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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.
Probability on Finite and Discrete Set and Uniform Distribution
"... Summary. A pseudorandom number generator plays an important role in ..."
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Cited by 3 (2 self)
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Summary. A pseudorandom number generator plays an important role in
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.
The Perfect Number Theorem and Wilson’s Theorem
, 2008
"... This article formalizes proofs of some elementary theorems of number theory (see [26, 1,?]): Wilson’s theorem (that n is prime iff n> 1 and (n − 1)! ∼ = −1(modn)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The arti ..."
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This article formalizes proofs of some elementary theorems of number theory (see [26, 1,?]): Wilson’s theorem (that n is prime iff n> 1 and (n − 1)! ∼ = −1(modn)), 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 knφ(k) = n.
Uniqueness of Factoring an Integer and Multiplicative Group ZZ∗
, 2008
"... In the [25], 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 ..."
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In the [25], 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 [28]. 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.
Let Xbeanemptyset.ObservethatcardXisempty. Onecancheckthateverybinaryrelationwhichisnaturalyieldingisalso
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"... polynomialsanddemonstratetherelationshipbetweencyclotomicpolynomials andunitalpolynomials. ..."
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polynomialsanddemonstratetherelationshipbetweencyclotomicpolynomials andunitalpolynomials.
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"... Date: Abstract Formalization of mathematics and automated proof checking open new possibilities in how everyday mathematics is done. While the entire area is still in early stages of development, it already promises higher levels of trust in mathematical proofs and also offers a clear benefit when s ..."
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Date: Abstract Formalization of mathematics and automated proof checking open new possibilities in how everyday mathematics is done. While the entire area is still in early stages of development, it already promises higher levels of trust in mathematical proofs and also offers a clear benefit when searching through electronic proof databases. We have decided to run an extensive experiment in formalization of graph theory, with the goal being the formal verification of several graph algorithms. We used the Mizar (http://mizar.org) proof assistant system; among many computerized proof assistants Mizar offers an environment closest to how everyday mathematics is done.