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2016 Amicable pairs and aliquot cycles for elliptic curves over number fields
Jim Brown, David Heras, Kevin James, Rodney Keaton, Andrew Qian
Rocky Mountain J. Math. 46(6): 1853-1866 (2016). DOI: 10.1216/RMJ-2016-46-6-1853


Let $E/\mathbb{Q} $ be an elliptic curve. Silverman and Stange define primes $p$ and $q$ to be an elliptic, amicable pair if $\#E(\mathbb{F} _p) = q$ and $\#E(\mathbb{F} _q) = p$. More generally, they define the notion of aliquot cycles for elliptic curves. Here, we study the same notion in the case that the elliptic curve is defined over a number field~$K$. We focus on proving the existence of an elliptic curve~$E/K$ with aliquot cycle $(\mathfrak{p} _1, \ldots , \mathfrak{p} _{n})$ where the $\mathfrak{p} _{i}$ are primes of~$K$ satisfying mild conditions.


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Jim Brown. David Heras. Kevin James. Rodney Keaton. Andrew Qian. "Amicable pairs and aliquot cycles for elliptic curves over number fields." Rocky Mountain J. Math. 46 (6) 1853 - 1866, 2016.


Published: 2016
First available in Project Euclid: 4 January 2017

zbMATH: 06673134
MathSciNet: MR3591263
Digital Object Identifier: 10.1216/RMJ-2016-46-6-1853

Primary: 11G05

Keywords: aliquot cycles , amicable pairs , Elliptic curves

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium


Vol.46 • No. 6 • 2016
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