Integer solutions to the equation $a^3+b^3+c^3=30$

The following problem was posed to me but I could not do much about it:

Determine if there are any integer solutions to the equation $a^3+b^3+c^3=30$

I made a computer search that shows that there are no integers $a,b,c$ such that $a^3+b^3+c^3=30$ and $|a|,|b|,|c|<51$

Thank you a lot.

Solution 1:

($\color{red}{Update:}$ March 2019. The case $N=33$ has been found.)

($\color{red}{Update:}$ September 2019. The case $N=42$ has been found.)

The equation

$$x^3+y^3+z^3 = N$$

has been oft-discussed in both MSE and MO. For example, see this, this, and this.

Searching a low range $|x,y,z|$ just won't do. It's quite interesting to see how search ranges have increased over the years using ever more clever algorithms.

I. 1955

Many $N\leqslant 100$ with search bound 3200.

J.C.P. Miller and M.F.C. Woollett. Solutions of the Diophantine Equation $x^3+y^3+z^3 = k$. J London Math Soc 30 (1955), p.111-113.

II. 1964

$$4271^3 -4126^3 -1972^3=87$$

with search bound 65536.

V.L. Gardiner; R.B. Lazarus; P.R. Stein. Solutions of the Diophantine Equation $x^3+y^3 = z^3 - d$. Mathematics of Computation, vol. 18, no. 87 (Jul 1964), pp.408-413.

III. 1992

$$134476^3+ 117367^3 -159380^3 = 39$$

$$40500964^3+ 22894759^3-42805979^3 = 84$$

B. Conn and L. Vaserstein, "On sums of three integral cubes". Penn State Department of Mathematics report PM 131 (1992). Contemporary Mathematics 166 (1994), p.285-294. MR1284068 (95g:11128)

IV. 1993

$$134476^3+ 117367^3 -159380^3 = 39$$

D. Heath-Brown, W. Lioen, and H. Te Riele, "On Solving the Diophantine Equation $x^3+y^3+z^3=k$ on a Vector Computer".

V. 1995

$$-435203231^3 +435203083^3 +4381159^3 = 75$$

Andrew Bremner. "On sums of three cubes". Canadian Mathematical Society Conference Proceedings 15 (1995), p.87-91.

VI. 1999

$$2220422932^3 -283059965^3 -2218888517^3 = 30$$

$$-61922712865^3+60702901317^3 +23961292454^3 = 52$$

Michael Beck, Eric Pine, Wayne Tarrant, and Kim Yarbrough Jensen (see p.18 of Noam Elkies, "Rational points near curves and small non-zero $|x^3-y^2|$ via lattice reduction").

VII. 2001

$$25585441403^3 + 47272468418^3 - 49649244505^3 = 834$$

with search bound $10^{11}$ found by D.J. Bernstein.

VIII. 2009

$$2322626411251^3 + 19868127639556^3 - 19878702430997^3 =894$$

with search bound $10^{14}$ found by A. Elsenhans and J. Jahnel.

IX. 2016

$$66229832190556^3 + 283450105697727^3 −284650292555885^3 = 74$$

with search bound $10^{15}$ found by S. Huisman.

X. Mar 2019

$$8866128975287528^3 -8778405442862239^3 -2736111468807040^3 = 33$$

with search bound $10^{16}$ found by Andrew Booker.

XI. Sep 2019

$$-80538738812075974^3+80435758145817515^3+12602123297335631^3 = 42$$

found by Andrew Booker at Bristol and Andrew Sutherland at MIT.

Papers

In "New integer representations as the sum of three cubes" (2007) by Beck, Pine, Tarrant, and Yarbrough-Jensen they give a list of 28 $N<1000$ with no $x,y,z$ decomposition.

In "New sums of three cubes" by A. Elsenhans and J. Jahnel (2009) this has been reduced to just 14 unsolved $N$ (also quoted in Mathworld) namely,

$$N = \color{red}{33, 42, 74}, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975$$

Update: Numbers in red have been solved, so there are now just 11 unsolved $N$. Hopefully, over the years, we can slowly complete this list.

Note: Relevant data are also given by Leonid Durman (inc. $x_1^4+x_2^4+x_3^4 = z^4$), by Mishima, while other solutions can be found in Elsenhans and Jahnel's site.