Deterministic primality test for numbers of the form 𝐴².3ⁿ+1, 𝑛≥3 odd

Berrizbeitia, Pedro; Iskra, Boris

doi:10.1090/S0002-9939-01-06100-7

Deterministic primality test for numbers of the form $A^2.3^n+1$, $n \ge 3$ odd
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by Pedro Berrizbeitia and Boris Iskra PDF

Proc. Amer. Math. Soc. 130 (2002), 363-365 Request permission

Abstract:

We use a result of E. Lehmer in cubic residuacity to find an algorithm to determine primality of numbers of the form $A^23^n+1$, $n$ odd, $A^2<4(3^n+1)$. The algorithm represents an improvement over the more general algorithm that determines primality of numbers of the form $A.3^n \pm 1$, $A/2<4.3^n-1$, presented by Berrizbeitia and Berry (1999).

References

Pedro Berrizbeitia and T. G. Berry, Cubic reciprocity and generalised Lucas-Lehmer tests for primality of $A\cdot 3^n\pm 1$, Proc. Amer. Math. Soc. 127 (1999), no. 7, 1923–1925. MR 1487359, DOI 10.1090/S0002-9939-99-04786-3
Andreas Guthmann, Effective primality tests for integers of the forms $N=k\cdot 3^n+1$ and $N=k\cdot 2^m3^n+1$, BIT 32 (1992), no. 3, 529–534. MR 1179238, DOI 10.1007/BF02074886
C. Kirfel and Ø. Rødseth. On the primality of $2h\cdot 3^n+1$. To appear. Discrete Math.
Emma Lehmer, Criteria for cubic and quartic residuacity, Mathematika 5 (1958), 20–29. MR 95162, DOI 10.1112/S0025579300001303
E. Lucas. Théorie des functions numériques simplement périodiques Amer. J. Math. 1 (1878) 184–214, 289–321.
H. C. Williams, The primality of $N=2A3^{n}-1$, Canad. Math. Bull. 15 (1972), 585–589. MR 311559, DOI 10.4153/CMB-1972-101-7
H. C. Williams, A note on the primality of $6^{2^n}+1$ and $10^{2^n}+1$, Fibonacci Quart. 26 (1988), no. 4, 296–305. MR 967648