JP Journal of Algebra, Number Theory and Applications
Volume 38, Issue 2, Pages 129 - 143
(April 2016) http://dx.doi.org/10.17654/NT038020129 |
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PRIME MAGMAS AND A CYCLICITY CONJECTURE
Mihai Caragiu and Paul A. Vicol
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Abstract: Let A be the set of all primes, and be the greatest prime factor of the integer We investigate the cyclic character or lack of it thereof for the general class of magma structures defined by where are positive integers. As a motivating example, we provide computational evidence suggesting the cyclicity of the structure with for any The validity of the cyclicity conjecture for would imply that any prime can be expressed as a non-associative product involving the constant 2, the operation symbol ‘*’, and parentheses ),(. However, in our main result we derive a set of fairly restrictive necessary conditions for the cyclicity of which severely limits the range of cyclic magmas of that type. We provide computational evidence suggesting cyclicity for a representative set of structures satisfying the established constraints. The data obtaining process was finalized with a Julia program executed on Google’s servers using Google Compute Engine. |
Keywords and phrases: magmas, primes, cyclic structures, greatest prime factor, computational number theory. |
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