Journal of Integer Sequences, Vol. 21 (2018), Article 18.6.2

A Proof of the Lucas-Lehmer Test and its Variations by Using a Singular Cubic Curve

Ömer Küçüksakallı
Mathematics Department
Middle East Technical University
06800 Ankara


We give another proof of the Lucas-Lehmer test by using a singular cubic curve. We also illustrate a practical way to choose a starting term for the Lucas-Lehmer-Riesel test by trial and error. Moreover, we provide a nondeterministic test for determining the primality of integers of the form N = hpn - 1 for any odd prime p. We achieve these by using the group structure on a singular cubic curve induced from the group law of elliptic curves.

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Received May 29 2018; revised versions received May 30 2018; July 5 2018. Published in Journal of Integer Sequences, July 11 2018.

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