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The Primality of N=2A3n-1

Published online by Cambridge University Press:  20 November 2018

H. C. Williams
H. C. Williams*
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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Lehmer [3] and Reisel [7] have devised tests for determining the primality of integers of the form A2n—1. Tables of primes of these forms may be found in [7] and Williams and Zarnke [10]. Little work, however, seems to have been done on integers of the form N=2A3n—1. Lucas [6] gave conditions that were only sufficient for the primality of N. Recently Lehmer [4] has given a method for determining the primality of an integer N if the factorization of N+1 is known.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 4 , December 1972 , pp. 585 - 589
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Cailler, C., Sur les congruences du troisième degré, L'Enseig. Math., 10 (1908), 474-487.Google Scholar
2. Cunningham, A. J. C., Quadratic Partitions, London, 1907.Google Scholar
3. Lehmer, D. H., An extended theory of Lucas'functions, Ann. of Math. (2), 31 (1930), 419-448.Google Scholar
4. Lehmer, D. H., Computer technology applied to the theory of numbers, M.A.A. studies in Mathematics, 6 (1969), 117-151.Google Scholar
5. Emma, Lehmer, Criteria for cubic and quartic residuacity, Mathematika 5 (1958), 20-29.Google Scholar
6. Edouard, Lucas, Théorie des functions numériques simplement périodiques, Amer. J. Math. 1 (1878), 184-240, 289-321.Google Scholar
7. Reisel, H., Lucasian criteria for the primality of N= h ⋅-2n—1, Math. Comp. 23 (1969). 869-875.Google Scholar
8. Robinson, R. M., The converse of Fermais theorem, Amer. Math. Monthly, 64 (1957), 703-710.Google Scholar
9. Smith, H. J. S., Report on the Theory of Numbers, Chelsea, New York, (1965), 103-105.Google Scholar
10. Williams, H. C. and Zarnke, C. R., A reporton primenumbersof'theforms M=(6a + 1)22m-1—1 and M' = (6a - 1)22m-1—1, Math. Comp. 22 (1968), 420-422.Google Scholar