Previous |  Up |  Next

Article

Title: Arithmetic progressions formed by pseudoprimes (English)
Author: Rotkiewicz, Andrzej
Language: English
Journal: Acta Mathematica et Informatica Universitatis Ostraviensis
ISSN: 1211-4774
Volume: 8
Issue: 1
Year: 2000
Pages: 61-74
.
Category: math
.
MSC: 11A07
MSC: 11B39
idZBL: Zbl 1075.11003
idMR: MR1800223
.
Date available: 2009-01-30T09:08:07Z
Last updated: 2013-10-22
Stable URL: http://hdl.handle.net/10338.dmlcz/120560
.
Reference: [1] Alford W.R., Granville A., Pomerance C.: There are infinitely many Carmichael numbers.Ann. of Math. 140 (1994), 703-722. Zbl 0816.11005, MR 1283874, 10.2307/2118576
Reference: [2] Baillie R., Wagstaff S., Jr.: Lucas pseudoprimes.Math. Comp. 35 (1980), 1391-1417. Zbl 0458.10003, MR 0583518, 10.1090/S0025-5718-1980-0583518-6
Reference: [3] BEEGER N.G.W.R.: On even numbers $m$ dividing $2m - 2$.Amer. Math. Monthly 58 (1951), 553-555. MR 0043798, 10.2307/2306320
Reference: [4] Carmichael R.D.: On the numerical factors of the arithmetic forms $\alpha^n ±\beta^n$.Ann. of Math. (2) 15 (1913), 30-70. MR 1502458, 10.2307/1967797
Reference: [5] Chowla S.: There exists an infinity of 3-combinations of primes in A.P..Proc Lahore Philos. Ser. 6, no 2 (1944), 15-16. Zbl 0063.00875, MR 0014125
Reference: [6] ClPOLLA M.: Sui numeri composti P che verificiano la congruenza di Fermat $a^{P-1} \equiv 1( \mod P)$.Annali di Matematica (3) 9 (1904), 139-160.
Reference: [7] Conway J.H., Guy R.K., Schneeberger W.A., Sloane N.J.A.: The primary pretenders.Acta Arith. 78 (1997), 307-313. Zbl 0863.11005, MR 1438588
Reference: [8] Dickson L.E.: A new extension of Dirichlet's theorem on prime numbers.Messenger Math. 33 (1904), 155-161.
Reference: [9] Dickson L.E.: History of the Theory of Numbers.3 vols., Washington 1919- 1923, reprint New York 1966.
Reference: [10] Duparc H.J.A.: On almost primes of the second order.Math. Centrum Amsterdam. Rap. ZW 1955-013, (1955), 1-13 . Zbl 0067.27303
Reference: [11] Duparc H.J.A.: A remark to report Z.W.-013.Math. Centrum Amsterdam, Rap. Z.W. 1956-008.
Reference: [12] Durst L.K.: Exceptional real Lehmer sequences.Pacific J. Math. 9 (1959), 437-441. Zbl 0091.04204, MR 0108465, 10.2140/pjm.1959.9.437
Reference: [13] Erdos P.: On almost primes.Amer. Math. Monthly 57 (1950), 404-407. MR 0036259, 10.2307/2307640
Reference: [14] GRANVILLE A.J.: The prime k-tuplets conjecture implies that there are arbitrarily long arithmetic progressions of Carmichael numbers.(written communication of December 1995).
Reference: [15] Heath-Brown D.R.: Three primes and an almost prime in arithmetic progression.J. London Math. Soc, (2) 23 (1981), 396-414. Zbl 0425.10051, MR 0616545, 10.1112/jlms/s2-23.3.396
Reference: [16] Jeans J.A.: The converse of Fermat's theorem.Messenger of Mathematics 27 (1898), p. 174.
Reference: [17] Kernbaum S.: O szeregu Fibonacciego i jego uogolnieniach.Wiadom. Mat. 24 (1920), 203-217, II ibid. 25 (1921), 49-68.
Reference: [18] Korselt A.: Probleme chinois.L'intermediare des mathematiciens 6 (1899), 142-143.
Reference: [19] Lehmer D.H.: An extended theory of Lucas functions.Ann. of Math. 31 (1930), 419-448. MR 1502953, 10.2307/1968235
Reference: [20] Lehmer E.: On the infinitude of Fibonacci pseudoprimes.Fibonacci Quart. 2 (1964), 229-230.
Reference: [21] Lehmer D.H.: Strong Carmichael numbers.J. Austral. Math. Soc Ser. A 21 (1978), 508-510. MR 0417032, 10.1017/S1446788700019364
Reference: [22] McDaniel W.L.: Some pseudoprimes and related numbers having special forms.Math. Comp. 53 (1989), 407-409. Zbl 0678.10003, MR 0968152, 10.1090/S0025-5718-1989-0968152-6
Reference: [23] Mahnke D.: Leibniz and der Suché nach einer allgemeinem Primzahlgleichung.Bibliotheca Math. Vol. 13 (1913), 29-61.
Reference: [24] Needham J.: Science and Civilization in China, vol. 3: Mathematics and Sciences of the Heavens and the Earth.Cambridge 1959, p. 54, footnote A. MR 0139507
Reference: [25] Niewiadomski R.: Spostrzezenia nad liczbami szeregu Fibonacciego.Wiadom. Mat. 15 (1911), 225-233.
Reference: [26] RlBENBOIM P.: The New Book of Prime Number Records.Springer-Verlag, New York - Heidelberg - Berlin, 1996. MR 1377060
Reference: [27] Rotkiewicz A.: Sur les formules donnant des nombres pseudopremiers.Colloq. Math. 12 (1964), 69-72. Zbl 0129.02703, MR 0166138
Reference: [28] ROTKIEWICZ A.: Sur les progressions arithmétiques et géométriques formées de trois nombres pseudopremiers distincts.Acta Arith. 10 (1964), 325-328. Zbl 0125.02304, MR 0171768
Reference: [29] Rotkiewicz A.: On arithmetical progressions formed by k different pseudo-primes.J. Math. Sci. 4 (1969), 5-10. MR 0250987
Reference: [30] Rotkiewicz A.: Pseudoprime numbers and their generalizations.Student Association of the Faculty of Sciences, University of Novi Sad, Novi Sad 1972, pp. i+169. Zbl 0324.10007, MR 0330034
Reference: [31] Rotkiewicz A.: On the pseudoprimes of the form $ax + b$ with respect to the sequence of Lehmer.Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 20 (1972), 349-354. Zbl 0249.10012, MR 0309843
Reference: [32] Rotkiewicz A.: The solution of W. Sierpinskťs problem.Rend. Circ Mat. Palermo (2) 28 (1979), 62-64. MR 0564551, 10.1007/BF02849586
Reference: [33] Rotkiewicz A.: Arithmetical progression formed from three different Euler pseudoprimes for the odd base a.Rend. Circ. Mat. Palermo (2) 29 (1980), 420-426. MR 0638680, 10.1007/BF02849758
Reference: [34] Rotkiewicz A.: On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters $L,Q$ in arithmetic progression.Math. Comp. 39 (1982), 239-247. MR 0658229
Reference: [35] Rotkiewicz A.: On strong Lehmer pseudoprimes in the case of negative discriminant in arithmetic progressions.Acta Arith. 68 (1994), 145-151. Zbl 0822.11016, MR 1305197
Reference: [36] Rotkiewicz A.: Arithmetical progressions formed by k different pseudoprimes.Rend. Circ. Mat. Palermo (2) 43 (1994), 391-402. MR 1344876
Reference: [37] Rotkiewicz A., Ziemak K.: On even pseudoprimes.The Fibonacci Quarterly, 33 (1995), 123-125. Zbl 0827.11003, MR 1329016
Reference: [38] Rotkiewicz A.: There are infinitely many arithmetical progressions formed by three different Fibonacci pseudoprimes.Applications of Fibonacci Numbers, Volume 7, Edited by G.E. Bergum, A.N. Philippou and A.F. Horadam, Kluwer Academic Publishers, Dordrecht, the Netherlands 1998, 327-332. Zbl 0926.11004, MR 1638459, 10.1007/978-94-011-5020-0_37
Reference: [39] ROTKIEWICZ A.: Arithmetical progression formed by Lucas pseudoprimes.Number Theory, Diophantine, Computational and Algebraic Aspects, Editors: Kálmán Gyóry, Attila Pethó and Vera T. Sos, Walter de Gruyter GmbH & Co., Berlin, New York 1998, 465-472. MR 1628862
Reference: [40] Rotkiewicz A.: Periodic sequences of pseudoprimes connected with Carmichael numbers and the least period of the function $l_x^C$.Acta Arith. 91 (1999), 75-83. MR 1726476
Reference: [41] Rotkiewicz A., Schinzel A.: Lucas pseudoprimes with a prescribed value of the Jacobi symbol.Bull. Polish Acad. Sci. Math. 48 (2000), 77-80. Zbl 0951.11002, MR 1751157
Reference: [42] Schinzel A.: The intrinsic divisors of Lehmer numbers in the case of negative discriminant.Ark. Mat. 4 (1962), 413-416. Zbl 0106.03105, MR 0139567, 10.1007/BF02591623
Reference: [43] Schinzel A.: On primitive prime factors of Lehmer numbers I.Acta Arith. 8 (1963), 213-223. Zbl 0118.27901, MR 0151423
Reference: [44] Schinzel A., Sierpiňski W.: Sur certaines hypotheses concernant les nombres premiers.Acta Arith. 4 (1958), 185-208, and corrigendum, ibidem 5 (1960), 259. MR 0106202
Reference: [45] SlERPlNSKl W.: Remarque sur une hypothěse des Chinois concernant les nombres $(2n - 2)/n$.Colloq. Math. 1 (1947), 9.
Reference: [46] SlERPlNSKl W.: Elementary Theory of Numbers.Monografie Matematyczne 42, PWN, Warsaw 1964 (second edition: North-Holland, Amsterdam, New York, Oxford 1987). MR 0930670
Reference: [47] Steuerwald R.: Über die Kongruenz $2^{n-1} \equiv 1 (\mod n)$.Sitz.-Ber. math. naturw. Kl. Bayer. Akad. Wiss. Munchen 1947, 177. MR 0030541
Reference: [48] Szymiczek K.: Kilka twierdzen o liczbach pseudopierwszych.Zeszyty naukowe Wyzszej Szkoly Pedagogicznej w Katowicach, Sekcja Matematyki, Zeszyt Nr 5 (1966), 39-46.
Reference: [49] Szymiczek K.: Note on Fermat numbers.Elem. Math. 21 (1966), 59. Zbl 0142.28904, MR 0193056
Reference: [50] Van der Corput J.G.: Über Summen von Primzahlen und Primzahlquadraten.Math. Ann. 116 (1939), 1-50. MR 1513216, 10.1007/BF01597346
Reference: [51] Ward M.: The intrinsic divisor of Lehmer numbers.Ann. of Math. (2) 62 (1955), 230-236. MR 0071446, 10.2307/1969677
Reference: [52] ZSIGMONDY K.: Zur Theorie der Potenzreste.Monatsh. Math. 3 (1892), 265-284. MR 1546236, 10.1007/BF01692444
.

Files

Files Size Format View
ActaOstrav_08-2000-1_7.pdf 1.847Mb application/pdf View/Open
Back to standard record
Partner of
EuDML logo