On Mordell's equation : a problem of Stolarsky
Author:
Ray P. Steiner
Journal:
Math. Comp. 46 (1986), 703714
MSC:
Primary 11D25; Secondary 1104, 11Y50
MathSciNet review:
829640
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: On page 1 of his book Algebraic Numbers and Diophantine Approximation, K. B. Stolarsky posed the problem of solving the equation in positive integers. In the present paper we refine some techniques of Ellison and Pethö and show that the complete set of integer solutions of Stolarsky's equation is and
 [1]
V.
I. Baulin, On an indeterminate equation of the third degree with
least positive discriminant, Tul′sk. Gos. Ped. Inst.
Učen. Zap. Fiz.Mat. Nauk Vyp. 7 (1960),
138–170 (Russian). MR 0199149
(33 #7298)
 [2]
A.
I. Borevich and I.
R. Shafarevich, Number theory, Translated from the Russian by
Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press,
New YorkLondon, 1966. MR 0195803
(33 #4001)
 [3]
B.
N. Delone and D.
K. Faddeev, The theory of irrationalities of the third degree,
Translations of Mathematical Monographs, Vol. 10, American Mathematical
Society, Providence, R.I., 1964. MR 0160744
(28 #3955)
 [4]
W. J. Ellison, "Recipes for solving Diophantine problems by Baker's method," Publ. Mathématiques, v. Ann. 1, Fasc. 1, 1972.
 [5]
W.
J. Ellison, F.
Ellison, J.
Pesek, C.
E. Stahl, and D.
S. Stall, The Diophantine equation
𝑦²+𝑘=𝑥³, J. Number Theory
4 (1972), 107–117. MR 0316376
(47 #4923)
 [6]
Ove
Hemer, On the solvability of the Diophantine equation
𝑎𝑥²+𝑏𝑦²+𝑐𝑧²=0
in imaginary Euclidean quadratic fields, Ark. Mat. 2
(1952), 57–82. MR 0049917
(14,247d)
 [7]
Wilhelm
Ljunggren, Einige Bemerkungen über die Darstellung ganzer
Zahlen durch binäre kubische Formen mit positiver Diskriminante,
Acta Math. 75 (1943), 1–21 (German). MR 0017303
(8,135k)
 [8]
A.
Pethő, Full cubes in the Fibonacci sequence, Publ.
Math. Debrecen 30 (1983), no. 12, 117–127. MR 733078
(85j:11027)
 [9]
V.
G. Sprindzhuk, Klassicheskie diofantovy uravneniya ot dvukh
neizvestnykh, “Nauka”, Moscow, 1982 (Russian). MR 685430
(85d:11022)
 [10]
Kenneth
B. Stolarsky, Algebraic numbers and Diophantine approximation,
Marcel Dekker, Inc., New York, 1974. Pure and Applied Mathematics, No. 26.
MR
0374041 (51 #10241)
 [11]
Nicholas
Tzanakis, The Diophantine equation
𝑥³3𝑥𝑦²𝑦³=1 and related
equations, J. Number Theory 18 (1984), no. 2,
192–205. MR
741950 (86d:11023), http://dx.doi.org/10.1016/0022314X(84)900532
 [12]
Michel
Waldschmidt, A lower bound for linear forms in logarithms,
Acta Arith. 37 (1980), 257–283. MR 598881
(82h:10049)
 [1]
 V. I. Baulin, "On the indeterminate equation of the third degree with least positive discriminant," Tulsk. Gos. Ped. Inst., Učen. Zap. Fiz. Mat. Nauk, v. 7, 1960, pp. 138170. (Russian) MR 0199149 (33:7298)
 [2]
 Z. I. Borevich & R. I. Shafarevich, Number Theory, English Transl., Academic Press, New York, 1966. MR 0195803 (33:4001)
 [3]
 B. N. Delone & D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Math. Mono., no. 10, Amer. Math. Soc., Providence, RI, 1964. MR 0160744 (28:3955)
 [4]
 W. J. Ellison, "Recipes for solving Diophantine problems by Baker's method," Publ. Mathématiques, v. Ann. 1, Fasc. 1, 1972.
 [5]
 W. J. Ellison, J. F. Ellison, J. Pesek, C. E. Stahl & D. S. Stahl, "The Diophantine equation ," J. Number Theory, v. 4, 1972, pp. 107117. MR 0316376 (47:4923)
 [6]
 O. Hemer, On the Diophantine Equation , Doctoral Dissertation, Uppsala, 1952. MR 0049917 (14:247d)
 [7]
 W. Ljunggren, "Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante," Acta Math., v. 75, 1942, pp. 121. MR 0017303 (8:135k)
 [8]
 A. Pethö, "Full cubes in the Fibonacci sequence," (Debrecen) Publ. Math., v. 30, Fasc. 12, 1983. MR 733078 (85j:11027)
 [9]
 V. G. Sprindzhuk, Classical Diophantine Equations in Two Unknowns, "Nauka", Moscow, 1982. (Russian) MR 685430 (85d:11022)
 [10]
 K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation, Marcel Dekker, New York, 1974. MR 0374041 (51:10241)
 [11]
 N. Tzanakis, "The Diophantine equation and related equations," J. Number Theory, v. 18, 1984, pp. 192205. MR 741950 (86d:11023)
 [12]
 M. Waldschmidt, "A lower bound for linear forms in logarithms," Acta Arith., v. 37, 1980, pp. 257283. MR 598881 (82h:10049)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
11D25,
1104,
11Y50
Retrieve articles in all journals
with MSC:
11D25,
1104,
11Y50
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608296403
PII:
S 00255718(1986)08296403
Keywords:
Mordell's equation,
Ellison's method,
Davenport's lemma,
linear forms in logarithms,
Thue equations,
cubic fields
Article copyright:
© Copyright 1986
American Mathematical Society
