Abstract
The recent ideas of Agrawal, Kayal, and Saxena have produced a milestone in the area of deterministic primality testing. Unfortunately, their method, as well as their successors are mainly of theoretical interest, as they are much too slow for practical applications.
Via a totally different approach, Lukes et al. developed a test which is conjectured to prove the primality of N in time only O((lg N)3 + 0(1)). Their (plausible) conjecture concerns the distribution of pseudosquares. These are numbers which locally behave like perfect squares but are nevertheless not perfect squares.
While squares are easy to deal with, this naturally gives rise to the question of whether the pseudosquares can be replaced by more general types of numbers. We have succeeded in extending the theory to the cubic case. To capture pseudocubes we rely on interesting properties of elements in the ring of Eisenstein integers and suitable applications of cubic residuacity. Surprisingly, the test itself is very simple as it can be formulated in the integers only. Moreover, the new theory appears to lead to an even more powerful primality testing algorithm than the one based on the pseudosquares.
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Berrizbeitia, P., Müller, S., Williams, H.C. (2004). Pseudocubes and Primality Testing. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_7
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DOI: https://doi.org/10.1007/978-3-540-24847-7_7
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