Компьютер как новая реальность математики: VI. Числа ферма и их родственники
Аннотация
В этой части, составляющей пандан к части, посвященной числам Мерсенна, я продолжаю обсуждать фантастический прогресс в решении классических задач теории чисел, достигнутый в последние десятилетия с использованием компьютеров. Здесь будет рассказано о проверке простоты, факторизациях и поиске простых делителей чисел специального вида, в первую очередь, чисел Ферма, их друзей и родственников, таких как обобщенные числа Ферма, простые Прота и т. д. Кроме того, мы детально обсудим роль чисел Ферма и чисел Пирпойнта в циклотомии.
Литература
V. Y. Bunyakovsky, “On a new case of divisibility of numbers of the form 2 2 m + 1, reported to the Academy by
Father Pervushin (read at the meeting of the Physics and Mathematics Department on April 4, 1878),” Zapiski Imperatorskoi Akademii Nauk, vol. 31, no. 1, pp. 223–224, 1878 (in Russian).
N. A. Vavilov, “Numerology of square equations,” Algebra i Analiz, vol. 20, no. 5, pp. 9–40, 2008; doi:10.1090/S1061-0022-09-01068-1
N. A. Vavilov, “Computers as novel mathematical reality. I. Personal Account,” Computer tools in education, no. 2, pp. 5–26, 2020 (in Russian); doi: 10.32603/2071-2340-2020-2-5-26
N. A. Vavilov, “Computers as novel mathematical reality. II. Waring Problem,” Computer tools in education, no. 3, pp. 5–55, 2020 (in Russian); doi: 10.32603/2071-2340-2020-3-5-55
N. A. Vavilov, “Computers as novel mathematical reality. III. Mersenne numbers and divisor sums,” Computer tools in education, no. 4, pp. 5–58, 2020 (in Russian); doi: 10.32603/2071-2340-2020-4-5-58
N. A. Vavilov, “Computers as novel mathematical reality. IV. Goldbach Problem,” Computer tools in education, no. 4, pp. 5–71, 2021 (in Russian); doi: 10.32603/2071-2340-2021-4-5-71
N. A. Vavilov, “Computers as novel mathematical reality. V. Easier Waring problem,” Computer tools in education, no. 3, pp. 5–63, 2022 (in Russian); doi: 10.32603/2071-2340-2022-3-5-63
N. A. Vavilov and V. G. Khalin, Zadachi po kursu Matematika i Komp’yuter. Vyp. 1. Arifmetika i teoriya chisel [Exercises for the course “Mathematics and Computers”. Issue 1, Arithmetics and Number Theory], St. Petersburg, Russia: OTsEiM, 2005 (in Russian).
N. A. Vavilov, V. G. Khalin, and A. V. Yurkov, Mathematica dlya nematematika [Mathematica for nonmathematician], Moscow: MCCME, 2021 (in Russian).
B. L. van der Waerden, Science awakening. Egyptian, Babylonian and Greek mathematics, Moscow: GIFML, 1959 (in Russian).
O. N. Vasilenko, “On some properties of Fermat numbers,” Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., no. 5, pp. 56–58, 1998 (in Russian).
A. M. Vershik, “The asymptotic distribution of factorizations of natural numbers into prime divisors,” Sov. Math. Dokl., vol. 34, pp. 57–61, 1987.
N. Wiener, I am mathematician, Izhevsk, Russia: Nelin. Dinam., 2001 (in Russian).
C. F. Gauss, Disquisitiones arithmeticae, I. M. Vinorradov ed., Moscow: Iz-vo AN SSSR, 1959 (in Russian).
C. F. Gauss, “Explanation of the possibility of constructing a seventeenagon,” E. P. Ozhigova ed., Istorikomatematicheskie issledovaniya, vol. 21, pp. 285–291, 1976 (in Russian).
J. L. Heiberg, Natural science and mathematics in classical antiquity, Leningrad, USSR: ONTI, 1936 (in Russian).
S. G. Gindikin, Stories about physicists and mathematicians, Moscow: Nauka, 1981 (in Russian).
J. M. Ziman, Elements of advanced quantum theory, Moscow: Mir, 1971 (in Russian).
G. P. Matvievskaya, E. P. Ozhigova, N. I. Nevskaya, and Yu. Kh. Kopelevich, Unpublished materials of L. Euler on number theory, St Petersburg, Russia: Nauka, 1997 (in Russian).
I. G. Melnikov, “On some questions of number theory in Euler’s correspondence with Goldbach,” Istoriya i metodologiya estestvennykh nauk, vol. 5, pp. 15–30, 1966 (in Russian).
I. G. Melnikov, “Questions of number theory in the works of Fermat and Euler,” Istoriko-matematicheskie issledovaniya, vol. 19, pp. 9–38, 1974 (in Russian).
О. Neugebauer, The exact sciences in antiquity, Moscow: Nauka, 1968 (in Russian).
J. Needham, A history of embryology, Moscow: Inostr. literatura, 1947 (in Russian).
M. M. Postnikov, Galois theory, Moscow: GIFML, 1963 (in Russian).
I. Braginskii ed., Poetry and prose of the Ancient East, BVL, ser. I, vol. 1, Moscow: Khudozhestvennaya literatura, 1973 (in Russian).
V. V Prasolov and Yu. P. Solov’ev, Elliptic functions and algebraic equations, Moscow: Faktorial, 1997 (in Russian).
E. V. Sadovnik, “Testing numbers of the form N = 2kpm − 1 for primality,” Discrete Math. Appl., vol. 16:2, pp. 99–108, 2006; doi:10.1515/156939206777344610
E. V. Sadovnik, “Testing numbers of the form N = 2kpm1 1 p m2 2 ...p mn n − 1 for primality,” Discrete Math. Appl., vol. 18, no. 3, pp. 239–249, 2008; doi:10.1515/DMA.2008.019
S. B. Stechkin, “Lucas’s criterion for the primality of numbers of the form N = h2 n −1,” Math. Notes, vol. 10:3, pp. 578–584, 1971; doi:197110.1007/BF01464716
D. K. Faddeev, Algebra Lectures. Moscow, Nauka, 1984 (in Russian).
P. Fermat, Studies in number theory and diophantine analysis, Moscow, Nauka, 1992 (in Russian).
G. G. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Moscow: In-t Komp’yuternykh Issl., 2002 (in Russian).
N. G. Chebotarev, Galois theory, Leningrad, USSR: ONTI, 1936 (in Russian).
H. G. Zeuthen, History of mathematics in antiquity and the Middle Ages, Leningrad, USSR: GTTI, 1932 (in Russian).
F. Adler, Theorie der geometrischen Konstruktionen, Sammlung Schubert, vol. 52, Leipzig, Germany: G¨oschen,1906 (in German).
F. J. Affolter, “Zur Staudt—Schr¨oter’schen Construction des regul¨aren Vielecks,” Math. Ann., vol. 6, pp. 582–591, 1873 (in German).
R. C. Agarwal and C. S. Burrus, “Fast digital convolution using Fermat transforms,” in Southwest IEEE Conf. Rec., Houston, Texas, pp. 538–543. 1973.
R. C. Agarwal and C. S. Burrus, “Fast convolution using Fermat number transforms with applications to digital filtering,” in IEEE Trans. Acoust. Speech Signal Processing, vol. 22, pp. 87–97, 1974.
A. Aigner, “Uber Primzahlen, nach denen (fast) alle Fermatschen Zahlen quadratische Nichtreste sind,” ¨ Monatsh. Math., vol. 101, no. 2, pp. 85–93, 1986 (in German).
B. Amiot, “Memoire sur les polygones r ˊ eguliers,” ˊ Nouv. Annales de Math., vol. 4, pp. 264–278, 1844 (in French).
R. C. Archibald, “The history of the construction of the regular polygon of seventeen sides,” American M. S. Bull. vol. 22, pp. 239–246, 1916.
R. C. Archibald, “Gauss and the regular polygon of seventeen sides,” Am. Math. Monthly, vol. 27, pp. 323–326, 1920.
J. M. Arnaudies and P. Delezoide, “Nombres ˋ (2, 3)-constructibles,” Adv. Math., vol. 158, no. 2, pp. 169–252, 2001.
S. P. Arya, “Fermat numbers,” Math. Ed., vol. 6, pp. 5–6, 1989.
S. P. Arya, “More about Fermat numbers,” Math. Ed., vol. 7, pp. 139–141, 1990.
S. Asadulla, “A note on Fermat numbers,” J. Natur. Sci. Math., vol. 17, pp. 113–118, 1977.
Z. S. Aygin and K. S. Williams, “Why does a prime p divide a Fermat number?,” Math. Mag., vol. 93, no. 4, pp. 288–294, 2020.
P. Bachmann, Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie, Leipzig, Germany: Teubner, 1872 (in German).
D. Bardziahin, “Finding special factors of values of polynomials at integer points,” Int. J. Number Theory, vol. 13, no. 1, pp. 209–228, 2017.
S. Baek, I. Choe, Y. Jung, D. Lee, and J. Seo, “Constructions by ruler and compass, together with a fixed conic,” Bull. Aust. Math. Soc., vol. 88, no. 3, pp. 473–478, 2013.
R. Baille, “New primes of the form k · 2n +1,” Math. Comput., vol. 33, no. 148, pp. 1333–1336, 1979.
R. Baillie, G. Cormack, and H. C. Williams, “The problem of Sierpinski concerning ˊ k ·2n +1,” Math. Comp., vol. 37, no. 155, pp. 229–231, 1981.
R. Baillie and S. S. Wagstaff, “Lucas pseudoprimes,” Math. Comp., vol. 35, no. 152, pp. 1391–1417, 1980.
E. Bainville and B. Geneves, “Constructions using conics,” ˊ Math. Intelligencer, vol. 22, no. 3, pp. 60–72, 2000.
R. Ballinger and W. Keller, Proth Search Page, 1997.
A. Baragar, “Constructions using a compass and twice-notched straightedge,” MAA Month., vol. 109, pp. 151–164, 2002.
C. B. Barker, “Proof that the Mersenne number M167 is composite,” Bull. Amer. Math. Soc., vol. 51, pp. 389–389, 1945.
K. Barner, “Paul Wolfskehl und der Wolfskehl-Preis,” Mitt. Dtsch. Math.-Ver., vol. 5, no. 3, pp. 4–11, 1997 (in German).
Материал публикуется под лицензией:
