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Cubic reciprocity and generalised Lucas-Lehmer tests for primality of $A.3^n\pm 1$

Author(s): Pedro Berrizbeitia; T. G. Berry
Journal: Proc. Amer. Math. Soc. 127 (1999), 1923-1925.
MSC (1991): Primary 11A51, 11Y11
Posted: February 18, 1999
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Abstract: Cubic reciprocity is used to derive primality tests analogous to the Lucas-Lehmer test for integers of the form $A.3^n \pm 1$. The test for $A.3^n-1$ is a minor improvement on a test derived by Williams by other means; the test for $A.3^n+1$ seems to be new.

References:

[G]
A. Guthmann. Effective primality tests for $N=k\cdot 3^{n+1}$ and $N=k\cdot 2^m 3^n +1$. BIT 32 (1992) 529-534. MR 93h:11008

[IR]
K. Ireland and M. Rosen. A classical Introduction to Modern Number Theory. Springer-Verlag, Berlin, 1982. MR 83g:12001

[R]
M. Rosen. A proof of the Lucas-Lehmer test. Amer. Math. Monthly 95 (1988) 855-856. MR 89i:11011

[W1]
H. C. Williams The primality of $N=2A3^n-1$. Can. Math. Bull. 15 (1972) 585-589. MR 47:121

[W2]
H. C. Williams. A note on the primality of $6\sp{2\sp n}+1$ and $10 \sp{2\sp n}+1$. Fibonacci Quart. 26 (1988) 296-305. MR 89i:11013

[W3]
H.C. Williams A class of primality tests for trinomials which includes the Lucas-Lehmer test. Pacific J. Math 98 (1982) 477-494. MR 83f:10008

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Additional Information:

Pedro Berrizbeitia
Affiliation: Departamento de Matematicas Puras y Aplicadas Universidad Simón Bolívar Caracas, Venezuela
Email: pedrob@usb.ve

T. G. Berry
Affiliation: Departamento de Matematicas Puras y Aplicadas Universidad Simón Bolívar Caracas, Venezuela
Email: berry@usb.ve

DOI: 10.1090/S0002-9939-99-04786-3
PII: S 0002-9939(99)04786-3
Received by editor(s): September 24, 1997
Posted: February 18, 1999
Communicated by: David E. Rohrlich
Copyright of article: Copyright 1999, American Mathematical Society

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