofPlato's numbers. Such partial quadratic form parametrizations can be found using the identity

(ax^2+v_1xy+bwy^2)^3+(bx^2-v_1xy+awy^2)^3+(cx^2+v_2xy+dwy^2)^3+(dx^2-v_2xy+cwy^2)^3 =(a^3+b^3+c^3+d^3)(x^2+wy^2)^3,
(9)

wherev_1=-(c^2-d^2),v_2=a^2-b^2, andw=(a+b)(c+d), and is reduced to finding solutions toa^3+b^3+c^3+d^3=0 (or the sum may be any number of cubes), which is just a special case of an even more general identity (Piezas 2005).

The 22 smallest integer solutions are

3^3+4^3+5^3=6^3
(10)
1^3+6^3+8^3=9^3
(11)
3^3+10^3+18^3=19^3
(12)
7^3+14^3+17^3=20^3
(13)
4^3+17^3+22^3=25^3
(14)
18^3+19^3+21^3=28^3
(15)
11^3+15^3+27^3=29^3
(16)
2^3+17^3+40^3=41^3
(17)
6^3+32^3+33^3=41^3
(18)
16^3+23^3+41^3=44^3
(19)
3^3+36^3+37^3=46^3
(20)
27^3+30^3+37^3=46^3
(21)
29^3+34^3+44^3=53^3
(22)
12^3+19^3+53^3=54^3
(23)
15^3+42^3+49^3=58^3
(24)
22^3+51^3+54^3=67^3
(25)
36^3+38^3+61^3=69^3
(26)
7^3+54^3+57^3=70^3
(27)
14^3+23^3+70^3=71^3
(28)
34^3+39^3+65^3=72^3
(29)
38^3+43^3+66^3=75^3
(30)
31^3+33^3+72^3=76^3.
(31)

Other small solutions include

28^3+53^3+75^3=84^3
(32)
26^3+55^3+78^3=87^3
(33)
33^3+70^3+92^3=105^3
(34)
1^3+71^3+138^3=144^3
(35)
1^3+135^3+138^3=172^3
(36)
1^3+372^3+426^3=505^3
(37)
1^3+426^3+486^3=577^3
(38)
1^3+566^3+823^3=904^3
(39)
1^3+242^3+720^3=729^3
(40)
1^3+791^3+812^3=1010^3
(41)
1^3+236^3+1207^3=1210^3
(42)
1^3+575^3+2292^3=2304^3
(43)
1^3+1938^3+2820^3=3097^3
(44)
1^3+2676^3+3230^3=3753^3
(45)
1^3+1124^3+5610^3=5625^3
(46)
1^3+2196^3+5984^3=6081^3
(47)
1^3+1943^3+6702^3=6756^3
(48)
1^3+1851^3+8675^3=8703^3
(49)

(Fredkin 1972; Madachy 1979, pp. 124 and 141; Dutch). A database with sumz^3 for allz<1000000 is maintained by Wroblewski.

Other general solutions have been found by Binet (1841) and Schwering (1902), although Ramanujan's formulation is the simplest. No general solution givingallpositive integral solutions is known (Dickson 2005, pp. 550-561). Y. Kohmoto has found a3.1.3^9 solution,

2100000^3=2046000^3+882000^3+216000^3
(50)
=1979600^3+1145400^3+85000^3
(51)
=2081100^3+628110^3+1890^3
(52)
=2043150^3+901200^3+30450^3
(53)
=2002280^3+1072480^3+30360^3
(54)
=1960480^3+1199520^3+15200^3
(55)
=1948800^3+1229760^3+30240^3
(56)
=2078160^3+658812^3+13188^3
(57)
=2009112^3+1048040^3+13888^3.
(58)

3.1.4 equations include

11^3+12^3+13^3+14^3=20^3
(59)
5^3+7^3+9^3+10^3=13^3.
(60)

3.1.5 equations include

1^3+3^3+4^3+5^3+8^3=9^3
(61)
3^3+4^3+5^3+8^3+10^3=12^3,
(62)

and a 3.1.6 equation is given by

1^3+5^3+6^3+7^3+8^3+10^3=13^3.
(63)

The 3.2.2 equation

A^3+B^3=C^3+D^3
(64)

has a known parametric solution (Guy 1994, p. 140; Dickson 2005, pp. 550-554), and 10 solutions with sum<10^5,

1729=1^3+12^3=9^3+10^3
(65)
4104=2^3+16^3=9^3+15^3
(66)
13832=2^3+24^3=18^3+20^3
(67)
20683=10^3+27^3=19^3+24^3
(68)
32832=4^3+32^3=18^3+30^3
(69)
39312=2^3+34^3=15^3+33^3
(70)
40033=9^3+34^3=16^3+33^3
(71)
46683=3^3+36^3=27^3+30^3
(72)
64232=17^3+39^3=26^3+36^3
(73)
65728=12^3+40^3=31^3+33^3
(74)

(OEISA001235; Moreau 1898). The first number (Madachy 1979, pp. 124 and 141) in this sequence, the so-calledHardy-Ramanujan number, is associated with a story told about Ramanujan by G. H. Hardy, but was known as early as 1657 (Berndt and Bhargava 1993). The smallest number representable inn ways as a sum of cubes is called thenthtaxicab number.

Ramanujan gave a general solution to the 3.2.2 equation as

(alpha+lambda^2gamma)^3+(lambdabeta+gamma)^3=(lambdaalpha+gamma)^3+(beta+lambda^2gamma)^3
(75)

where

alpha^2+alphabeta+beta^2=3lambdagamma^2
(76)

(Berndt and Bhargava 1993; Berndt 1994, p. 107). Another form due to Ramanujan is

(A^2+7AB-9B^2)^3+(2A^2-4AB+12B^2)^3 =(2A^2+10B^2)^3+(A^2-9AB-B^2)^3.
(77)

Hardy and Wright (1979, Theorem 412) prove that there are numbers that are expressible as the sum of two cubes inn ways for anyn (Guy 1994, pp. 140-141). The proof is constructive, providing a method for computing such numbers: givenrationals numbersr ands, compute

t=(r(r^3+2s^3))/(r^3-s^3)
(78)
u=(s(2r^3+s^3))/(r^3-s^3)
(79)
v=(t(t^3-2u^3))/(t^3+u^3)
(80)
w=(u(2t^3-u^3))/(t^3+u^3).
(81)

Then

r^3+s^3=t^3-u^3=v^3+w^3
(82)

Thedenominators can now be cleared to produce an integer solution. Ifr/s is picked to be large enough, thev andw will bepositive. Ifr/s is still larger, thev/w will be large enough forv andw to be used as the inputs to produce a third pair, etc. However, the resulting integers may be quite large, even forn=2. E.g., starting with3^3+1^3=28, the algorithm finds

28=((28340511)/(21446828))^3+((63284705)/(21446828))^3,
(83)

giving

28·21446828^3=(3·21446828)^3+21446828^3
(84)
=28340511^3+63284705^3.
(85)

The numbers representable in three ways as a sum of two cubes (a3.2^3 equation) are

87539319=167^3+436^3=228^3+423^3=255^3+414^3
(86)
119824488=11^3+493^3=90^3+492^3=346^3+428^3
(87)
143604279=111^3+522^3=359^3+460^3=408^3+423^3
(88)
175959000=70^3+560^3=198^3+552^3=315^3+525^3
(89)
327763000=300^3+670^3=339^3+661^3=510^3+580^3
(90)

(Guy 1994, OEISA003825). Wilson (1997) found 32 numbers representable in four ways as the sum of two cubes (a3.2^4 equation). The first is

6963472309248=2421^3+19083^3=5436^2+18948^3 =10200^3+18072^3=13322^3+16630^3.
(91)

The smallest known numbers so representable are 6963472309248, 12625136269928, 21131226514944, 26059452841000, ... (OEISA003826). Wilson also found six five-way sums,

48988659276962496=38787^3+365757^3
(92)
=107839^3+362753^3
(93)
=205292^3+342952^3
(94)
=221424^3+336588^3
(95)
=231518^3+331954^3
(96)
490593422681271000=48369^3+788631^3
(97)
=233775^3+781785^3
(98)
=285120^3+776070^3
(99)
=543145^3+691295^3
(100)
=579240^3+666630^3
(101)
6355491080314102272=103113^3+1852215^3
(102)
=580488^3+1833120^3
(103)
=788724^3+1803372^3
(104)
=1150792^3+1690544^3
(105)
=1462050^3+1478238^3
(106)
27365551142421413376=167751^3+3013305^3
(107)
=265392^3+3012792^3
(108)
=944376^3+2982240^3
(109)
=1283148^3+2933844^3
(110)
=1872184^3+2750288^3
(111)
1199962860219870469632=591543^3+10625865^3
(112)
=935856^3+10624056^3
(113)
=3330168^3+10516320^3
(114)
=6601912^3+9698384^3
(115)
=8387550^3+8480418^3
(116)
111549833098123426841016=1074073^3+48137999^3
(117)
=8787870^3+48040356^3
(118)
=13950972^3+47744382^3
(119)
=24450192^3+45936462^3
(120)
=33784478^3+41791204^3,
(121)

and a single six-way sum

8230545258248091551205888 =11239317^3+201891435^3 =17781264^3+201857064^3 =63273192^3+199810080^3 =85970916^3+196567548^3 =125436328^3+184269296^3 =159363450^3+161127942^3.
(122)

A solution to the 3.4.4 equation is

2^3+3^3+10^3+11^3=1^3+5^3+8^3+12^3
(123)

(Madachy 1979, pp. 118 and 133).

3.6.6 equations also exist:

1^3+2^3+4^3+8^3+9^3+12^3=3^3+5^3+6^3+7^3+10^3+11^3
(124)
87^3+233^3+264^3+396^3+496^3+540^3
(125)
=90^3+206^3+309^3+366^3+522^3+523^3.
(126)

(Madachy 1979, p. 142; Chen Shuwen).

In 1756-1757, Euler (1761, 1849, 1915) gave a parametric solution to

A^3+B^3=C^2
(127)

as

A=3n^3+6n^2-n
(128)
B=-3n^3+6n^2+n
(129)
C=6n^2(3n^2+1),
(130)

althoughrelatively prime solutions require the use of fractional values ofn (Dickson 2005, p. 578). To avoid this, Euler also gave the solutions

A=4mn(3m^2-3mn+n^2)
(131)
B=(m-n)(3m-n)(3m^2+n^2)
(132)
C=(3m^2-n^2)(9m^4-18m^3n+18m^2n^2-6mn^3+n^4)
(133)

forGCD(A+B,A^2-AB+B^2)=1, and

A=3m^4+6m^2n^2-n^4
(134)
B=-3m^4+6m^2n^2+n^4
(135)
C=6mn(3m^4+n^4)
(136)

forGCD(A+B,A^2-AB+B^2)=3 (Dickson 2005, p. 579).

See also

Cannonball Problem,Cubic Number,Hardy-Ramanujan Number,Multigrade Equation,Super-d Number,Taxicab Number,Trimorphic Number,Waring's Problem

Portions of this entry contributed byTitoPiezas III

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References

Berndt, B. C.Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.Berndt, B. C. and Bhargava, S. "Ramanujan--For Lowbrows."Amer. Math. Monthly100, 645-656, 1993.Binet, J. P. M. "Note sur une question relative à la théorie des nombres."C. R. Acad. Sci. (Paris)12, 248-250, 1841.Dickson, L. E.History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005.Dutch, S. "Cubic Quartets witha,b,c<1000."http://www.uwgb.edu/dutchs/RECMATH/Cubesums.htm.Elkies, N. "Every Even Number Greater Than 454 Is the Sum of Seven Cubes." 21 Sep 2010.https://arxiv.org/abs/1009.3983.Euler, L.Novi. Comm. Acad. Petrop.6 (ad annos 1756-1757), p. 181, 1761.Euler, L.Comm. Arith. Coll.1, 207, 1849.Euler, L.Opera Omnia, Series Prima, Vol. 2. Leipzig, Germany: Teubner, p. 454, 1915.Fredkin, E. Item 58 in Beeler, M.; Gosper, R. W.; and Schroeppel, R.HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 23, Feb. 1972.http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item58.Gardiner, V. L.; Lazarus, R. B.; and Stein, P. R. "Solutions of the Diophantine Equationx^3+y^3=z^3-d."Math. Comput.18, 408-413, 1964.Guy, R. K. "Sums of Like Powers. Euler's Conjecture." §D1 inUnsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-144, 1994.Hardy, G. H.Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Jacobi, C. G. J. "Über die zusammensetzung der zahlen aus ganzen positiven cuben; nebst einer tabelle für die kleinste cubenanzahl, aus welcher jede zahl bis 12000 zusammengesetzt werden kann."J. reine angew. Math.42, 322-354, 1851.Koyama, K.; Tsuruoka, Y.; and Sekigawa, H. "On Searching for Solutions of the Diophantine Equationx^3+y^3+z^3=n."Math. Comput.66, 841-851, 1997.Kraus, A. "Sur l'équationa^3+b^3=c^p."Experim. Math.7, 1-13, 1998.Linnik, U. V. "On the Representation of Large Numbers as Sums of Seven Cubes."Mat. Sbornik N.S.12, 218-224, 1943.Madachy, J. S.Madachy's Mathematical Recreations. New York: Dover, 1979.Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers."http://euler.free.fr.Miller, J. C. P. and Woollett, M. F. C. "Solutions of the Diophantine Equationx^3+y^3+z^3=k."J. London Math. Soc.30, 101-110, 1955.Moreau, C. "Plus petit nombre égal à la somme de deux cubes de deux façons."L'Intermediaire Math.5, 66, 1898.Nagell, T. "The Diophantine Equationxi^3+eta^3+zeta^3=0 and Analogous Equations" and "Diophantine Equations of the Third Degree with an Infinity of Solutions." §65 and 66 inIntroduction to Number Theory. New York: Wiley, pp. 241-248, 1951.Piezas, T. "Ramanujan and the Cubic Equation3^3+4^3+5^3=6^3." 2005.http://www.geocities.com/titus_piezas/RamCube.pdf.Rivera, C. "Problems & Puzzles: Puzzle 048-p^3=a^3+b^3+c^3,pa,b,c Prime."http://www.primepuzzles.net/puzzles/puzz_048.htm.Schwering, K. "Vereinfachte Lösungen des Eulerschen Aufgabe:x^3+y^3+z^3+v^3=0."Arch. Math. Phys.2, 280-284, 1902.Shanks, D.Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 157, 1993.Siksek, S. "Every Integer Greater Than 454 Is the Sum of at Most Seven Positive Cubes." 30 Dec 2015.https://arxiv.org/abs/1505.00647.Sloane, N. J. A. SequencesA001235,A003825, andA018888 in "The On-Line Encyclopedia of Integer Sequences."Wilson, D. Personal communication, Apr. 17, 1997.Wroblewski, J. "Equal Sums of Powers--Tables."http://www.math.uni.wroc.pl/~jwr/eslp/tables.htm.

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Diophantine Equation--3rd Powers

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Piezas, Tito III andWeisstein, Eric W. "Diophantine Equation--3rd Powers." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/DiophantineEquation3rdPowers.html

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