Sets in which is always a square
Author:
Ezra Brown
Journal:
Math. Comp. 45 (1985), 613620
MSC:
Primary 11D57
MathSciNet review:
804949
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Abstract: A set of size n is a set of distinct positive integers such that is a perfect square, whenever ; a set X can be extended if there exists such that is still a set. The most famous result on sets is due to Baker and Davenport, who proved that the set 1, 3, 8, 120 cannot be extended. In this paper, we show, among other things, that if , then there does not exist a set of size 4, and that the set 1, 2, 5 cannot be extended.
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 A. Baker & H. Davenport, "The equations and ," Quart. J. Math. Oxford Ser. (3), v. 20, 1969, pp. 129137. MR 0248079 (40:1333)
 [2]
 L. E. Dickson, History of the Theory of Numbers, vol. II, Carnegie Institute, Washington, 1920; reprinted, Chelsea, New York, 1966.
 [3]
 C. M. Grinstead, "On a method of solving a class of Diophantine equations," Math. Comp., v. 32, 1978, pp. 936940. MR 0491480 (58:10724)
 [4]
 P. Heichelheim, "The study of positive integers (a, b) such that is a square," Fibonacci Quart., v. 17, 1979, pp. 269274. MR 549810 (81e:10014)
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 P. Kanagasabapathy & T. Ponnudurai, "The simultsneous Diophantine equations and ," Quart. J. Math. Oxford Ser. (3), v. 26, 1975, pp. 275278. MR 0387182 (52:8027)
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 S. P. Mohanty & A. M. S. Ramasamy, "The simultaneous Diophantine equations and ," J. Number Theory, v. 18, 1984, pp. 356359. MR 746870 (85h:11013)
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 T. Nagell, Introduction to Number Theory, Wiley, New York, 1951. MR 0043111 (13:207b)
 [8]
 G. Sansone, "Il sistema diofanteo , , ," Ann. Mat. Pura Appl. (4), v. 111, 1976, pp. 125151. MR 0424672 (54:12631)
 [9]
 N. Thamotherampillai, "The set of numbers 1, 2, 7," Bull. Calcutta Math. Soc., v. 72, 1980, pp. 195197. MR 669583 (83h:10032)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198508049497
PII:
S 00255718(1985)08049497
Article copyright:
© Copyright 1985 American Mathematical Society
