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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.

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    $\begingroup$ ams.org/journals/mcom/2007-76-259/S0025-5718-07-01947-3/… $\endgroup$ – Will Jagy Aug 5 '15 at 23:42
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    $\begingroup$ From Wolfram: "... all numbers $N<1000$ and not of the form $9n \pm 4$ are known to be expressible as the sum $N=A^3+B^3+C^3$ of three (positive or negative) cubes with the exception of $N=33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921$, and $975$ (Miller and Woollett 1955; Gardiner et al. 1964; Guy 1994, p. 151; Mishima; Elsenhaus and Jahnel 2007). Examples include $30 = (-283059965)^3+(-2218888517)^3+2220422932^3$ ..." $\endgroup$ – 727 Aug 5 '15 at 23:42
  • $\begingroup$ @WillJagy - a great find! $\endgroup$ – hypergeometric Aug 7 '15 at 16:16
  • $\begingroup$ Table 2 in the paper cited by Jagy above needs an update. $\endgroup$ – Tito Piezas III Dec 8 '15 at 6:25
  • $\begingroup$ As of 2016, three solutions are now known for $N=30$, $$2220422932^3 - 283059965^3 - 2218888517^3 =30\\ 3982933876681^3 - 636600549515^3 - 3977505554546^3 = 30\\ -662037799708799^3 + 190809268841284^3 + 656711689254565^3 = 30$$ with the last found by S. Huisman. $\endgroup$ – Tito Piezas III Dec 28 '17 at 7:00
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($\color{red}{Update:}$ March 2019. The case $N=33$ 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. 2019

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

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

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,

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

Update: Numbers in red have been solved, so there are now just 12 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.

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    $\begingroup$ You might want to update your table as $N=74$ is solved. $$ (-284650292555885)^3 + 66229832190556^3 + 283450105697727^3 = 74 $$ $\endgroup$ – Yong Hao Ng Dec 28 '17 at 5:58
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    $\begingroup$ 33 is solved as well! $33=8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3$ $\endgroup$ – mkocabas Mar 10 at 6:22
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    $\begingroup$ According to Gil Kalai, credit for finding that solution for 33 is due to Andrew Booker. Booker documented the search in his paper Cracking the Problem with 33. $\endgroup$ – Rosie F Mar 11 at 16:42
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    $\begingroup$ @RosieF. I've looked again at Heath-Brown et al's 1993 paper and they explicitly say they found the first solution for $N=39$. And I looked at Lazarus et al's 1964 paper. Their range was only about 66,000, so they couldn't have found $N=39$ back then. So the possibility is that your copy of Gardner's Knotted Doughnuts is not the 1986 first printing. $\endgroup$ – Tito Piezas III Mar 12 at 1:39
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    $\begingroup$ @RosieF: Gardner in p. 229 of Knotted Doughnuts remarks "Some were not easy to come by, notably the expression for 87 in which each cube has four digits." I find it strange that he makes this remark, while ignoring 39 in the same table with solutions of six digits. It's almost as if 39 was added later. $\endgroup$ – Tito Piezas III Mar 12 at 2:36

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