zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On sums of three integral cubes. (English) Zbl 0808.11075
Andrews, George E. (ed.) et al., The Rademacher legacy to mathematics. The centenary conference in honor of Hans Rademacher, July 21-25, 1992, Pennsylvania State University, University Park, PA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 166, 285-294 (1994).
The authors report on calculations to find integral solutions of $x\sp 3+ y\sp 3+ z\sp 3=k$, for small values of $k$. Representations for $k= 39$, 84, 556 and 870 were discovered. Previous works [see {\it V. L. Gardiner}, {\it R. B. Lazarus} and {\it P. R. Stein}, Math. Comput. 18, 408-413 (1964; Zbl 0121.284), for example] had failed to find solutions for these $k$. A solution for $k=2$, not belonging to the family $6t\sp 3+1$, $-6t\sp 3+1$, $-6t\sp 2$ has also been found. Some of these results had been found independently by the reviewer, {\it W. M. Lioen} and {\it H. J. J. te Riele} [Math. Comput. 61, 235-244 (1993; Zbl 0783.11046)]. Different search strategies were used for different ranges of the variables, but in each case $x+y$ was first fixed, and the congruence $z\sp 3\equiv k \pmod {x+y}$ solved. The search region was somewhat complicated, and the largest solutions found had 10 digits. However the largest cube in the search region is roughly $\vert x\vert,\vert y\vert,\vert z\vert\leq 2\cdot 2\times 10\sp 6$. This compares with the cube $\vert x\vert,\vert y\vert,\vert z\vert\leq 1\cdot 3\times 10\sp 8$ covered by the reviewer et al. (loc. cit.) for the case $k=3$. Unfortunately the present paper does not specify the amount of computer time used. The paper concludes with the conjecture that, for fixed $k\not\equiv \pm 4\pmod 9$, the number of solutions in a cube of side $M$ grows as $c(k)\log M$. This agrees with the heuristics presented by the reviewer [Math. Comput. 59, 613-623 (1992; Zbl 0778.11017)]. For the entire collection see [Zbl 0798.00010].
MSC:
11Y50Computer solution of Diophantine equations
11P05Waring’s problem and variants
11D25Cubic and quartic diophantine equations