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Lehmer", title = "A factorization theorem applied to a test for primality", journal = BAMS, volume = 45, year = 1939, pages = "132-137"} @article{Lehmer:1949, key = "D. H. Lehmer 1949", author = "D. H. Lehmer", title = "On the converse of {Fermat's} theorem {II}", journal = AMM, volume = 56, year = 1949, pages = "300-309", note = "Errata in {\it Math. Comp.} {\bf 25} (1971), 943--944"} @article{Lucas:1878a, key = "Lucas 1878a", author = "E. Lucas", title = {Th\'eorie des fonctions num\'eriques simplement p\'eriodiques}, journal = AJM, volume = 1, year = "{\noopsort{1878a}}1878", pages = "289-321"} @article{Western:1932, key = "Western 1932", author = "A. E. Western", title = "On {Lucas's} and {Pepin's} tests for the primeness of {Mersenne} numbers", journal = JLMS, volume = 7, year = 1932, pages = "130-137"} @article{Ankeny:1952, key = "Ankeny 1952", author = "N. C. 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Morrison", title = "A note on primality testing using {Lucas} sequences", journal = MC, volume = 29, year = 1975, pages = "181-182"} @article{Pocklington:1914, key = "Pocklington 1914", author = "H. C. Pocklington", title = "The determination of the prime or composite nature of large numbers by {Fermat's} theorem", journal = PCPS, volume = 18, year = "1914-1916", pages = "29-30"} @article{Pollard:1971b, key = "Pollard 1971b", author = "J. M. Pollard", title = "An algorithm for testing the primality of any integer", journal = BLMS, volume = 3, year = "{\noopsort{1971b}}1971", pages = "337-340"} @article{Jones&Sato&Wada&Wiens:1976, key = "J. Jones, Sato, Wada, and Wiens 1976", author = "J. P. Jones and D. Sato and H. Wada and D. Wiens", title = "Diophantine representation of the set of prime numbers", journal = AMM, volume = 83, year = 1976, pages = "449-464"} @article{Chaitin&Schwartz:1978, key = "Chaitin and Schwartz 1978", author = "G. J. Chaitin and J. T. Schwartz", title = "A note on {Monte} {Carlo} primality tests and algorithmic information theory", journal = CPAM, volume = 31, year = 1978, pages = "521-527"} @techreport{Baratz:1978, key = "Baratz 1978", author = "A. E. Baratz", title = "An analysis of the {Solovay} and {Strassen} test for primality", institution = "MIT Laboratory for Computer Science", number = "MIT/LCS/TM-108", month = "July", year = 1978} @article{Wagon:1986, key = "Wagon 1986", author = "S. Wagon", title = "Primality testing", journal = MI, volume = 8, number = 3, year = 1986, pages = "58-61"} @article{Pomerance:1981a, key = "Pomerance 1981a", author = "C. Pomerance", title = "Recent developments in primality testing", journal = MI, volume = 3, year = "{\noopsort{1981a}}1981", pages = "97-105"} @article{Cohen&Lenstra:1987, key = "H. Cohen and A. K. Lenstra 1987", author = "H. Cohen and A. K. Lenstra", title = "Implementation of a new primality test", journal = MC, volume = 48, year = 1987, pages = "103-121"} @article{Riesel:1969, key = "Riesel 1969", author = "H. Riesel", title = "Lucasian criteria for the primality of {$N = h \cdot 2^n - 1$}", journal = MC, volume = 23, year = 1969, pages = "869-875"} @article{Adams&Shanks:1982, key = "W. Adams and Shanks 1982", author = "W. W. Adams and D. Shanks", title = "Strong primality tests that are not sufficient", journal = MC, volume = 39, year = 1982, pages = "255-300"} @article{Adleman&Leighton:1981, key = "Adleman and Leighton 1981", author = "L. M. Adleman and F. T. Leighton", title = "An {$O(n^{1/10.89})$} primality testing algorithm", journal = MC, volume = 36, year = 1981, pages = "261-266"} @article{Wunderlich:1983a, key = "Wunderlich 1983a", author = "M. C. Wunderlich", title = "A performance analysis of a simple prime-testing algorithm", journal = MC, volume = 40, year = "{\noopsort{1983a}}1983", pages = "709-714"} @article{Beauchemin&Brassard&Crepeau&Goutier&Pomerance:1988, key = "Beauchemin, Brassard, Crepeau, Goutier, and Pomerance 1988", author = "P. Beauchemin and G. Brassard and C. Cr\'epeau and C. Goutier and C. Pomerance", title = "The generation of random numbers that are probably prime", journal = JC, volume = 1, year = 1988, pages = "53-64"} @inproceedings{Mihailescu:1988, key = "Mihailescu 1988", author = "P. Mihailescu", title = "A primality test using cyclotomic functions", booktitle = "AAECC-6", editor = "T. Mora", series = LNICS, volume = 357, year = 1988, publisher = SV, pages = "310-323"} @article{Kim&Pomerance:1989, key = "Kim and Pomerance 1989", author = "S. H. Kim and C. Pomerance", title = "The probability that a random probable prime is composite", journal = MC, volume = 53, year = 1989, pages = "721-741"} @article{Fendel:1985, key = "Fendel 1985", author = "D. Fendel", title = "Prime-producing polynomials and principal ideal domains", journal = MM, volume = 58, year = 1985, pages = "204-210"} @article{Frobenius:1912, key = "Frobenius 1912", author = "F. G. Frobenius", title = "{\"Uber} quadratische {Formen}, die viele {Primzahlen} darstellen", journal = "Sitzungsber. d. {K\"onigl.} Akad. d. Wiss. zu Berlin", year = 1912, pages = "966-980", note = "Reprinted in {\it Gesammelte Abhandlungen}, Vol.~III, pp.~573--587"} @incollection{Rabinovitch:1912, key = "Rabinovitch 1912", author = "G. Rabinovitch", title = "Eindeutigkeit der {Zerlegung} in {Primzahlfaktoren} in quadratischen {Zahlk\"orpern}", booktitle = "Proc. 5th Int. Cong. of Mathematicians 1912", editor = "E. W. Hobson and A. E. H. Love", volume = 1, year = 1912, pages = "418-421"} @article{Ribenboim:1988a, key = "Ribenboim 1988a", author = "P. Ribenboim", title = "{Euler}'s famous prime generating polynomial and the class number of imaginary quadratic fields", journal = EM, volume = 34, year = "{\noopsort{1988a}}1988", pages = "23-42"} @incollection{Boyd:1988, key = "Boyd 1988", author = "C. Boyd", title = "Probabilistic prime tests", booktitle = "Computers in Mathematical Research", editor = "N. M. Stephens and M. P. Thorne", publisher = "Clarendon Press", address = "Oxford", year = 1988, pages = "57-68"} @article{Dudley:1969, key = "Dudley 1969", author = "U. Dudley", title = "History of a formula for primes", journal = AMM, volume = 76, year = 1969, pages = "23-28"} @article{Zagier:1977, key = "Zagier 1977", author = "D. Zagier", title = "The first 50 million prime numbers", journal = MI, volume = 0, month = "August", year = 1977, pages = "7-19"} @article{Stechkin:1971, key = "Stechkin 1971", author = "S. B. Stechkin", title = "Lucas's criterion for the primality of numbers of the form {$N = h 2^n - 1$}", journal = MN, volume = 10, year = 1971, pages = "578-584"} @article{Bach:1990a, key = "Bach 1990a", author = "E. Bach", title = "Explicit bounds for primality testing and related problems", journal = MC, volume = 55, year = "{\noopsort{1990a}}1990", pages = "355-380"} @article{Richards:1974, key = "Richards 1974", author = "I. Richards", title = "On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem", journal = BAMS, volume = 80, year = 1974, pages = "419-438"} @article{Couvreur&Quisquater:1982, key = "Couvreur and Quisquater 1982", author = "C. Couvreur and J. J. Quisquater", title = "An introduction to fast generation of large prime numbers", journal = PJR, volume = 37, year = 1982, pages = "231-264", note = "Errata in {\it Philips J. Res.} {\bf 38} (1983), 77"} @article{Huthnance&Warndof:1988, key = "Huthnance and Warndof 1988", author = "E. D. Huthnance and J. Warndof", title = "On using primes for public key encryption systems", journal = AML, volume = 1, year = 1988, pages = "225-227"} @article{Ogiwara:1987, key = "Ogiwara 1987", author = "M. Ogiwara", title = "A primality criteria based on an improvement of {Lenstra's} residue class algorithm", journal = "Res. Rep. Inf. Sci. C, Comput. Sci. (Japan)", number = 88, month = "November", year = 1987, pages = "1-14"} @article{Szekeres:1974, key = "Szekeres 1974", author = "G. Szekeres", title = "On the number of divisors of {$x^2 + x + A$}", journal = JNT, volume = 6, year = 1974, pages = "434-442"} @article{Ayoub&Chowla:1981, key = "Ayoub and Chowla 1981", author = "R. G. Ayoub and S. Chowla", title = "On {Euler's} polynomial", journal = JNT, volume = 13, year = 1981, pages = "443-445"} % more papers on primality testing & pseudoprimes @incollection{Lenstra:1985, key = "H. W. Lenstra 1985", author = "Lenstra, Jr., H. W.", title = "Galois theory and primality testing", booktitle = "Orders and their Applications", series = LNIM, volume = 1142, editor = "I. Reiner and K. W. Roggenkamp", publisher = SV, year = 1985, pages = "169-189"} @article{Adams:1987, key = "W. Adams 1987", author = "W. W. Adams", title = "Characterizing pseudoprimes for third-order linear recurrences", journal = MC, volume = 48, year = 1987, pages = "1-15"} @article{McDaniel:1989, key = "McDaniel 1989", author = "W. L. McDaniel", title = "Some pseudoprimes and related numbers having special forms", journal = MC, volume = 53, year = 1989, pages = "407-409"} @article{Pintz&Steiger&Szemeredi:1989, key = "Pintz, Steiger, and {Szemer\'edi} 1989", author = "J. Pintz and W. L. Steiger and E. Szemer\'edi", title = "Infinite sets of primes with fast primality tests and quick generation of large primes", journal = MC, volume = 53, year = 1989, pages = "399-406"} @article{Jammalamadaka&Uppuluri:1989, key = "Jammalamadaka and Uppuluri 1989", author = "S. R. Jammalamadaka and V. R. R. Uppuluri", title = "Is $p$ a prime number? {Some} probabilistic tests for primality", journal = MATHSCI, volume = 14, year = 1989, pages = "55-61"} @article{Williams:1982b, key = "H. Williams 1982b", author = "H. C. Williams", title = "A class of primality tests for trinomials which includes the {Lucas-Lehmer} test", journal = PJM, volume = 98, year = "{\noopsort{1982b}}1982", pages = "477-494"} @article{Calude&Zimand:1984, key = "Calude and Zimand 1984", author = "C. Calude and M. Zimand", title = "A relation between correctness and randomness in the computation of probabilistic algorithms", journal = IJCM, volume = 16, year = 1984, pages = "47-53"} @article{Borho&Buhl&Hoffmann&Mertens&Nebgen&Reckow:1983, key = "Borho, Buhl, Hoffmann, Mertens, Nebgen, and Reckow 1983", author = "W. Borho and J. Buhl and H. Hoffmann and S. Mertens and E. Nebgen and R. Reckow", title = "{Gro\ss e} {Primzahlen} und befreundete {Zahlen}: {\"Uber} den {Lucas-Test} und {Thabit-Regeln}", journal = MMGH, volume = 11, year = 1983, pages = "232-256"} @article{Pajunen:1980, key = "Pajunen 1980", author = "S. Pajunen", title = "On two theorems of {Lenstra}", journal = IPL, volume = 11, year = 1980, pages = "224-228"} @article{Harris:1969, key = "V. Harris 1969", author = "V. C. Harris", title = "A test for primality", journal = NMT, volume = 17, year = 1969, pages = "82"} @article{Mignotte:1980b, key = "Mignotte 1980b", author = "M. Mignotte", title = "Tests de {primalit\'e}", journal = TCS, volume = 12, year = "{\noopsort{1980b}}1980", pages = "109-117"} @article{Ward:1959, key = "Ward 1959", author = "M. Ward", title = "Tests for primality based on {Sylvester's} cyclotomic numbers", journal = PJM, volume = 9, year = 1959, pages = "1269-1272"} @article{Norman:1979, key = "Norman 1979", author = "A. C. Norman", title = "Testing word-sized numbers for primality", journal = ASB, volume = 13, number = 4, year = "Nov. 1979", pages = "19-20"} @article{Jonsson:1972, key = "{J\"onsson} 1972", author = "I. {J\"onsson}", title = "On certain primes of {Mersenne-type}", journal = BIT, volume = 12, year = 1972, pages = "117-118"} @article{Morehead&Western:1909, key = "Morehead and Western 1909", author = "J. C. Morehead and A. E. Western", title = "Note on {Fermat's} numbers", journal = BAMS, volume = 16, year = 1909, pages = "1-6"} @inproceedings{Pin:1981, key = "Pin 1981", author = "J.-E. Pin", title = "Test de primalit\'e pour calculatrice de poche, d'apr\`es {L. Monier}", booktitle = "Laboratoire Informatique Th\'eorique, Institut de Programmation, Institut Henri Poincar\'e", year = "1980/81", pages = "251-262"} @techreport{Morain:1988, key = "Morain 1988", author = "F. Morain", title = "Implementation of the {Goldwasser-Kilian-Atkin} primality testing algorithm", institution = "INRIA", month = "October", year = 1988, number = "911"} @article{Young&Buell:1988, key = "Young and Buell 1988", author = "J. Young and D. A. Buell", title = "The twentieth {Fermat} number is composite", journal = MC, volume = 50, year = 1988, pages = "261-263"} @inproceedings{DiPorto&Filipponi:1988, key = "Di Porto and Filipponi 1988", author = "A. Di Porto and P. Filipponi", title = "A probabilistic primality test based on the properties of certain generalized {Lucas} numbers", booktitle = EUROCRYPT88, volume = 330, series = LNICS, publisher = SV, editor = "C. G. {G\"unther}", year = 1988, pages = "211-223"} @inproceedings{GordonJ:1984a, key = "J. Gordon 1984a", author = "J. A. Gordon", title = "Strong primes are easy to find", booktitle = EUROCRYPT84, volume = 209, year = "{\noopsort{1984a}}1984", series = LNICS, editor = "T. Beth and N. Cot and I. Ingemarsson", publisher = SV, pages = "216-223"} @article{Proth:1878, key = "Proth 1878", author = "F. Proth", title = "Th\'eor\`emes sur les nombres premiers", journal = CRASP, volume = 87, year = 1878, pages = "926"} @article{Cunningham&Western:1904, key = "Cunningham and Western 1904", author = "A. Cunningham and A. E. Western", title = "On {Fermat's} numbers", journal = PLMS, volume = 1, year = 1904, pages = "175"} @article{Lawrence:1897, key = "Lawrence 1897", author = "F. W. Lawrence", title = "Determination of certain primes", journal = PLMS, volume = 28, year = 1897, pages = "465-475"} @article{Richards:1982, key = "Richards 1982", author = "I. 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