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Primitive roots of unity and cyclotomic polynomials
 Journal of Formalized Mathematics
"... Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials. ..."
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Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.
On the Properties of the Möbius Function
"... Summary. We formalized some basic properties of the Möbius function which is defined 8 classically as>< 1, if n = 1, µ(n) = 0, if p 2 n for some prime p, (−1) r, if n = p1p2 · · · pr, where pi are distinct primes. as e.g., its multiplicativity. To enable smooth reasoning about the sum of this nu ..."
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Summary. We formalized some basic properties of the Möbius function which is defined 8 classically as>< 1, if n = 1, µ(n) = 0, if p 2 n for some prime p, (−1) r, if n = p1p2 · · · pr, where pi are distinct primes. as e.g., its multiplicativity. To enable smooth reasoning about the sum of this numbertheoretic function, we introduced an underlying manysorted set indexed by the set of natural numbers. Its elements are just values of the Möbius function. The second part of the paper is devoted to the notion of the radical of number, i.e. the product of its all prime factors. The formalization (which is very much like the one developed in Isabelle proof assistant connected with Avigad’s formal proof of Prime Number Theorem) was done according to the book [13].
Probability on Finite and Discrete Set and Uniform Distribution
"... Summary. A pseudorandom number generator plays an important role in ..."
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Summary. A pseudorandom number generator plays an important role in
The Perfect Number Theorem and Wilson’s Theorem
"... Summary. This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson’s theorem (that n is prime iff n> 1 and (n − 1)! ∼ = −1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The ..."
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Summary. This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson’s theorem (that n is prime iff n> 1 and (n − 1)! ∼ = −1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler’s sum of divisors function φ, proves that φ is multiplicative and that ∑ φ(k) = n. kn
Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ ∗
"... Summary. In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if p is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ ∗ = Z/pZ\{0} is a multiplicative cyclic group, too. The former has been prove ..."
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Summary. In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if p is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ ∗ = Z/pZ\{0} is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/p ∗. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ ∗ and prove it is cyclic.
Recognizing Chordal Graphs: Lex BFS and MCS
, 2006
"... We are formalizing the algorithm for recognizing chordal graphs by lexicographic breadthfirst search as presented in [13, Section 3 of Chapter 4, pp. 81–84]. Then we follow with a formalization of another algorithm serving the same end but based on maximum cardinality search as presented by Tarjan ..."
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We are formalizing the algorithm for recognizing chordal graphs by lexicographic breadthfirst search as presented in [13, Section 3 of Chapter 4, pp. 81–84]. Then we follow with a formalization of another algorithm serving the same end but based on maximum cardinality search as presented by Tarjan and Yannakakis [25]. This work is a part of the MSc work of the first author under supervision of the second author. We would like to thank one of the anonymous reviewers for very useful suggestions.