Publication:
The Support Problem and Its Elliptic Analogue

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Publication Date
1997
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Academic Press
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Abstract
Let F be a number field, Suppose x, y Є F* have the property that for all n Є Z and almost all prime ideals p of the ring of integers of F* one has that yn =1 (mod p) whenever xn=1 (mod p). We show that then y is a power of x. This answers a question of Erdos. We also prove an elliptic analogue of this result.
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J. W. S. Cassels and A. Frohlich, Eds., ``Algebraic Number Theory,'' Academic Press,London/New York, 1967. G. Janusz, ``Algebraic Number Theory,'' Academic Press, New York/London, 1973. J. Silverman, ``The Arithmetic of Elliptic Curves,''Graduate Texts in Mathematics,Vol. 106, Springer-Verlag, Heidelberg/New York, 1986. A. Schinzel, On exponential congruences, Mat. Zametki, to appear. J.-P. Serre, Proprietes galoisiennes des points d'ordre fini des courbes elliptiques, Invent.Math. 15 (1972),259-331. (1uvres, III, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1986, 1-73.)
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