Sum of three cube numbers

Single-logarithmic graph of the solutions to the equation x ³  + y ³  + z ³  = n with integer x , y and z , and n from [0, 100]. Green bars mean that there are no proven solutions for these problems .

Which property must an integer have so that it can be represented as the sum of three cube numbers and with integer bases ? For a given number, what are the possible number triples and such that is satisfied? How many solutions are there for a given number ? ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle x^{3},\ y^{3}}$ ${\displaystyle z^{3}}$ ${\displaystyle x,y,z\in \mathbb {Z} }$
${\displaystyle n}$ ${\displaystyle x,\ y}$ ${\displaystyle z}$ ${\displaystyle n=x^{3}+y^{3}+z^{3}}$ ${\displaystyle n}$

The solution of this Diophantine equation for a given is an unsolved problem of number theory for 160 years . ${\displaystyle n}$

Solutions to the equation

Representations for n  = 0

The simplest trivial representation for the sum of three cube numbers is: ${\displaystyle n=0}$

${\displaystyle 0=0^{3}+0^{3}+0^{3}}$.

Other trivial representations are:

${\displaystyle 0=0^{3}+a^{3}+(-a)^{3}}$  with  . ${\displaystyle a\in \mathbb {Z} }$

Non-trivial representations do not exist.

Proof:

Suppose there is a nontrivial representation of the shape with . If you move to the right side, you get an integer solution for the equation with . However, this contradicts Fermat's Great Theorem applied to cubic numbers , which says that the equation has no solutions for positive integers . Thus the assumption has to be dropped, which means that there cannot be a nontrivial representation of the form .  ∎  ${\displaystyle 0=x^{3}+y^{3}+z^{3}}$ ${\displaystyle x,y,z\not =0}$ ${\displaystyle z^{3}}$ ${\displaystyle x^{3}+y^{3}=-z^{3}}$ ${\displaystyle x^{3}+y^{3}=z'^{3}}$ ${\displaystyle z'=-z}$ ${\displaystyle x^{3}+y^{3}=z^{3}}$ ${\displaystyle x,y,z\in \mathbb {Z} }$ ${\displaystyle x^{3}+y^{3}+z^{3}=0}$

Representations for n  = 1

The trivial representation for the sum of three cubic numbers is: ${\displaystyle n=1}$

${\displaystyle 1=0^{3}+0^{3}+1^{3}}$

The simplest nontrivial representations are:

${\displaystyle 1=1^{3}+b^{3}+(-b)^{3}}$  with  . ${\displaystyle b\in \mathbb {Z} \setminus 0}$

But there are also other representations, such as B .:

${\displaystyle 1=(-6)^{3}+(-8)^{3}+9^{3}}$

${\displaystyle ...}$

${\displaystyle 1=131769^{3}+(-131802)^{3}+11980^{3}}$

A more complicated family of representations is that discovered by the mathematician Kurt Mahler in 1936 :

${\displaystyle 1=(9b^{4})^{3}+(3b-9b^{4})^{3}+(1-9b^{3})^{3}}$  with  . ${\displaystyle b\in \mathbb {Z} }$

as well as the following:

${\displaystyle 1=(1-9b^{3}+648b^{6}+3888b^{9})^{3}+(-135b^{4}+3888b^{10})^{3}+(3b-81b^{4}-1296b^{7}-3888b^{10})^{3}}$  With  ${\displaystyle b\in \mathbb {Z} }$

For Lehmer delivered an infinite number of polynomial solution families. Be ${\displaystyle n=1}$

${\displaystyle (x_{0},y_{0},z_{0})=(9b^{4},3b-9b^{4},1-9b^{3})}$,

${\displaystyle (x_{1},y_{1},z_{1})=(9b^{4},-3b-9b^{4},1+9b^{3})}$.

Now applies

${\displaystyle x_{k+1}=2(216b^{6}-1)x_{k}-x_{k-1}-108b^{4}}$,

${\displaystyle y_{k+1}=2(216b^{6}-1)y_{k}-y_{k-1}-108b^{4}}$,

${\displaystyle z_{k+1}=2(216b^{6}-1)z_{k}-z_{k-1}+216b^{4}+4}$.

Representations for n  = 2

The trivial representation for the sum of three cubic numbers is: ${\displaystyle n=2}$

${\displaystyle 2=0^{3}+1^{3}+1^{3}}$.

A nontrivial representation family discovered in 1908 is

${\displaystyle 2=(1+6b^{3})^{3}+(1-6b^{3})^{3}+(-6b^{2})^{3}}$  with  . ${\displaystyle b\in \mathbb {Z} }$

Other well-known representations that do not belong to the above family are:

${\displaystyle {\begin{array}{lrcrcr}2=&1214928^{3}&+&3480205^{3}&+&(-3528875)^{3}\\2=&37404275617^{3}&+&(-25282289375)^{3}&+&(-33071554596)^{3}\\2=&3737830626090^{3}&+&1490220318001^{3}&+&(-3815176160999)^{3}\end{array}}}$

Representations for n  = 3

Until September 2019, the only known representations for the sum of three cubic numbers were as follows: ${\displaystyle n=3}$

${\displaystyle 3=1^{3}+1^{3}+1^{3}}$  and

${\displaystyle 3=4^{3}+4^{3}+(-5)^{3}}$

Surprisingly, another representation was discovered in September 2019:

${\displaystyle 3=569936821221962380720^{3}+(-569936821113563493509)^{3}+(-472715493453327032)^{3}}$

One does not know whether there are only three, finitely many or infinitely many representations for . ${\displaystyle n=3}$

Representations for n  = 4 and 5

There are no solutions for and . ${\displaystyle n=4}$ ${\displaystyle 5}$

Representations for n  = 6

There are several representations; those for are: ${\displaystyle |x|\leq |y|\leq |z|\leq 1000}$

${\displaystyle 6=(-1)^{3}+(-1)^{3}+2^{3}}$

${\displaystyle 6=(-43)^{3}+(-58)^{3}+65^{3}}$

${\displaystyle 6=(-55)^{3}+(-235)^{3}+236^{3}}$

${\displaystyle 6=(-205)^{3}+(-637)^{3}+644^{3}}$

Representations for n  = 7

There are several representations; those for are: ${\displaystyle |x|\leq |y|\leq |z|\leq 1000}$

${\displaystyle 7=0^{3}+(-1)^{3}+2^{3}}$

${\displaystyle 7=32^{3}+104^{3}+(-105)^{3}}$

${\displaystyle 7=44^{3}+168^{3}+(-169)^{3}}$

Constructible solutions for n  =  k ³ m

Represented as the product of a cube number and a number , that number inherits all the solutions to the number in the following way: ${\displaystyle n}$ ${\displaystyle k^{3}}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m}$

${\displaystyle m=x^{3}+y^{3}+z^{3}\quad \longrightarrow \quad n=k^{3}m=k^{3}(x^{3}+y^{3}+z^{3})\quad \longrightarrow \quad n=k^{3}m=(kx)^{3}+(ky)^{3}+(kz)^{3}}$

example

${\displaystyle 1=(-6)^{3}+(-8)^{3}+9^{3}\quad \longrightarrow \quad 8=(-12)^{3}+(-16)^{3}+18^{3}}$

Smallest representations for n  = 0 to 107

The following table contains, for each smallest solutions of the equation with , : ${\displaystyle 0\leq n\leq 107,\ n\in \mathbb {N} _{0}}$ ${\displaystyle n=x^{3}+y^{3}+z^{3}}$ ${\displaystyle |x|\leq |y|\leq |z|}$ ${\displaystyle \ x,y,z\in \mathbb {Z} }$

A more extensive table can be found in the discussion section of this article .

Chronology of the discoveries

1954

Miller and Woolet found 69 of the 78 possible solutions for brute force searches of all combinations . ${\displaystyle 1\leq n\leq 100}$ ${\displaystyle |x|\leq |y|\leq |z|\leq 2164}$

The solutions to the nine numbers and remained unknown . ${\displaystyle 30,\ 33,\ 39,\ 42,\ 52,\ 74,\ 75,\ 84}$ ${\displaystyle 87}$

The last 5 of the 69 decompositions found are:

${\displaystyle 70=11^{3}+20^{3}+(-21)^{3}}$

${\displaystyle 96=14^{3}+20^{3}+(-22)^{3}}$

${\displaystyle 79=(-19)^{3}+(-33)^{3}+35^{3}}$

${\displaystyle 78=26^{3}+53^{3}+(-55)^{3}}$

${\displaystyle 51=602^{3}+659^{3}+(-796)^{3}}$

1963

Gardiner, Lazarus and Stein continued to search with and for . ${\displaystyle |x|\leq |y|\leq 2^{16}}$ ${\displaystyle 0\leq z-x\leq 2^{16}}$ ${\displaystyle 1\leq n\leq 1000}$

For they found the following further solution: ${\displaystyle n\leq 100}$

${\displaystyle 87=(-1972)^{3}+(-4126)^{3}+4271^{3}}$

For they found 708 of the 778 solutions. ${\displaystyle n\leq 1000}$

The last 5 of the 708 decompositions found are:

${\displaystyle 978=8666^{3}+40169^{3}+(-40303)^{3}}$

${\displaystyle 402=37685^{3}+41378^{3}+(-49915)^{3}}$

${\displaystyle 583=(-17419)^{3}+(-48223)^{3}+(48969)^{3}}$

${\displaystyle 227=24579^{3}+51748^{3}+(-53534)^{3}}$

${\displaystyle 971=7423^{3}+55643^{3}+(-55687)^{3}}$

1992

Heath-Brown , Lioen and te Riele found the following further solution:

${\displaystyle 39=117367^{3}+134476^{3}+(-159380)^{3}}$

1994

Conn and Vaseršteĭn found the following further solution:

${\displaystyle 84=(-8241191)^{3}+(-41531726)^{3}[phone]^{3}}$

1999

For were already known for 75 different solutions. ${\displaystyle n\leq 100}$ ${\displaystyle n}$

There were also:

${\displaystyle 75=4381159^{3}[phone]^{3}+(-435203231)^{3}}$

${\displaystyle 30=(-283059965)^{3}+(-2218888517)^{3}[phone]^{3}}$

${\displaystyle 52=23961292454^{3}[phone]^{3}+(-61922712865)^{3}}$

The only thing missing was the solutions for and . ${\displaystyle n=33,\ 42}$ ${\displaystyle 74}$

For they found 751 of the 778 solutions. ${\displaystyle n\leq 1000}$

2007

were only missing for the following representations between and above: ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle 1000}$

${\displaystyle 33,\ 42,\ 74,\ 114,\ 165,\ 390,\ 579,\ 627,\ 633,\ 732,\ 795,\ 906,\ 921}$ and ${\displaystyle 975}$

2016

the problem was solved for by Sander Huisman: ${\displaystyle n=74}$

${\displaystyle 74=(-284650292555885)^{3}[phone]^{3}[phone]^{3}}$

2019

the problem was solved for by the mathematician Andrew Booker through massive use of computers: ${\displaystyle n=33}$

${\displaystyle 33=8866128975287528^{3}+(-8778405442862239)^{3}+(-2736111468807040)^{3}}$

September 2019

the problem was solved for the last remaining number , namely for also by Andrew Booker and the mathematician Andrew Sutherland: ${\displaystyle n\leq 100}$ ${\displaystyle n=42}$

${\displaystyle 42=(-80538738812075974)^{3}+80435758145817515^{3}+12602123297335631^{3}}$

Since the last unsolved problem was up for this type of equation, a connection was made with answer 42 from the multi-filmed novel and radio play series The Hitchhiker's Guide to the Galaxy by the English author Douglas Adams for fun . ${\displaystyle n=42}$ ${\displaystyle n\leq 100}$

until mid-2020

three more cases were resolved, also by Andrew Booker and Andrew Sutherland:

${\displaystyle 165=(-385495523231271884)^{3}+383344975542639445^{3}+98422560467622814^{3}}$

${\displaystyle 795=(-14219049725358227)^{3}+14197965759741571^{3}+2337348783323923^{3}}$

${\displaystyle 906=(-74924259395610397)^{3}+72054089679353378^{3}+35961979615356503^{3}}$

A representation as the sum of three cube numbers is therefore only unknown for the following eight values (as of June 1, 2020): ${\displaystyle n\leq 1000}$

${\displaystyle 114,\ 390,\ 579,\ 627,\ 633,\ 732,\ 921}$ and ${\displaystyle 975}$

At the moment the equation is the one with the smallest natural one , for which no integer solution is known. ${\displaystyle 114=x^{3}+y^{3}+z^{3}}$ ${\displaystyle n\in \mathbb {N} _{0}}$

properties

• Be solvable as an integer. Then is a necessary condition for the following: ${\displaystyle n=x^{3}+y^{3}+z^{3}}$ ${\displaystyle n}$

${\displaystyle n\not \equiv \pm 4{\pmod {9}}}$

Unfortunately, it is not known whether this property is also sufficient (then the hitherto unsolved problem of number theory to which this article is dedicated would be solved). However, it was suggested by Heath-Brown that the Diophantine equation has an infinite number of integer solutions for all . ${\displaystyle n}$ ${\displaystyle n=x^{3}+y^{3}+z^{3}}$ ${\displaystyle n\not \equiv \pm 4}$

• There are some special relationships between and , such as the following: ${\displaystyle x}$ ${\displaystyle n}$

Be solvable as an integer. Given the following conditions apply to : ${\displaystyle n=x^{3}+y^{3}+z^{3}}$ ${\displaystyle n}$ ${\displaystyle x}$

If is, must apply: or . ${\displaystyle n\equiv +2{\pmod {7}}}$ ${\displaystyle x\equiv 0,+1,+2}$ ${\displaystyle +4{\pmod {7}}}$

If is, must apply: or . ${\displaystyle n\equiv -2{\pmod {7}}}$ ${\displaystyle x\equiv 0,-1,-2}$ ${\displaystyle -4{\pmod {7}}}$

If is, must apply: or . ${\displaystyle n\equiv +3{\pmod {7}}}$ ${\displaystyle x\equiv +1,+2}$ ${\displaystyle +4{\pmod {7}}}$

If is, must apply: or . ${\displaystyle n\equiv -3{\pmod {7}}}$ ${\displaystyle x\equiv -1,-2}$ ${\displaystyle -4{\pmod {7}}}$

When is must apply: . ${\displaystyle n\equiv +2{\pmod {9}}}$ ${\displaystyle x,y,z\not \equiv +2{\pmod {3}}}$

When is must apply: . ${\displaystyle n\equiv -2{\pmod {9}}}$ ${\displaystyle x,y,z\not \equiv -2{\pmod {3}}}$

When is must apply: . ${\displaystyle n\equiv +3{\pmod {9}}}$ ${\displaystyle x,y,z\equiv +1{\pmod {3}}}$

When is must apply: . ${\displaystyle n\equiv -3{\pmod {9}}}$ ${\displaystyle x,y,z\equiv -1{\pmod {3}}}$

Each smallest representations for n  = 0 to 91 of the OEIS found

The following describes how the smallest solutions for larger n can be found in the lists Sequence A060464 in OEIS ... Sequence A060467 in OEIS . The four lists each contain in the same order the values ​​for n , x , y and z for values ​​of n for which a solution exists and is known. The solution is always included. ${\displaystyle |x|\leq |y|\leq |z|}$

Episode A060464 in OEIS contains the : ${\displaystyle n}$

00, 01, 02, 03,  06, 07, 08,

09, 10, 11, 12, 15, 16, 17,

18, 19, 20, 21,  24 , 25, 26,

27, 28, 29, 30, 33, 34, 35,

36, 37, 38, 39, 42, 43, 44, ...

Episode A060465 in OEIS contains the : ${\displaystyle x}$

0, 0, 0, 1, −1, 0, 0,

0, 1, −2, 7, −1, −511, 1,

−1, 0, 1, −11, −2901096694 , −1, 0,

0, 0, 1, −283059965, −2736111468807040, −1, 0,

1, 0, 1, 117367, 12602123297335631, 2, −5, ...

Episode A060466 in OEIS contains the : ${\displaystyle y}$

0, 0, 1, 1, −1, −1, 0,

1, 1, −2, 10, 2, −1609, 2,

−2, −2, −2, −14, −15550555555 , −1, −1,

0, 1, 1, −2218888517, −8778405442862239, 2, 2,

2, −3, −3, 134476, 80435758145817515, 2, −7,

Episode A060467 in OEIS contains the : ${\displaystyle z}$

0, 1, 1, 1, 2, 2, 2,

2, 2, 3, −11, 2, 1626, 2,

3, 3, 3, 16, 15584139827 , 3, 3,

3, 3, 3, 2220422932, 8866128975287528, 3, 3,

3, 4, 4, −159380, −80538738812075974, 3, 8, ...

Example for n  = 24, the 19th entry

In each of the four lists above, the 19th entry was marked in bold. The values ​​are:

n = 24

x = −2901096694

y = −15550555555

z = 15584139827

The smallest possible representation for n = 24 is thus:

${\displaystyle 24=(-2901096694)^{3}+(-15550555555)^{3}[phone]^{3}}$

Trivia

• For always exist solutions. For a given number and a freely selectable parameter one obtains solutions e.g. B. by: ${\displaystyle x,y,z\in \mathbb {Q} }$ ${\displaystyle n>0}$ ${\displaystyle b=\mathbb {Z} \setminus \{0\}}$

${\displaystyle x={\frac {(9b^{6}-30n^{2}b^{3}+n^{4})(3b^{3}+n^{2})+72n^{4}b^{3}}{6bn\ (3b^{3}+n^{2})^{2}}}}$

${\displaystyle y={\frac {30n^{2}b^{3}-9b^{6}-n^{4}}{6bn\ (3b^{3}+n^{2})}}}$

${\displaystyle z={\frac {18nb^{5}-6n^{3}b^{2}}{(3b^{3}+n^{2})^{2}}}}$

• As soon as it is one of the bases , any solutions can be constructed directly without detours: ${\displaystyle x,y,z}$ ${\displaystyle \in \mathbb {R} }$

${\displaystyle x^{3}+y^{3}+z^{3}=n\ \longrightarrow \ x={\sqrt[{3}]{n-y^{3}-z^{3}}}}$,   given, any ${\displaystyle n}$ ${\displaystyle y,z}$

Web links

• W. Conn, LN Vaseršteĭn : On Sums of Three Integral Cubes. Contemporary Mathematics 166 , March 1992, pp. 1–11 , accessed September 19, 2019 . 
• Roger Heath-Brown , Herman te Riele , Walter M. Lioen: On solving the Diophantine equation x³ + y³ + z³ = k on a vector computer. Mathematics of Computation 61 (203), July 1993, pp. 235–244 , accessed September 19, 2019 . 
• Kenji Koyama: Tables of solutions of the Diophantine equation x³ + y³ + z³ = n . Mathematics of Computation 62 (206), April 1994, pp. 941-942 , accessed September 24, 2019 . 
• Eric Rowland: Koyama's table of integer solutions of n = x³ + y³ + z³ . Retrieved September 24, 2019 (5417 solutions from n = 2 to 999 , of which 521 solutions from n = 2 to 100 ). 
• Erik Dofs: Solution of x³ + y³ + z³ = nxyz . Acta Arithmetica LXXIII.3, 1995, pp. 201-213 , accessed September 19, 2019 . 
• Kenji Koyama, Yukio Tsuruoka, Hiroshi Sekigawa: On searching for solutions of the diophantine equation x³ + y³ + z³ = n . Mathematics of Computation 66 (218), April 1997, pp. 841-851 , accessed September 19, 2019 . 
• Eric Rowland: Known families of integer solutions of x³ + y³ + z³ = n . February 28, 2005, pp. 1–6 , accessed September 19, 2019 . 
• Michael Beck, Eric Pine, Wayne Tarrant, Kin Yarbrough Jensen: New Integer Representations as the Sum of three Cubes. Mathematics of Computation 76 (259), July 2007, pp. 1683–1690 , accessed September 19, 2019 . 
• Sander G. Huisman: Newer Sums of three Cubes. April 26, 2016, pp. 1–3 , accessed September 19, 2019 . 
• Armen Avagyan, Gurgen Dallakyan: A new method in the problem of three cubes. Armenian State Pedagogical University after Khachatur Abovyan, February 21, 2018, pp. 1–23 , accessed September 19, 2019 . 

Individual evidence

1. ↑ ^{a b c d e f g h} Armen Avagyan, Gurgen Dallakyan: A new method in the problem of three cubes. Armenian State Pedagogical University after Khachatur Abovyan, February 21, 2018, pp. 1–23 , accessed September 18, 2019 . 
2. ↑ ^{a b} Eric S. Rowland: Known families of integer solutions of x ^ 3 + y ^ 3 + z ^ 3 = n . ( psu.edu [PDF]).
3. ↑ Mark McAndrew: Insanely huge Sum-Of-Three-Cubes für 3 discovered - After 66 year search. Twitter, September 16, 2018, accessed September 18, 2019 . 
4. ↑ Hisanori Mishima: Solutions of n = x³ + y³ + z³, 0 <= n <= 99. Retrieved September 18, 2019 . 
5. ↑ Tito Piezas III: Integer solutions to the equation a³ + b³ + c³ = 30. Retrieved September 18, 2019 . 
6. ^ ^{A b} Sander G. Huisman: Newer Sums of three Cubes. April 26, 2016, pp. 1–3 , accessed September 19, 2019 . 
7. ↑ W. Conn, LN Vaseršteĭn : On Sums of Three integral cubes. Contemporary Mathematics 166 , March 1992, pp. 1–11 , accessed September 19, 2019 . 
8. ↑ Eric Rowland: Koyama's table of integer solutions of n = x³ + y³ + z³ . Retrieved September 24, 2019 (5417 solutions from n = 2 to 999 , of which 521 solutions from n = 2 to 100 ). 
9. ↑ DJ Bernstein: threecubes. Retrieved September 29, 2019 (other solutions). 
10. ↑ Andrew R. Booker, Cracking the problem with 33rd University of Bristol , 2019, pp. 1-6 , accessed September 18, 2019 . 
11. ↑ Lance Fortnow, Bill Gasarch: x³ + + Y³ z³ = 33 has a solution in Z. And its big! Computational Complexity.org, April 28, 2019, accessed September 18, 2019 . 
12. ↑ ^{a b} Robin Houston: 42 is the answer to the question "what is (-80538738812075974) ³ + 80435758145817515³ + 12602123297335631³?" The Aperiodical, September 6, 2019, accessed September 18, 2019 . 
13. ↑ Michelle Starr: Mathematicians Solve '42' Problem With Planetary Supercomputer. science alert, September 9, 2019, accessed September 18, 2019 . 
14. ↑ Mathematicians solve puzzles around the number 42
15. ^ DR Heath-Brown: The Density of Zeros of forms for which weak approximation fails . tape 59 , no. 200 . mathematics of computation, October 1992, p. 613-623 ( ams.org [PDF]).
16. ↑ Kenji Koyama, Yukio Tsuruoka, Hiroshi Sekigawa: On searching for solutions of the diophantine equation x³ + y³ + z³ = n , Property 1 and 2. Mathematics of Computation 66 (218), April 1997, pp. 843-844 , accessed on September 28, 2019 .