Publikacje prof. dr. hab. A. Rotkiewicza z teorii liczb
Przed doktoratem:
1. Sur l’équation \(x^z-y^t=a^t\), oú \(|x-y|=a\), Ann. Polon. Math. 3 (1956), 7–8; MR 18 p. 561, Zbl. 70 (1957), p. 272.
2. Sur les nombres composés \(n\) qui divisent \(a^{n-1}-b^{n-1}\), Rend. Circ. Mat. Palermo (2) 8 (1959), 115–116, MR 23 # A1579, Zbl. 90 (1961), p. 258.
3. Sur les nombres pairs \(n\) pour lesquels les nombres \(a^nb - ab^n\) respectivement \(a^{n-1}-b^{n-1}\), sont divisibles par \(n\), ibidem, 341–342; MR 22 # 3703, Zbl. 93 (1962), p. 257, 258.
4. Sur les nombres pairs \(n\) qui divisent \((a+2)^{n-1}-a^{n-1}\), Rend. Circ. Mat. Palermo (2) 9 (1960), 78–80, MR 27 # 94, Zbl. 104 (1964), p. 266.
5. Elementary proof of the existence of the primitive prime divisor of the number \(a^n-b^n\), Prace Mat. 4 (1960), 21–28 (Polish); MR 24 # A1237, Zbl. 101 (1963), p. 31.
6. Une remarque sur le dernier théorème de Fermat, Mathesis 69 (1960), 135–140; MR 22 # 9474.
7. Sur le problème de Catalan, Elem. Math. 15 (1960), 121–124; MR 22 # 9468, Zbl. 96 (1962), p. 28.
8. Sur le problème de Catalan II, ibidem 16 (1961), 25–27; MR 23 # A1597, Zbl. 102 (1968), p. 35.
9. On the numbers of the form \({(4k+1)^{4k+1}-1\over 4k}\), \({(4k+3)^{4k+3}+1\over 4k+4}\), Prace Mat. 5 (1961), 95–99 (Polish); MR 23 # A3700, Zbl. 111 (1964), p. 249, 250.
10. On the properties of the expression \(a^n\pm b^n\), ibidem 6 (1961), 1–20 (Polish); MR 26 # 4960, Zbl. 107 (1964), p. 40.
11. On the numbers \(\phi(a^n\pm b^n)\), Proc. Amer. Math. Soc. 12 (1961), 419–421; MR 23 # A2353, Zbl. 99 (1963), p. 28.
12. Démonstration arithmétique de l’existence d’une infinité de nombres premiers de la forme \(nk+1\), Enseignement Math. 7 (1962), 277–280; MR 25 # 38, Zbl. 109 (1964), p. 30.
13. Sur quelques généralisations des nombres pseudopremiers, Colloq. Math. 9 (1962), 109–113; MR 24 # A2553, Zbl. 100 (1964), p. 27.
14. On Lucas numbers with two intrinsic prime divisors, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 10 (1962), 229–232; MR 25 # 3024, Zbl. 136 (1967), p. 325.
15. Sur les nombres premiers \(p\) et \(q\) tels que \(pq|2^{pq}-2\), Rend. Circ. Mat. Palermo (2) 11 (1962), 280–282; MR 29 # 3415, Zbl. 119 (1960–64), p. 39.
16. Sur les diviseurs composés des nombres \(a^n-b^n\), Bull. Soc. Roy. Sci. Liège 32 (1963), 191–195; MR 26 # 3645, Zbl. 117 (1960–64), p. 30.
17. Sur les nombres pseudopremiers relativement à un nombre naturel a contenus dans les progressions arithmétiques, ibidem 32 (1963), 456–458; MR 27 # 3577, Zbl. 116 (1965), p. 269.
18. Sur les nombres composés \(n\) de la forme \(cx+d\) pour lesquels \(n|a^{n-1}-b^{n-1}\), ibidem 32 (1963), 823–829; MR 29 # 62, Zbl. 141 (1968), p. 40.
19. Remarque sur les nombres parfaits pairs de la forme \(a^n\pm b^n\), Elem. Math. 18 (1963), 76–78, Zbl. 115 (1965), p. 263.
20. Sur les nombres composés tels que \(n|2^n-2\) et \(n\nmid 3^n-3\), Bull. Soc. Math. Phys. Serbie 15 (1963), 7–11; MR 32 # 1161, Zbl. 125 (1966), p. 23.
21. Sur les chiffres initiaux et finals des nombres \(a^n\) et \(a^n\pm b^n\), Rend. Circ. Mat. Palermo (2) 12 (1963), 150–154; MR 29 # 2209, Zbl. 117 (1960–64), p. 30.
22. Sur les nombres pseudopremiers de la forme \(ax+b\), C.R. Acad. Sci. Paris 257 (1963), 2601–2604; MR 29 # 61, Zbl. 116 (1965), p. 35.
Po doktoracie a przed habilitacja:
23. with A. Schinzel, Sur les nombres pseudopremiers de la forme \(ax^2+bxy+cy^2\), ibidem 258 (1964), 3617–3620; MR 28 # 5032, Zbl. 117 (1960–64), p. 282.
24. Sur les nombres pseudopremiers pentagonaux, Bull. Soc. Roy. Liège 33 (1964), 261–263; MR # 4749, Zbl. 125 (1968), p. 23.
25. Sur les nombres á la fois triangulaires, carrés et pentagonaux, ibidem 33 (1964), 518–519; MR 30 # 1971, Zbl. 125 (1966), p. 23.
26. Remarques sur un théorème de F. Proth, Mat. Vesnik 1 (16) (1964), 244–245; MR 32 # 7483b, Zbl. 125 (1966), p. 295.
27. with W. Sierpiński, Sur l’équation diophantienne \(2^x-xy=2\), Publ. Inst. Math. (Beograd) (N.S.) 4 (18) (1964), 135–137; MR 30 # 1972, Zbl. 129 (1967), p. 26.
28. Quelques conséquences de l’existence d’une infinité des nombres pseudopremiers de la forme \(ax+b\), ibidem 4 (1964), 139–140; MR 30 # 1973, Zbl. 129 (1967), p. 27.
29. Une formule explicite pour un nombre pseudopremier par rapport à un nombre entier donné \(a>1\), Elem. Math. 19 (1964), 36, Zbl. 122 (1966), p. 48.
30. Sur les nombres pseudopremiers triangulaires, ibidem 19 (1964), 82–83; MR 29 # 4724, Zbl. 117 (1960–1964), p. 30.
31. Sur les polynômes en \(x\) qui pour une infinité de nombres naturels \(x\) donnent des nombres pseudopremiers, Atti Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 36 (1964), 136–140, MR 30 # 1107, Zbl. 132 (1967), p. 284.
32. Sur les nombres naturels \(n\) et \(k\) tels que les nombres \(n\) et \(nk\) sont à la fois pseudopremiers, ibidem 36 (1964), 816–818; MR 30 # 4719, Zbl. 173 (1969), p. 38.
33. Sur les formules donnant des nombres pseudopremiers, Colloq. Math. 12 (1964), 69–72; MR 29 # 3416, Zbl. 129 (1967), p. 27.
34. Sur les progressions arithmétiques et géométriques formées de trois nombres pseudopremiers distincts, Acta Arith. 10 (1964), 325–328; MR 30 # 1995, Zbl. 125 (1966), p. 23.
35. Sur les nombres pseudopremiers tels que \(n|2^{(n-1)/k}-1\), Rend. Circ. Mat. Palermo (2) 13 (1964), 154–156, MR 34 # 7446, Zbl. 131 (1967), p. 284.
36. Sur les progressions géométriques formées de \(k\) nombres pseudopremiers distincts, ibidem (2) 13 (1964), 369–372; MR 32 # 5581, Zbl. 134 (1967), p. 275.
37. Une généralisation d’un théorème de Cipolla, Glasnik Math. Fiz. Astronom. Ser. II Drušvo Mat. Fiz. Hrvatske 19 (1964), 187–188; MR 30 # 4755, Zbl. 131 (1967), p. 284.
38. Sur les nombres pseudopremiers carrés, Elem. Math. 20 (1965), 39–40; MR 32 # 2369, Zbl. 125 (1966), p. 23.
39. Sur les nombres pseudopremiers de la forme \(M_pM_q\), ibidem 20 (1965), 108–109; MR 30 # 4756, Zbl. 134 (1967), p. 275.
40. Sur les nombres de Mersenne dépourvus de diviseurs carrés et sur les nombres naturels \(n\), tels que \(n^2|2^n-2\), Mat. Vesnik 2 (17) (1965), 78–80; MR 33 # 2596, Zbl. 134 (1967), p. 275.
41. Sur les polynômes en \(x\) du premier degré qui, pour une infinité de valeurs de \(x\) donnent des nombres pseudopremiers, ibidem 2 (17) (1965), 157–161; MR 33 # 111, Zbl. 146 (1968), p. 51, 52.
42. On the divisibility of the number \(a^n+1\) by \(n\), Wiadom. Mat. (2) 8 (1965), 141–142 (Polish); MR 32 # 7490, Zbl. 152 (1968), p. 31.
43. Les intervalles contenants les nombres pseudopremiers, Rend. Circ. Mat. Palermo (2) 14 (1965), 278–280; MR 35 # 2818, Zbl. 152 (1968), p. 31.
44. Sur les nombres pseudopremiers de la forme \(nk+1\), Elem. Mat. 21 (1966), 32–33; MR 33 # 112, Zbl. 151 (1968), p. 26.
45. with A. Makowski, On pseudoprime numbers of the form \(M_pM_t\), Elem. Mat. 21 (1966), 133–134; MR 34 # 4191, Zbl. 147 (1968), p. 23.
46. On pseudoprimes of the form \(ax+b\), Proc. Cambridge Philos. Soc. 63 (1967), 389–392; MR 35 # 122, Zbl. 152 (1968), p. 31.
47. On pseudoprime numbers, Publ. Math. Debrecen 14 (1967), 69–74; MR 36 # 3719, Zbl. 178 (1970), p. 45, 46.
48. On the prime factors of the number \(2^{p-1}-1\), Glasgow Math. J. 9 (1968), 82–86; MR 38 # 2078, Zbl. 165 (1969), p. 57.
49. with H. Halberstam, A gap theorem for pseudoprimes in arithmetic progression, Acta Arith. 13 (1967/68), 395–404; MR 37 # 1329, Zbl. 155 (1968), p. 92.
50. On arithmetical progressions formed by \(k\) different pseudoprimes, J. Math. Sci. 4 (1969), 5–10; MR 40 # 4218, Zbl. 186 (1970), p. 86.
51. with A. Makowski, On pseudoprime numbers of special form, Colloq. Math. 20 (1969), 269–271; MR 39 # 5458, Zbl. 207 (1971), p. 352.
52. Un problème sur les nombres pseudopremiers, Nederl. Akad. Wetensch. Proc. Ser. A = Indag. Math. 34 (1972), 86–91; MR 46 # 1689, Zbl. 227 (1972), 10006.
53. Pseudoprime numbers and their generalizations, Student Association of the Faculty of Sciences, University of Novi Sad, Novi Sad 1972, pp. i+169; MR 48 # 8373, Zbl. 324.10007.
54. 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; MR 46 # 8948, Zbl. 249 (1973), 10012.
55. On a problem of W. Sierpiński, Elem. Math. 27 (1972), 83–85; MR 46 # 7139, Zbl. 249 (1973), 10013.
56. On some problems of W. Sierpiński, Acta Arith. 21 (1972), 251–259; MR 46 # 1695, Zbl. 253 (1973), 10014.
57. W. Sierpiński’s works on the theory of numbers, Rend. Circ. Mat. Palermo (2) 21 (1972), 5–24; MR 48 # 2032, Zbl. 269 (1974), 01014.
58. On the number of pseudoprimes \(\le x\), Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 381–409 (1972), 43–45; MR 48 # 256, Zbl. 257 (1973), 10008.
59. On pyramidal numbers of order \(4\), Elem. Math. 28 (1973), 15–16; MR 48 # 5986, Zbl. 266 (1973), 10014.
60. On the pseudoprimes with respect to the Lucas sequences, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 21 (1973), 793–797; MR 48 # 10966, Zbl. 267 (1974), 10008.
61. with R. Wasén, On a theorem of Cipolla, Elem. Math. 30 (1975), 127–128; MR 52 # 10581, Zbl. 315 (1976), 10010.
Po habilitacji:
62. with R. Wasén, On a number theoretical series, Publ. Math. Debrecen 26 (1979), 1–4; MR 84a: 10050, Zbl. 424 (1980), 10012.
63. The solution of W. Sierpiński’s problem, Rend. Circ. Mat. Palermo (2) 28 (1979), 62–64; MR 81g: 10008, Zbl. 425 (1980), 10009.
64. with A.I. van der Poorten, On strong pseudoprimes in arithmetic progressions, J. Austral. Math. Soc. Ser. A 29 (1980), 316–321; MR 81h: 10010, Zbl. 428 (1980), 16001.
65. with R. Wasén, Lehmer’s numbers, Acta Arith. 36 (1980), 203–217; MR 82a: 10009, Zbl. 436 (1981), 10007.
66. Arithmetical progression formed from three different Euler pseudoprimes for the odd base \(a\), Rend. Circ. Mat. Palermo (2) 29 (1980), 420–426; MR 83b: 1002, Zbl. 471 (1982), 10005.
67. On Fermat’s equation with exponent \(2p\), Colloq. Math. 45 (1981), 101–102; MR 84h: 10024, Zbl. 483 (1983), 10017.
68. On the equation \(x^p+y^p=z^2\), Bull. Acad. Polon. Sci. Ser. Math. 30 (1982), 211–214; MR 83k: 10034, Zbl. 484 (1983), 10011.
69. On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters \(L,Q\) in arithmetic progression, Math. Comp. 39 (1982), 239–247; MR 83k: 10004, Zbl. 492 (1983), 10002.
70. Applications of Jacobi’s symbol to Lehmer’s numbers, Acta Arith. 42 (1983), 163–187; MR 84m: 10010, Zbl. 519 (1984), 11004.
71. On the congruence \(2^{n-2}\equiv 1 (\mod n)\), Math. Comp. 43 (1984), 271–272; MR 85e: 11005, Zbl. 542 (1985), 10003.
72. Problems on Fibonacci numbers and their generalizations, Applications of Fibonacci Numbers, Edited by A.N. Philippou, B.E. Bergum and A.F. Horadam, D. Reidel Publ. Comp. 1986, 241–255; MR 87k: 11025, Zbl. 594 (1987), 10004.
73. with A. Schinzel, On the diophantine equation \(x^p+y^{2p}=z^2\), Colloq. Math. 53 (1986), 146–153; MR 88e: 11017, Zbl. 622 (1988), 10012.
74. Note on the diophantine equation \(1+x+x^2+\ldots+x^n=y^m\), Elem. Math. 42 (1987), 76; MR 88e: 11017, Zbl. 703 (1991), 11016.
75. with W. Złotkowski, On the Diophantine equation \(1+p^{\alpha_1} +p^{\alpha_2}+\ldots+p^{\alpha_k}=y^2\) in Number Theory, vol. 11 (Budapest, 1987), North-Holland, Colloq. Math. Soc. János Bolyai, 51 (1990), 917–937; MR 91e: 11032, Zbl. 705 (1991), 11012.
76. On the diophantine equation \(x^{2p}+y^{2p}=z^p\), Colloq. Math. 62 (1991), 15–19; MR 92d: 11025, Zbl. 747 (1992), 11015.
77. On strong Lehmer pseudoprimes in the case of negative discriminant in arithmetic progressions, Acta Arith. 68 (1994), 145–151; MR 96h: 11008, Zbl. 822 (1995), 11016.
78. Arithmetical progressions formed by \(k\) different pseudoprimes, Rend. Circ. Mat. Palermo (2) 43 (1994), 391–402; MR 96h: 11008.
79. with K. Ziemak, On even pseudoprimes, The Fibonacci Quarterly, 33 (1995), 123–125, MR 96c: 11005, Zbl. 827 (1996), 11003.
80. On Lucas pseudoprimes of the form \(ax^2+bxy+cy^2\), Applications of Fibonacci Numbers, Volume 6, Edited by G.E. Bergum, A.N. Philippou and A.F. Horadam, Kluwer Academic Publishers, Dordrecht, the Netherlands 1996, 409–421, MR 97d: 11008, Zbl. 852 (1997), 11006.
81. On the theorem of Wójcik, Glasgow Math. J. 38 (1996), 157–162, MR 97c: 11010, Zbl. 858 (1997), 11006.
82. 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, MR 2000a: 11027.
83. Arithmetical progression formed by Lucas pseudoprimes, Number Theory, Diophantine, Computational and Algebraic Aspects, Editors: Kálmán Gyory, Attila Petho and Vera T. Sós, Walter de Gruyer GmbH & Co., Berlin, New York 1998, 465–472; MR 99f: 11020, Zbl. 917 (1999), 11010.
84. Periodic sequences of pseudoprimes connected with Carmichael numbers and the least period of the function \(l^C_x\), Acta Arith. 91 (1999), 75–83, MR 2000h: 11006.
85. Solved and unsolved problems on pseudoprime numbers and their generalizations, Applications of Fibonacci Numbers, Volume 8, Edited by Frederic T. Howard, Kluwer Academic Publishers, Dordrecht, the Netherlands 1999, 293–306, MR 2000j: 11006.
86. with A. Schinzel, Lucas pseudoprimes with a prescribed value of the Jacobi symbol, Bull. Polish Acad. Sci. Math. 48 (2000), 77–80, MR 2001a: 11013.
87. Lucas pseudoprimes, Funct. Approximatio Comment. Math. 28 (2000), 97–104.
88. Arithmetic progressions formed by pseudoprimes, Acta Math. Univ. Ostraviensis 8 (2000), 61–74, \(\{\)A review for this item is in process\(\}\).
89. On Lucas pseudoprimes of the form \(ax^2+bxy+cy^2\) in arithmetic progression \(AX+B\) with a prescribed value of the Jacobi symbol, Acta Math. Univ. Ostraviensis 10 (2002), 103–109.
90. Lucas and Frobenius pseudoprimes, Ann. Math. Siles. 17 (2003), 1–21.
91. On a problem of H. J. A. Duparc, Tatra Mt. Math. Publ. 32 (2005), 15–32.
92. On pseudoprimes having special forms and a solution of K. Szymiczek’s problem, Acta Math. Univ. Ostrav. 13 (2005), no. 1, 57–71.
93. On Lucas numbers, Lucas pseudoprimes and a number theoretical series involving Lucas pseudoprimes and Carmichael numbers, Ann. Math. Sil. No. 21 (2007), 49-60 (2008).
94. On pseudoprimes of the form \(a^n-a\), Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, Congr. Numer. 194 (2009), 191–197.
95. On Lucas cyclotomic pseudoprimes having special forms, Congr. Numer. 200 (2010), 239–244.
96. On Jacobi’s symbol \((\sqrt{P_n\over P_m})\) of Lehmer’s numbers in the case of negative discriminant, Congr. Numer. 201 (2010), 289–295.
Inne publikacje prof. dr. hab. A. Rotkiewicza
1. Remarks on solving trigonometric equations, Matematyka 12 (1959) no. 3, 99–102 (Polish).
2. with M. Mancewicz, Solution of problem 73 from the textbook of Prof. W. Janowski for the tenth grade, Matematyka 26 (1973), no.4, 253–255 (Polish).
3. On Fermat’s last theorem, Delta 1974, no. 10, 1–3 (Polish).
4. On a problem of Catalan, Delta 1975, no. 8, 12–13 (Polish).
5. Remarks on paper: “The decomposition of the expression \(a^n\pm b^n\)”, Matematyka 30 (1977) no. 2, 71–73 (Polish).
6. On the greatest prime number, Delta 1977, no. 2, 6–7 (Polish).
7. On famous problems of the theory of numbers in simple terms, School of didactics of mathematics Karpacz 1977, IKNiBO Warszawa 1979, 83–91 (Polish).
8. Professor Sierpiński and year 1920, Biuletyn Informacyjny Nr 37, 12.08.1981 NSZZ “Solidarność”, Region Białystok (Polish).
9. On the equation \(x^k+y^k=z^k\) once more, Matematyka 36 (1983) no. 4, 256 (Polish).
10. Wybrakowany klucz, Matematyka 45 (2002), no. 1, 20–22.