On searching for solutions of the Diophantine equation x3 + y3 +2z3 = n
@article{Koyama2000OnSF, title={On searching for solutions of the Diophantine equation x3 + y3 +2z3 = n}, author={Kenji Koyama}, journal={Math. Comput.}, year={2000}, volume={69}, pages={1735-1742} }
We propose an efficient search algorithm to solve the equation x3 + y3 + 2z3 = n for a fixed value of n > 0. By parametrizing |z|, this algorithm obtains |x| and |y| (if they exist) by solving a quadratic equation derived from divisors of 2|z|3±n. Thanks to the use of several efficient numbertheoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of n below 1000 for which no solution had previously…
10 Citations
A New Method in the Problem of Three Cubes
- Mathematics
- 2017
In the current paper we are seeking P1(y); P2(y); P3(y) with the highest possible degree polynomials with integer coefficients, and Q(y) via the lowest possible degree polynomial, such that = Q(y).…
90.34 On triples of integers having the same sum and the same product
- MathematicsThe Mathematical Gazette
- 2006
x Sx + mx P = 0, where m is an arbitrary complex number. For each m, d, e and/ will be the roots of this equation. But the problem starts to be more interesting and difficult if we wish to have (a,…
New Solutions of $d=2x^3+y^3+z^3$
- Mathematics
- 2011
We discuss finding large integer solutions of $d=2x^3+y^3+z^3$ by using Elsenhans and Jahnel's adaptation of Elkies' LLL-reduction method. We find 28 first solutions for $|d|<10000$.
On sums of figurate numbers by using techniques of poset representation theory
- Mathematics
- 2008
We use representations and differentiation algorithms of posets, in order to obtain results concerning unsolved problems on figurate numbers. In particular, we present criteria for natural numbers…
A Complete Bibliography of Publications in Mathematics of Computation , 2010 – 2019
- Chemistry
- 1999
(1; e) [Sij12]. (2, 1) [SSV14]. (2k + 1)n≥1 [CLPM16]. (`, `) [CR15]. 1 [ABBM18, BFZ10, CZ11, DKMW13, DHMG11, EHR18, IPZ15, KMPW10, Ort11, Rau16, Sør16, Tao14, XZ10]. 1, 2, 3 [LWCI13]. 1, 2, 4…
Cracking the problem with 33
- MathematicsResearch 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…
Groupe de Brauer et points entiers de deux surfaces cubiques affines
- Mathematics
- 2009
Il est connu depuis Ryley [Ryl25] que tout entier, et meme tout nombre rationnel, peut s’ecrire comme somme de trois cubes de nombres rationnels. La question de savoir quels entiers s’ecrivent comme…
JSGrid: An Environment for Heterogenous Cluster Computing
- Computer ScienceSixth International Conference on Parallel and Distributed Computing Applications and Technologies (PDCAT'05)
- 2005
JSGrid makes good use of idle computational resources without installing any software, and enables various kinds of computers to be a part of a cluster, for instance, educational terminals at schools and universities, computers for development, office computers at companies and laboratories, and information retrieval terminals at libraries.
On partitions into four cubes
- Mathematics
- 2009
We use partially ordered sets (posets) and graphs in order to obtain a formula for the number of partitions of a positive integer n into four cubes with two of them equal.
異機種混合並列計算ミドルウェア JSGrid の開発と評価 ∗ Development and Evaluation of JSGrid : A Middleware for Parallel Computing with Heterogenous Cluster 武田和大 1 ,小野智司 1 ,中山 茂 1
- Computer Science
- 2006
Experimental results have shown that JSGrid could manage various operating systems such as Windows, Linux, and Mac OS, and keep high throughput almost equal to theoretical prediction even under a circumstance in which dynamic join and termination of computers occur.
References
SHOWING 1-10 OF 34 REFERENCES
On searching for solutions of the Diophantine equation x3 + y3 + z3 = n
- Mathematics, Computer ScienceMath. Comput.
- 1997
A new search algorithm to solve the equation x 3 + y 3 + z 3 = n for a fixed value of n > 0.5 is proposed, using several efficient number-theoretic sieves and much faster on average than previous straightforward algorithms.
Nonexistence conditions of a solution for the congruence x1k + ... + xsk = N (mod pn)
- MathematicsMath. Comput.
- 1999
We obtain nonexistence conditions of a solution for of the congruence x k 1 +... + x k s ≡ N (mod p n ), where k ≥ 2, s ≥ 2 and N are integers, and p n is a prime power. We give nonexistence…
A Note on the Diophantine Equation: x n + y n + z n = 3
- Mathematics
- 1984
In this note solutions for the Diophantine equation 3 v3 + 9 = 3 are sought along planes x + v + z = 3m, m E Z. This was done for Iml < 50000, and no new solutions were found.
Computational number theory at CWI in 1970--1994
- Mathematics
- 1994
A concise survey of the research in Computational Number Theory, carried out at CWI in the period 1970 to 1994, with updates to the present state-of-the-art of the various subjects, if necessary.
Solutions of the Diophantine equation
- Mathematics
- 1983
. Let ( F n ) n ≥ 0 be the Fibonacci sequence given by F 0 = 0 , F 1 = 1 and F n +2 = F n +1 + F n for n ≥ 0. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a…
Unsolved problems in number theory
- MathematicsPeriod. Math. Hung.
- 2001
The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences, arithmetic functions and the greatest prime factor func- tion and mixed problems.
On solving the Diophantine equation x3 + y3 + z3 = k on a vector processor
- Math. Comp
- 1993
Nonexistence conditions of a solution for the congruence x k 1 +· · ·+x k s ≡ N (mod p n ), to appear in Math
- Comp. MR
- 1999