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 |
. |