-------------------------------------------------
$${Published}$$  $${Online}$$  $${First}$$  $${(15/2/2024)}$$
$${Latest}$$  $${additions}$$  $${(15/2/2024)}$$
-------------------------------------------------
$${x^3+y^3+z^3=n}$$
$${n  \not\equiv \pm 4  (mod   9)}$$
$${n  \in  \mathbb{Z}}$$
-------------------------------------------------
$${REFERENCES}$$
-------------------------------------------------
今回は以下の論文を転記しました。

$${Armen~Avagyan, Gurgen~Dallakyan}$$

$${「A~New~Method~in~the~Problem\\of~Three~Cubes」}$$

$${\small Universal~Journal~of~Computational\\Mathematics~5(3):(p~45-56), 2017}$$
-------------------------------------------------
Abstract
In the current paper we are seeking $${P_1(y),P_2(y),P_3(y)}$$with the highest possible degree polynomials with integer coefficients, and $${Q(y)}$$ via the lowest possible degree polynomial, such that $${P_1^3(y)+P_2^3(y)+P_3^3(y) = Q(y)}$$.
Actually, the solution of this problem has close relation with the problem of the sum of three cubes $${a^3 + b^3 + c^3 = d, }$$ since deg $${Q(y) = 0}$$ case coincides with above mentioned problem.
It has been considered estimation of possibility of minimization of deg $${Q(y)}$$.
As a conclusion, for specific values of d we survey a new algorithm for finding integer solutions of $${a^3 + b^3 + c^3 = d}$$.

-------------------------------------------------
1.  Some facts on the history of the equation
$${a^3 +b^3 +c^3 =d}$$

The question as to which integers are expressible as a sum of three integer cubes is over $${160}$$ years old.
The first known reference to this problem was made by Fermat, who has offered to find the three nonzero integers, so that the sum of their nth powers is equal to zero.

This framework of the problem makes the beginning of survey of equation
$${a^3 + b^3 + c^3 = d}$$ for many mathematicians.
We present some significant result schedule.

$${1825}$$ year
S. Ryley in [1] gave a parametrization of rational solutions for$${ d ∈ \mathbb {Z}}$$;

$${\small x =\dfrac{(9k^6 − 30d^2k^3 + d^4)(3k^3 + d^2) + 72d^4k^3}{6kd(3k^3 + d^2)^2}}$$

$${\small y =\dfrac{30d^2k^3 −9k^6 −d^4}{6kd(3k^3 + d^2)}}$$

$${\small z =\dfrac{18dk^5 − 6d^3k^2}{(3k^3 + d^2)^2}}$$

$${1908}$$year
A.S. Werebrusov in [2] found the following parametric family for $${d = 2}$$:
$${(6t^3+1)^3 −(6t^3−1)^3 −(6t^2)^3=2}$$

$${1936}$$year
Later in [3] Mahler discovered a first parametric solution for $${d = 1}$$:
$${(9t^4)^3 +(3t−9t^4)^3 +(1−9t^3)^3=1}$$

$${1942}$$year
Mordell in [2] proved that for any other d a parametric solution with rational coefficients must have degree at least $${5}$$.

$${1954}$$year
Miller and Woollett [4] discovered explicit representations for $${69}$$ values of d between $${1}$$and $${100}$$. Their search exhausted the region $${|a|, |b|, |c| \leqq 3164.}$$

$${1963}$$ year
$${1963}$$ Gardiner, Lazarus, and Stein [5] looked at the equation $${x^3+y^3=z^3-d}$$ in the range $${0 \leq x \leq y \leq 2^{16}}$$, where $${0 \leq (z−x) \leq 2^{16}}$$ and $${0 \leq |d| \leq 999}$$.
Their search left only $${70}$$ values of d between $${1}$$ and $${1000}$$ without a known representation including eight values less than $${100}$$.

$${1992}$$ year
the first solution for $${d = 39}$$ was found. Heath-Brown, Lioen, and te Riele [8] determined that
$${39 = 134476^3 + 117367^3 + (−159380)^3}$$
with the rather deep algorithm of Heath-Brown [6]. This algorithm involved searching for solutions for a specific value of d using the class number of $${Q(3d)}$$ to eliminate values of $${a,b,c}$$ which would not yield a solution.

$${1994}$$ year
Koyama [7] used modern computers to expand the search region to$${ |a|, |b|, |c| \leq 221}$$ and successfully found first solutions for $${16}$$ integers between $${100}$$ and $${1000}$$ [9]. Also in $${1994}$$, Conn and Vaserstein [8] chose specific values of d to target, and then used relations implied by each chosen value to limit the number of triples$${(a, b, c)}$$ searched. So doing, they found first representations for $${84}$$ and $${960}$$. Their paper also lists a solution for each $${d < 100}$$ for which a representation was known.

$${1995}$$ year
Bremner [9] devised an algorithm which uses elliptic curve arguments to narrow the search space. He discovered a solution for $${75}$$ (and thus a solution for $${600}$$), leaving only five values less than $${100}$$ for which no solution was known. Lukes then extended this search method to also find the first representations for each of the values $${110, 435,}$$ and $${478}$$ [10].

$${1997}$$ year
Koyama, Tsuruoka, and Sekigawa [11] used a new algorithm to find first solutions for five more values between $${100}$$ and $${1000}$$ as well as independently finding the same solution for $${75}$$ that Bremner found. Also in the same paper, the authors discuss the complexity of the above algorithms.

$${1999}$$ year – Bernstein [12] had implemented the method of Elkies [13] and found solutions for $${11}$$ new values of d. Summarizing the above, it can be noted that up to $${21}$$st century only $${27}$$ values were left unresolved. These, together with the range of their search, are presented in the table below.

Only recently, in $${2007}$$, Elsenhans and Yahnel [14] found the solutions for the values
$${\small d = 156, 318, 366, 420, 564, 758, 789, 894, 948.}$$
$${\begin{aligned}\small 156 = &\small 26577110807569^3-18161093358005^3\\&\small -23381515025762^3 \end{aligned}}$$

$${\begin{aligned}\small 318 = &\small 47835963799^3 + 20549442727^3\\&\small -49068024704^3 \end{aligned}}$$

$${\begin{aligned}\small 318 = &\small 1970320861387^3 + 1750553226136^3\\ &\small − 2352152467181^3 \end{aligned}}$$

$${\begin{aligned}\small 318 = &\small 30828727881037^3 + 27378037791169^3 \\ &\small − 36796384363814^3 \end{aligned}}$$

$${\begin{aligned}\small 366 = &\small 241832223257^3 + 167734571306^3\\ &\small − 266193616507^3 \end{aligned}}$$

$${\begin{aligned}\small 420 = &\small 8859060149051^3 − 2680209928162^3\\ &\small − 8776520527687^3 \end{aligned}}$$

$${\begin{aligned}\small 564 = &\small 53872419107^3 − 1300749634^3\\ &\small − 53872166335^3 \end{aligned}}$$

$${\begin{aligned}\small 758 = &\small 662325744409^3 + 109962567936^3\\ &\small − 663334553003^3 \end{aligned}}$$

$${\begin{aligned}\small 789 = &\small 18918117957926^3 + 4836228687485^3 \\ &\small − 19022888796058^3 \end{aligned}}$$

$${\begin{aligned}\small 894 = &\small 19868127639556^3 + 2322626411251^3 \\ &\small − 19878702430997^3 \end{aligned}}$$

$${\begin{aligned}\small 948 = &\small 323019573172^3 + 63657228055^3 \\ &\small − 323841549995^3 \end{aligned}}$$

$${\begin{aligned}\small 948 = &\small 103458528103519^3+6604706697037^3\\ &\small − 103467499687004^3 \end{aligned}}$$

Thus, until $${1000}$$ there are only the numbers $${33, 42, 114, 165, 390, 579, 627, 633, 732,\\795, 906, 921, 975}$$ lefts, which have not yet been solved, all the other presentations are posted on the web, in particular, it was made by Sander Huisman [15].
-------------------------------------------------
$${March 26, 2019}$$
A number theorist with programming prowess has found a solution to $${33 = x³ + y³ + z³}$$, a much-studied equation that went unsolved for $${64}$$ years.
Andrew Booker, a mathematician at the University of Bristol, has finally cracked it: He discovered that
$${\begin{aligned}\small 33= &\small 8866128975287528^3 \\&\small–8778405442862239^3\\ &\small–2736111468807040^3 \end{aligned}}$$

$${\small September 6, 2019}$$
Booker and Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology, found a sum of three cubes for $${42}$$:
$${\begin{aligned}\small 42= &\small -80538738812075974^3 \\ &\small + 80435758145817515^3\\ &\small + 12602123297335631^3 \end{aligned}}$$
This leaves 114 as the lowest unsolved case.

$${\begin{aligned}\small 165= &\small-385495523231271884^3 \\ &\small + 383344975542639445^3 \\ &\small + 98422560467622814^3 \end{aligned}}$$

$${\begin{aligned}\small 579= &\small 143075750505019222645^3 \\ &\small -143070303858622169975^3 \\ &\small -6941531883806363291^3 \end{aligned}}$$

$${\begin{aligned}\small 906= &\small-74924259395610397^3 \\ &\small + 72054089679353378^3 \\ &\small + 35961979615356503^3 \end{aligned}}$$
-------------------------------------------------
残る未解決は$${1000}$$以下では
$${114, 390, 627, 633, 732, 795, 921, 975}$$
の$${8}$$個$${(Feb/15/2024}$$現在$${)}$$
-------------------------------------------------
続きは別の機会に…
-------------------------------------------------
$${REFERENCES}$$
-------------------------------------------------
$${L. J. Mordell\\Diophantine~Equations\\Academic~Press, London, 1969}$$

$${Armen~Avagyan, Gurgen~Dallakyan\\A~New~Method~in~the~Problem\\of~Three~Cubes\\Universal~Journal~of~\\Computational~Mathematics\\5(3): (p~45-56), 2017}$$
-------------------------------------------------
[1] S. Ryley, The Ladies’ Diary 122 (1825), 35.
[2] L.J. Mordell, On Sums of Three Cubes, Journal of the London Mathemat- ical Society 17 (1942), 139-144. MR0007761 (4:189d) .
[3] Kurt Mahler, Note On Hypothesis K of Hardy and Littlewood, Journal of the London Mathematical Society 11 (1936), 136-138.
[4] J.C.P. Miller and M.F.C. Woollett, Solution of the Diophantine Equation x3 + y3 + z3 = k, Journal of the London Mathematical Society 30 (1955), 101-110. MR0067916 (16:797e).
[5] V.L. Gardiner, R.B. Lazarus, and P.R. Stein, Solutions of the Diophantine Equation x3 +y3 = z3 −d, Mathematics of Computation 18 (1964), 408-413. MR0175843 (31:119).
[6] D.R. Heath-Brown, W.M. Lioen, and H.J.J. te Riele, On Solving the Dio- phantine Equation x3 + y3 + z3 = k on a Vector Computer, Mathematics of Computation 61 (1993), 235-244. MR1202610 (94f:11132).
[7] Kenji Koyama, Tables of solutions of the Diophantine equation x3 + y3 + z3 = n, Mathematics of Computation 62 (1994), 941-942.
[8] W. Conn and L.N. Vaserstein, On Sums of Three Integral Cubes, Contem- porary Mathematics 166 (1994), 285-294. MR1284068 (95g:11128).
[9] Andrew Bremner, On sums of three cubes, Canadian Mathematical Society Conference Proceedings 15 (1995), 87-91. MR1353923 (96g:11024).
[10] Richard F. Lukes, A Very Fast Electronic Number Sieve, University of Manitoba doctoral thesis, 1995.
[11] Kenji Koyama, Yukio Tsuruoka and Hiroshi Sekigawa, On Searching for Solutions of the Diophantine Equation x3 + y3 + z3 = n, Mathematics of Computation 66 (1997), 841-851. MR1401942 (97m:11041).
[12] Bernstein, D., Three cubes, available at: http://cr.yp.to/threecubes.html.
[13] Noam Elkies,¡elkies@abel.math.harvard.edu¿ ”x3 + y3 + z3 = d”, 9 July 1996,nmbrthry@listserv.nodak.edu via ¡http://listserv.nodak.edu/archives/nmbrthry.html¿.
[14] http://www.uni-math.gwdg.de/jahnel/Preprints/elk ants6c.pdf .
[15] Sander G. Huisman, Newer sums of three cubes, archive.org.
[16] Payne G., Vaserstein L.N., Sums of three cubes. Pages 443-454 in The Arithmetic of Function Fields, de Gruyter, 1992.
[17] Elkies, N. D., Rational points near curves and small nonzero |x3 − y2| via lattice reduction, in: Algorithmic number theory (Leiden 2000), Lecture Notes in Computer Science 1838, Springer, Berlin 2000, 33-63. MR1850598 (2002g:11035).
[18] Beck M., Pine E., Tarrant W. and Yarbrough Jensen K.: New integer representations as the sum of three cubes, Math. Comp. 76 (2007), 1683- 1690. MR2299795 (2007m:11170).23
-------------------------------------------------