Fermat’s Little Theorem via Divisibility of Newton’s Binomial

• Autorzy: Rafał Ziobro
• Języki publikacji: EN

Abstrakty

EN

Solving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant in its content (another proof of the theorem has earlier been included in [12]), the article provides a good example, how the application of registrations could shorten the length of Mizar proofs [9], [17].

Słowa kluczowe

EN

factorization; primes; Fermat

Kategorie tematyczne

• 11A51: Factorization; primality
• 03B35: Mechanization of proofs and logical operations
• 11Y55: Calculation of integer sequences

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2015
• Tom: 23
• Numer: 3
• Strony: 215-229

Daty

wydano

2015-09-01

otrzymano

2015-06-30

online

2015-09-30

Twórcy

autor

Rafał Ziobro

• Department of Carbohydrate Technology University of Agriculture Krakow, Poland

Bibliografia

• [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
• [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
• [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
• [5] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
• [7] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
• [8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [9] Marco B. Caminati and Giuseppe Rosolini. Custom automations in Mizar. Journal of Automated Reasoning, 50(2):147-160, 2013.
• [10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
• [11] Yoshinori Fujisawa and Yasushi Fuwa. The Euler’s function. Formalized Mathematics, 6 (4):549-551, 1997.
• [12] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Euler’s Theorem and small Fermat’s Theorem. Formalized Mathematics, 7(1):123-126, 1998.
• [13] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin’s test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317-321, 1998.
• [14] Jacek Gancarzewicz. Arytmetyka, 2000. In Polish.
• [15] Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.
• [16] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573-577, 1997.
• [17] Artur Korniłowicz. On rewriting rules in Mizar. Journal of Automated Reasoning, 50(2): 203-210, 2013.
• [18] Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004.
• [19] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.
• [20] Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part II: Examples and problems. Formalized Mathematics, 9(1):143-154, 2001.
• [21] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
• [22] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.
• [23] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
• [24] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125-130, 1991.
• [25] Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.
• [26] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.
• [27] Christoph Schwarzweller. Modular integer arithmetic. Formalized Mathematics, 16(3): 247-252, 2008. doi:10.2478/v10037-008-0029-8.[Crossref]
• [28] Wacław Sierpinski. Teoria liczb. 1950. In Polish.
• [29] Wacław Sierpinski. O rozwiazywaniu rownan w liczbach całkowitych, 1956. In Polish.
• [30] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
• [31] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.
• [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
• [33] Li Yan, Xiquan Liang, and Junjie Zhao. Gauss lemma and law of quadratic reciprocity. Formalized Mathematics, 16(1):23-28, 2008. doi:10.2478/v10037-008-0004-4.[Crossref]
• [34] Rafał Ziobro. Some remarkable identities involving numbers. Formalized Mathematics, 22(3):205-208, 2014. doi:10.2478/forma-2014-0023. [Crossref]

Identyfikatory

DOI

10.1515/forma-2015-0018