• Corpus ID: 119130221

Newer sums of three cubes

@article{Huisman2016NewerSO,
  title={Newer sums of three cubes},
  author={Sander G. Huisman},
  journal={arXiv: Number Theory},
  year={2016}
}
  • S. Huisman
  • Published 26 April 2016
  • Mathematics
  • arXiv: Number Theory
The search of solutions of the Diophantine equation $x^3 + y^3 + z^3 = k$ for $k<1000$ has been extended with bounds of $|x|$, $|y|$ and $|z|$ up to $10^{15}$. The first solution for $k=74$ is reported. This only leaves $k=33$ and $k=42$ for $k<100$ for which no solution has yet been found. A total of 966 new solutions were found. 
Hilbert's Tenth Problem: Refinements and Variants
Hilbert’s 10th problem, stated in modern terms, is Find an algorithm that will, given p ∈ Z [ x 1 , . . . , x n ] , determine if there exists a 1 , . . . , a n ∈ Z such that p ( a 1 , . . . , a n ) =
Cracking the problem with 33
  • A. Booker
  • Mathematics
    Research in Number Theory
  • 2019
Inspired by the Numberphile video "The uncracked problem with 33" by Tim Browning and Brady Haran, we investigate solutions to $x^3+y^3+z^3=k$ for a few small values of $k$. We find the first known
1 Hilbert ’ s Tenth Problem
Hilbert’s 10th problem, stated in modern terms, is Find an algorithm that will, given p ∈ Z[x1, . . . , xn], determine if there exists a1, . . . , an ∈ Z such that p(a1, . . . , an) = 0. Davis,
Hilbert's Tenth Problem
This column is a short version of a long version of an article based on a blog. What? I give the complete history.
On a question of Mordell
TLDR
This paper concludes a 65-y search with an affirmative answer to Mordell’s question and strongly supports a related conjecture of Heath-Brown and makes several improvements to methods for finding integer solutions to x3+y3+z3=k for small values of k.
Existential refinement on the search of integer solutions for the diophantine equation $x^3+y^3+z^3=n$
We propose a new algorithm, call S.A.M to determinate the existence of the solutions for the equation x 3 + y 3 + z 3 = n for a fixed value n > 0 unknown.

References

SHOWING 1-10 OF 10 REFERENCES
New integer representations as the sum of three cubes
TLDR
A new algorithm is described for finding integer solutions to x 3 + y 3 + z 3 = k for specific values of k and this is used to find representations forvalues of k for which no solution was previously known, including k = 30 and k = 52.
Mathematics of Computation
For each prime p, let p# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n!± 1 and p#± 1 are known and have found two new primes of
Rational Points Near Curves and Small Nonzero |x3-y2| via Lattice Reduction
TLDR
A new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C is given, and its proof also yields new estimates on the distribution mod 1 of (cx)3/2 for any positive rational c.
Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines
Soit~$a$ un entier non nul. Si~$a$ n'est pas de la forme $9n\pm 4$ pour un $n \in {\bf Z}$, il n'y a pas d'obstruction de Brauer-Manin \`a l'existence d'une solution de l'\'equation $x^3+y^3+z^3=a$
The density of zeros of forms for which weak approximation fails
The weak approximation principal fails for the forms x + y + z = kw, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these
American Journal of Mathematics
  • Mathematics
    Nature
  • 1886
WE are glad to note that the successive parts now appear with praiseworthy regularity, and the arrival of our number can be predicted to a very close order of approximation. The volume opens with a
(1) Proceedings of the London Mathematical Society (2) Journal of the London Mathematical Society
  • W. B.
  • Economics
    Nature
  • 1927
THE London Mathematical Society now issues its transactions in two volumes a year, the Proceedings and the Journal. Vol. 25 of the Proceedings contains 31 technical papers on various branches of
New Solutions of d = 2x3 + y3 + z3
  • ArXiv e-prints, September
  • 2011
Sums of Three Cubes