Enumerating solutions to

Author:
Daniel J. Bernstein

Journal:
Math. Comp. **70** (2001), 389-394

MSC (2000):
Primary 11Y50; Secondary 11D25, 11D41, 11P05, 11Y16

DOI:
https://doi.org/10.1090/S0025-5718-00-01219-9

Published electronically:
June 12, 2000

MathSciNet review:
1709145

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers satisfying . Numerical results include all small solutions to ; all small solutions to ; and the smallest positive integer that can be written in ways as a sum of two coprime cubes.

**1.**Svante Carlsson,*Average-case results on heapsort*, BIT**27**(1987), 2-17. MR**88b:68017****2.**Randy L. Ekl,*Equal sums of four seventh powers*, Mathematics of Computation**65**(1996), 1755-1756. MR**97a:11050****3.**Randy L. Ekl,*New results in equal sums of like powers*, Mathematics of Computation**67**(1998), 1309-1315. MR**98m:11023****4.**Noam D. Elkies,*On*, Mathematics of Computation**51**(1988), 825-835. MR**89h:11012****5.**Robert W. Floyd,*Algorithm 245: Treesort3*, Communications of the ACM**7**(1964), 701.**6.**Roger E. Frye,*Finding on the Connection Machine*,*in [15]*, 106-116.**7.**Richard K. Guy,*Unsolved problems in number theory*, second edition, Springer-Verlag, New York, 1994. MR**96e:11002****8.**D. R. Heath-Brown,*The density of zeros of forms for which weak approximation fails*, Mathematics of Computation**59**(1992), 613-623. MR**93a:11055****9.**Donald E. Knuth,*The art of computer programming, volume 3: sorting and searching*, Addison-Wesley, Reading, Massachusetts, 1973. MR**56:4281****10.**Donald E. Knuth,*The art of computer programming, volume 3: sorting and searching, second edition*, Addison-Wesley, Reading, Massachusetts, 1998.**11.**Leon J. Lander, Thomas R. Parkin,*Equal sums of biquadrates*, Mathematics of Computation**20**(1966), 450-451.**12.**Leon J. Lander, Thomas R. Parkin,*A counterexample to Euler's sum of powers conjecture*, Mathematics of Computation**21**(1967), 101-103. MR**36:3721****13.**Leon J. Lander, Thomas R. Parkin, John L. Selfridge,*A survey of equal sums of like powers*, Mathematics of Computation**21**(1967), 446-459. MR**36:5060****14.**John Leech,*Some solutions of Diophantine equations*, Proceedings of the Cambridge Philosophical Society**53**(1957), 778-780. MR**19:837f****15.**Joanne L. Martin, Stephen F. Lundstrom,*Supercomputing '88: proceedings, volume 2*, IEEE Computer Society Press, Silver Spring, Maryland, 1988.**16.**Emmanuel Peyre, Yuri Tschinkel,*Tamagawa numbers of diagonal cubic surfaces, numerical evidence*, this journal, previous article.**17.**E. Rosenstiel, J. A. Dardis, C. R. Rosenstiel,*The four least solutions in distinct positive integers of the Diophantine equation*, Bulletin of the Institute for Mathematics and its Applications**27**(1991), 155-157. MR**92i:11134****18.**Joseph H. Silverman,*Integer points and the rank of Thue elliptic curves*, Inventiones Mathematicae**66**(1982), 395-404. MR**83h:10036****19.**Joseph H. Silverman,*Integer points on curves of genus*, Journal of the London Mathematical Society**28**(1983), 1-7. MR**84g:10033****20.**Morgan Ward,*Euler's problem on sums of three fourth powers*, Duke Mathematical Journal**15**(1948), 827-837. MR**10:283f****21.**Ingo Wegener,*Bottom-up-heapsort, a new variant of heapsort, beating, on average, quicksort (if is not very small)*, Theoretical Computer Science**118**(1993), 81-98. MR**94c:68007****22.**John W. J. Williams,*Algorithm 232: Heapsort*, Communications of the ACM**7**(1964), 347-348.**23.**Aurel J. Zajta,*Solutions of the diophantine equation*, Mathematics of Computation**41**(1983), 635-659. MR**85d:11025**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11Y50,
11D25,
11D41,
11P05,
11Y16

Retrieve articles in all journals with MSC (2000): 11Y50, 11D25, 11D41, 11P05, 11Y16

Additional Information

**Daniel J. Bernstein**

Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249) The University of Illinois at Chicago, Chicago, IL 60607–7045

Email:
djb@pobox.com

DOI:
https://doi.org/10.1090/S0025-5718-00-01219-9

Received by editor(s):
July 10, 1998

Received by editor(s) in revised form:
January 4, 1999

Published electronically:
June 12, 2000

Additional Notes:
The author was supported by the National Science Foundation under grant DMS–9600083.

Article copyright:
© Copyright 2000
D. J. Bernstein