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Diophantine Equation--3rd Powers

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As a part of the study of Waring'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 every even number greater than 454 is the sum of at most seven positive cubes. Siksek (2015) subsequently proved that all 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 (OEIS A018888), as conjectured by Jacobi (1851).

It is known that every n can be written in the form

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

An elliptic curve of the form y^2=x^3+n for n an integer is known as a Mordell curve.

The 3.1.2 equation

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

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

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

for A, B, and l integers, has only finite many solutions (Hardy 1999, pp. 78-79).

Miller and Woollett (1955) and Gardiner et al. (1964) investigated integer solutions of

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

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

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 equation 3^3+4^3+5^3=6^3=216, which is one of Plato'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)

where v_1=-(c^2-d^2), v_2=a^2-b^2, and w=(a+b)(c+d), and is reduced to finding solutions to a^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 sum z^3 for all z<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 giving all positive integral solutions is known (Dickson 2005, pp. 550-561). Y. Kohmoto has found a 3.1.3^9 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)

(OEIS A001235; Moreau 1898). The first number (Madachy 1979, pp. 124 and 141) in this sequence, the so-called Hardy-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 in n ways as a sum of cubes is called the nth taxicab 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 in n ways for any n (Guy 1994, pp. 140-141). The proof is constructive, providing a method for computing such numbers: given rationals numbers r and s, 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)

The denominators can now be cleared to produce an integer solution. If r/s is picked to be large enough, the v and w will be positive. If r/s is still larger, the v/w will be large enough for v and w to be used as the inputs to produce a third pair, etc. However, the resulting integers may be quite large, even for n=2. E.g., starting with 3^3+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 (a 3.2^3 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, OEIS A003825). Wilson (1997) found 32 numbers representable in four ways as the sum of two cubes (a 3.2^4 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, ... (OEIS A003826). 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)

although relatively prime solutions require the use of fractional values of n (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)

for GCD(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)

for GCD(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 by Tito Piezas III

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References

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Cite this as:

Piezas, Tito III and Weisstein, Eric W. "Diophantine Equation--3rd Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquation3rdPowers.html

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