Journal articles: 'Fundamental theorem of arithmetic'

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Agargun, Ahmet G., and Colin R. Fletcher. "The Fundamental Theorem of Arithmetic Dissected." Mathematical Gazette 81, no. 490 (March 1997): 53. http://dx.doi.org/10.2307/3618768.

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Griffiths, Martin. "Intuiting the fundamental theorem of arithmetic." Educational Studies in Mathematics 82, no. 1 (May 1, 2012): 75–96. http://dx.doi.org/10.1007/s10649-012-9410-1.

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Baeth, Nicholas R., Brandon Burns, and James Mixco. "A fundamental theorem of modular arithmetic." Periodica Mathematica Hungarica 75, no. 2 (July 31, 2017): 356–67. http://dx.doi.org/10.1007/s10998-017-0205-0.

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Sprows, David J. "Irrationals and the Fundamental Theorem of Arithmetic." American Mathematical Monthly 96, no. 8 (October 1989): 732. http://dx.doi.org/10.2307/2324726.

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Sprows, David J. "Irrationals and the Fundamental Theorem of Arithmetic." American Mathematical Monthly 96, no. 8 (October 1989): 732. http://dx.doi.org/10.1080/00029890.1989.11972274.

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Sprows, David J. "THE UNIQUENESS OF THE FUNDAMENTAL THEOREM OF ARITHMETIC." PRIMUS 11, no. 3 (January 1, 2001): 286–88. http://dx.doi.org/10.1080/10511970108984006.

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Sprows, David. "Unique factorization and the fundamental theorem of arithmetic." International Journal of Mathematical Education in Science and Technology 48, no. 1 (June 24, 2016): 130–31. http://dx.doi.org/10.1080/0020739x.2016.1199059.

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Tianze, Yang. "The Fundamental Theorem of Arithmetic and Goldbach Conjecture." International Journal of Mathematics Trends and Technology 61, no. 2 (September 25, 2018): 142–43. http://dx.doi.org/10.14445/22315373/ijmtt-v61p520.

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Agargün, Ahmet G., and Colin R. Fletcher. "al-Fārisī and the fundamental theorem of arithmetic." Historia Mathematica 21, no. 2 (May 1994): 162–73. http://dx.doi.org/10.1006/hmat.1994.1015.

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Marshall, Stephen M. "An All-Inclusive Proof of Beal’s Conjecture." Bulletin of Society for Mathematical Services and Standards 7 (September 2013): 17–22. http://dx.doi.org/10.18052/www.scipress.com/bsmass.7.17.

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This paper presents a complete and exhaustive proof of the Beal Conjecture. The approach to this proof uses the Fundamental Theorem of Arithmetic as the basis for the proof of the Beal Conjecture. The Fundamental Theorem of Arithmetic states that every number greater than 1 is either prime itself or is unique product of prime numbers. The prime factorization of every number greater than 1 is used throughout every section of the proof of the Beal Conjecture. Without the Fundamental Theorem of Arithmetic, this approach to proving the Beal Conjecture would not be possible.

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SCHWAB, EMIL DANIEL, and PENTTI HAUKKANEN. "A UNIQUE FACTORIZATION IN COMMUTATIVE MÖBIUS MONOIDS." International Journal of Number Theory 04, no. 04 (August 2008): 549–61. http://dx.doi.org/10.1142/s1793042108001523.

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We show that any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative Möbius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative Möbius monoid.

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Phúc, Đặng Võ, and Shahid Nawaz. "A NEW PROOF OF THE FUNDAMENTAL THEOREM OF ARITHMETIC." Far East Journal of Mathematical Sciences (FJMS) 126, no. 2 (October 20, 2020): 99–104. http://dx.doi.org/10.17654/ms126020099.

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Ağargün, A. Göksel, and E. Mehmet Özkan. "A Historical Survey of the Fundamental Theorem of Arithmetic." Historia Mathematica 28, no. 3 (August 2001): 207–14. http://dx.doi.org/10.1006/hmat.2001.2318.

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Watson, Eric S. "Proof of the Beal Conjecture through the Fundamental Theorem of Arithmetic." Bulletin of Mathematical Sciences and Applications 12 (May 2015): 35. http://dx.doi.org/10.18052/www.scipress.com/bmsa.12.35.

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Gök, Mustafa. "Introducing the fundamental theorem of arithmetic through a mobile game." New Trends and Issues Proceedings on Humanities and Social Sciences 7, no. 1 (July 2, 2020): 249–62. http://dx.doi.org/10.18844/prosoc.v7i1.4940.

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There is an increasing trend towards the use of mobile games in education. Presenting knowledge with mobile games requires many variables to be employed. These processes should be made more rigorous in domains such as mathematics where knowledge is abstract. The aim of this study is to develop an application to introduce the Fundamental Theorem of Arithmetic through a mobile game. The findings of the research show that the Fundamental Theorem of Arithmetic can be introduced with the developed mobile game. When the mobile game is evaluated in terms of mathematical knowledge, it is determined that while the constraints and conditions determined in the game hide the mathematical knowledge. In this sense, some game examples are given in this study and some models related to feedbacks that students can take in these games are presented. Keywords: a-didactical situations, mobile games, mathematics teaching, mathematical concepts;

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Luttik, Bas, and Vincent van Oostrom. "Decomposition orders—another generalisation of the fundamental theorem of arithmetic." Theoretical Computer Science 335, no. 2-3 (May 2005): 147–86. http://dx.doi.org/10.1016/j.tcs.2004.11.019.

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Starni, Paolo. "87.05 A simple proof of the fundamental theorem of arithmetic." Mathematical Gazette 87, no. 508 (March 2003): 106. http://dx.doi.org/10.1017/s0025557200172195.

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Testov, V. A. "An analog of the fundamental theorem of arithmetic in ordered groupoids." Mathematical Notes 62, no. 6 (December 1997): 762–66. http://dx.doi.org/10.1007/bf02355465.

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Goldstein, Catherine. "On a seventeenth century version of the “fundamental theorem of arithmetic”." Historia Mathematica 19, no. 2 (May 1992): 177–87. http://dx.doi.org/10.1016/0315-0860(92)90075-m.

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MAYOR, GASPAR, and JAUME MONREAL. "THE GREATEST COMMON DIVISOR AND OTHER TRIANGULAR NORMS ON THE EXTENDED SET OF NATURAL NUMBERS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17, no. 01 (February 2009): 35–45. http://dx.doi.org/10.1142/s0218488509005723.

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This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, …, +∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characterization of those t-norms on N which are a direct product of t-norms on {0, 1, 2, …, +∞} is given and, after introducing the concept of T-prime (prime with respect to a t-norm T), a theorem about the existence of a T-prime decomposition is obtained. This result generalizes the fundamental theorem of arithmetic.

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Simpson, Stephen G. "Ordinal numbers and the Hilbert basis theorem." Journal of Symbolic Logic 53, no. 3 (September 1988): 961–74. http://dx.doi.org/10.2307/2274585.

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In [5] and [21] we studied countable algebra in the context of “reverse mathematics”. We considered set existence axioms formulated in the language of second order arithmetic. We showed that many well-known theorems about countable fields, countable rings, countable abelian groups, etc. are equivalent to the respective set existence axioms which are needed to prove them.One classical algebraic theorem which we did not consider in [5] and [21] is the Hilbert basis theorem. Let K be a field. For any natural number m, let K[x1,…,xm] be the ring of polynomials over K in m commuting indeterminates x1,…,xm. The Hilbert basis theorem asserts that for all K and m, every ideal in the ring K[x1,…,xm] is finitely generated. This theorem is of fundamental importance for invariant theory and for algebraic geometry. There is also a generalization, the Robson basis theorem [11], which makes a similar but more restrictive assertion about the ring K〈x1,…,xm〉 of polynomials over K in mnoncommuting indeterminates.In this paper we study a certain formal version of the Hilbert basis theorem within the language of second order arithmetic. Our main result is that, for any or all countable fields K, our version of the Hilbert basis theorem is equivalent to the assertion that the ordinal number ωω is well ordered. (The equivalence is provable in the weak base theory RCA0.) Thus the ordinal number ωω is a measure of the “intrinsic logical strength” of the Hilbert basis theorem. Such a measure is of interest in reference to the historic controversy surrounding the Hilbert basis theorem's apparent lack of constructive or computational content.

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Csányi, Petra, Kata Fábián, Csaba Szabó, and Zsanett Szabó. "Number theory vs. Hungarian high school textbooks : the fundamental theorem of arithmetic." Teaching Mathematics and Computer Science 13, no. 2 (2015): 209–23. http://dx.doi.org/10.5485/tmcs.2015.0397.

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Zhu, Hai, Liqun Pu, Hengzhou Xu, and Bo Zhang. "Construction of Quasi-Cyclic LDPC Codes Based on Fundamental Theorem of Arithmetic." Wireless Communications and Mobile Computing 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/5264724.

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Quasi-cyclic (QC) LDPC codes play an important role in 5G communications and have been chosen as the standard codes for 5G enhanced mobile broadband (eMBB) data channel. In this paper, we study the construction of QC LDPC codes based on an arbitrary given expansion factor (or lifting degree). First, we analyze the cycle structure of QC LDPC codes and give the necessary and sufficient condition for the existence of short cycles. Based on the fundamental theorem of arithmetic in number theory, we divide the integer factorization into three cases and present three classes of QC LDPC codes accordingly. Furthermore, a general construction method of QC LDPC codes with girth of at least 6 is proposed. Numerical results show that the constructed QC LDPC codes perform well over the AWGN channel when decoded with the iterative algorithms.

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Brauner, Nadia, Sylvain Gravier, Louis-Philippe Kronek, and Frédéric Meunier. "LAD models, trees, and an analog of the fundamental theorem of arithmetic." Discrete Applied Mathematics 161, no. 7-8 (May 2013): 909–20. http://dx.doi.org/10.1016/j.dam.2012.12.004.

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Khelif, Anatole. "The Bass-Milnor-Serre theorem for nonstandard models in Peano arithmetic." Journal of Symbolic Logic 58, no. 4 (December 1993): 1451–58. http://dx.doi.org/10.2307/2275153.

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The aim of this paper is to extend the Bass-Milnor-Serre theorem to the nonstandard rings associated with nonstandard models of Peano arithmetic, in brief to Peano rings.First, we recall the classical setting. Let k be an algebraïc number field, and let θ be its ring of integers. Let n be an integer ≥ 3, and let G be the group Sln(θ) of (n, n) matrices of determinant 1 with coefficients in θ.The profinite topology in G is the topology having as fundamental system of open subgroups the subgroups of finite index.Congruence subgroups of finite index of G are the kernels of the maps Sln(θ) → Sln(θ/I) for which all ideals I of θ are of finite index. By taking these subgroups as a fundamental system of open subgroups, one obtains the congruence topology on G. Every open set for this topology is open in the profinite topology.We denote by Ḡ (resp., Ĝ) the completion of G for the congruence (resp., profinite) topology.The Bass-Milnor-Serre theorem [1] consists of the two following statements:(A) If k admits a real embedding, then we have an exact sequenceThat is, Ĝ and Ḡ are isomorphic.(B) If k is totally imaginary, then one has an exact sequencewhere μ(k)is the group of the roots of unity of k.

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Wu, Yi, Lan Long, and Zheng Ping Zhang. "The Integer Solutions of Diophantine Equation Χ²− 21 = 4y⁵." Applied Mechanics and Materials 687-691 (November 2014): 1182–85. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.1182.

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In this paper, we studied the integer solutions of the typical Diophantine equations with some important theories in quadratic fields and the fundamental theorem of arithmetic in the ring of quadratic algebraic integers. We proved all the integer solutions of the Diophantine equation Χ2 − 21 = 4y5.

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Patterson, S. J. "Tori in metaplectic covers of GL2 and applications to a formula of Loxton–Matthews." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 2 (September 1986): 249–63. http://dx.doi.org/10.1017/s030500410006607x.

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In this paper we have two objectives. The first is to investigate the restriction of a metaplectic cover to an arbitrary torus in GL2. This will be explained at greater length below, and the main results are Theorems 1 and 2. The second is an application of the same ideas to introduce the arithmetic function P, which has already appeared in a special case in [9], and to prove the fundamental property given by Theorem 3. These theorems will be proved in §§ 2 and 3. In §§ 4 and 5 we remark on the appearence of the function P in the formula of Loxton and Matthews [5], [6] for the biquadratic Gauss sum and discuss the structure of this formula.

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Salehi, Saeed. "Herbrand consistency of some arithmetical theories." Journal of Symbolic Logic 77, no. 3 (September 2012): 807–27. http://dx.doi.org/10.2178/jsl/1344862163.

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AbstractGödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279–292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories IΔ0 + Ωm with m ≥ 2, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory T ⊇ IΔ0 + Ω2 in T itself.In this paper, the above results are generalized for Δ0 + Ω1. Also after tailoring the definition of Herbrand consistency for IΔ0 we prove the corresponding theorems for IΔ0. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories IΔ0 + Ω1 and IΔ0.

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Nepomuceno, Erivelton G., Márcia L. C. Peixoto, Samir A. M. Martins, Heitor M. Rodrigues, and Matjaž Perc. "Inconsistencies in Numerical Simulations of Dynamical Systems Using Interval Arithmetic." International Journal of Bifurcation and Chaos 28, no. 04 (April 2018): 1850055. http://dx.doi.org/10.1142/s0218127418500554.

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Over the past few decades, interval arithmetic has been attracting widespread interest from the scientific community. With the expansion of computing power, scientific computing is encountering a noteworthy shift from floating-point arithmetic toward increased use of interval arithmetic. Notwithstanding the significant reliability of interval arithmetic, this paper presents a theoretical inconsistency in a simulation of dynamical systems using a well-known implementation of arithmetic interval. We have observed that two natural interval extensions present an empty intersection during a finite time range, which is contrary to the fundamental theorem of interval analysis. We have proposed a procedure to at least partially overcome this problem, based on the union of the two generated pseudo-orbits. This paper also shows a successful case of interval arithmetic application in the reduction of interval width size on the simulation of discrete map. The implications of our findings on the reliability of scientific computing using interval arithmetic have been properly addressed using two numerical examples.

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Li, Chao, and Yihang Zhu. "Remarks on the arithmetic fundamental lemma." Algebra & Number Theory 11, no. 10 (December 31, 2017): 2425–45. http://dx.doi.org/10.2140/ant.2017.11.2425.

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THORNER, JESSE. "A variant of the Bombieri–Vinogradov theorem in short intervals and some questions of Serre." Mathematical Proceedings of the Cambridge Philosophical Society 161, no. 1 (February 22, 2016): 53–63. http://dx.doi.org/10.1017/s0305004116000050.

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AbstractWe generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extensionL/$\mathbb{Q}$exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modularL-functionL(s, f), the fundamental discriminantsdfor which thed-quadratic twist ofL(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

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HASHIMOTO, YASUFUMI. "ASYMPTOTIC FORMULAS FOR CLASS NUMBER SUMS OF INDEFINITE BINARY QUADRATIC FORMS ON ARITHMETIC PROGRESSIONS." International Journal of Number Theory 09, no. 01 (November 13, 2012): 27–51. http://dx.doi.org/10.1142/s1793042112501230.

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It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants on arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjugacy classes in the finite groups given by the modular group and its congruence subgroups, we can get the desired asymptotic formulas by arranging the Tchebotarev-type prime geodesic theorem. While such asymptotic formulas were already given by Raulf, the approaches are quite different, the expressions of the leading terms of our asymptotic formulas are simpler and the estimates of the remainder terms are sharper.

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Harada, Shinya, and Toshiro Hiranouchi. "Smallness of fundamental groups for arithmetic schemes." Journal of Number Theory 129, no. 11 (November 2009): 2702–12. http://dx.doi.org/10.1016/j.jnt.2009.03.010.

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Rapoport, Michael, Ulrich Terstiege, and Wei Zhang. "On the arithmetic fundamental lemma in the minuscule case." Compositio Mathematica 149, no. 10 (July 4, 2013): 1631–66. http://dx.doi.org/10.1112/s0010437x13007239.

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AbstractThe arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

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Rungtanapirom, Nithi. "Godeaux–Serre varieties with prescribed arithmetic fundamental group." Journal of Pure and Applied Algebra 222, no. 11 (November 2018): 3337–44. http://dx.doi.org/10.1016/j.jpaa.2017.12.010.

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McGee, Vann. "On the degrees of unsolvability of modal predicate logics of provability." Journal of Symbolic Logic 59, no. 1 (March 1994): 253–61. http://dx.doi.org/10.2307/2275263.

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The modal predicate logic of provability identifies the “□” of modal logic with the “Bew” of proof theory, so that, where “Bew” is a formula representing, in the usual way, provability in a consistent, recursively axiomatized theory Γ extending Peano arithmetic (PA), an interpretation of a language for the modal predicate calculus is a map * which associates with each modal formula an arithmetical formula with the same free variables which commutes with the Boolean connectives and the quantifiers and which sets (□ϕ)* equal to Bew(⌈ϕ*⌉). Where Δ is an extension of PA (all the theories we discuss will be extensions of PA), MPL(Δ) will be the set of modal formulas ϕ such that, for every interpretation *, ϕ* is a theorem of Δ. Most of what is currently known about the modal predicate logic of provability consists in demonstrations that MPL(Δ) must be computationally highly complex. Thus Vardanyan [11] shows that, provided that Δ is 1-consistent and recursively axiomatizable, MPL(Δ) will be complete , and Boolos and McGee [5] show that MPL({true arithmetical sentences}) is complete in {true arithmetical sentences}. All of these results take as their starting point Artemov's demonstration in [1] that {true arithmetical sentences} is 1-reducible to MPL({true arithmetical sentences}).The aim here is to consolidate these results by providing a general theorem which yields all the other results as special cases. These results provide a striking contrast with the situation in modal sentential logic (MSL); according to fundamental results of Solovay [8], provided Γ does not entail any falsehoods, MSL({true arithmetical sentences}) and MSL(PA) (which is the same as MSL(Γ)) are both decidable.

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Xuan, Ti Zuo. "Integers free of small prime factors in arithmetic progressions." Nagoya Mathematical Journal 157 (2000): 103–27. http://dx.doi.org/10.1017/s0027763000007212.

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For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

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Solomon, Reed. "– CA0 and order types of countable ordered groups." Journal of Symbolic Logic 66, no. 1 (March 2001): 192–206. http://dx.doi.org/10.2307/2694917.

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Reverse mathematics uses subsystems of second order arithmetic to determine which set existence axioms are required to prove particular theorems. Surprisingly, almost every theorem studied is either provable in RCA0 or equivalent over RCA0 to one of four other subsystems: WKL0, ACA0, ATR0 or – CA0. Of these subsystems, – CA0 has the fewest known equivalences. This article presents a new equivalence of – C0 which comes from ordered group theory.One of the fundamental problems about ordered groups is to classify all possible orders for various classes of orderable groups. In general, this problem is extremely difficult to solve. Mal'tsev [1949] solved a related problem by showing that the order type of a countable ordered group is ℤαℚε where ℤ is the order type of the integers, ℚ is the order type of the rationals, α is a countable ordinal, and ε is either 0 or 1. The goal of this article is to prove that this theorem is equivalent over RCA0 to – CA0.In Section 2, we give the basic definitions and notation for RCA0, ACA0 and CA0 as well as for ordered groups. For more information on reverse mathematics, see Friedman, Simpson, and Smith [1983] or Simpson [1999] and for ordered groups, see Kokorin and Kopytov [1974] or Fuchs [1963]. Our notation will follow these sources. In Section 3, we show that – CA0 suffices to prove Mal'tsev's Theorem and the reversal is done over RCA0 in Section 4.

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Li, Chao, and Yihang Zhu. "Arithmetic intersection on GSpin Rapoport–Zink spaces." Compositio Mathematica 154, no. 7 (May 16, 2018): 1407–40. http://dx.doi.org/10.1112/s0010437x18007108.

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We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport–Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan–Gross–Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic–geometric side of the arithmetic fundamental lemma proved by Rapoport–Terstiege–Zhang in the minuscule case.

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Kobal, Damjan. "Matrix zeros of polynomials." Mathematical Gazette 104, no. 559 (March 2020): 27–35. http://dx.doi.org/10.1017/mag.2020.4.

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The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analogue to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley-Hamilton theorem.

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Solomon, Reed. "Ordered Groups: A Case Study in Reverse Mathematics." Bulletin of Symbolic Logic 5, no. 1 (March 1999): 45–58. http://dx.doi.org/10.2307/421140.

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The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson (1985) gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups (including an equivalent of ACA0), groups and semigroup conditions which imply orderability (WKL0), the orderability of free groups (RCA0), Hölder's Theorem (RCA0), Mal'tsev's classification of the order types of countable ordered groups ( – CA0)

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Fesenko, Ivan. "Analysis on arithmetic schemes. II." Journal of K-theory 5, no. 3 (May 20, 2010): 437–557. http://dx.doi.org/10.1017/is010004028jkt103.

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AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

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Artal Bartolo, Enrique, José Ignacio Cogolludo-Agustín, Benoît Guerville-Ballé, and Miguel Marco-Buzunáriz. "An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group." Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 111, no. 2 (April 16, 2016): 377–402. http://dx.doi.org/10.1007/s13398-016-0298-y.

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Khan, Wilayat, Muhammad Kamran, Syed Rameez Naqvi, Farrukh Aslam Khan, Ahmed S. Alghamdi, and Eesa Alsolami. "Formal Verification of Hardware Components in Critical Systems." Wireless Communications and Mobile Computing 2020 (February 20, 2020): 1–15. http://dx.doi.org/10.1155/2020/7346763.

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Hardware components, such as memory and arithmetic units, are integral part of every computer-controlled system, for example, Unmanned Aerial Vehicles (UAVs). The fundamental requirement of these hardware components is that they must behave as desired; otherwise, the whole system built upon them may fail. To determine whether or not a component is behaving adequately, the desired behaviour of the component is often specified in the Boolean algebra. Boolean algebra is one of the most widely used mathematical tools to analyse hardware components represented at gate level using Boolean functions. To ensure reliable computer-controlled system design, simulation and testing methods are commonly used to detect faults; however, such methods do not ensure absence of faults. In critical systems’ design, such as UAVs, the simulation-based techniques are often augmented with mathematical tools and techniques to prove stronger properties, for example, absence of faults, in the early stages of the system design. In this paper, we define a lightweight mathematical framework in computer-based theorem prover Coq for describing and reasoning about Boolean algebra and hardware components (logic circuits) modelled as Boolean functions. To demonstrate the usefulness of the framework, we (1) define and prove the correctness of principle of duality mechanically using a computer tool and all basic theorems of Boolean algebra, (2) formally define the algebraic manipulation (step-by-step procedure of proving functional equivalence of functions) used in Boolean function simplification, and (3) verify functional correctness and reliability properties of two hardware components. The major advantage of using mechanical theorem provers is that the correctness of all definitions and proofs can be checked mechanically using the type checker and proof checker facilities of the proof assistant Coq.

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Böckle, Gebhard, and Chandrashekhar Khare. "Mod $\ell$ representations of arithmetic fundamental groups II: A conjecture of A. J. de Jong." Compositio Mathematica 142, no. 02 (March 2006): 271–94. http://dx.doi.org/10.1112/s0010437x05002022.

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Ahmed, Asad, Osman Hasan, Falah Awwad, and Nabil Bastaki. "Formalization of Cost and Utility in Microeconomics." Energies 13, no. 3 (February 6, 2020): 712. http://dx.doi.org/10.3390/en13030712.

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Cost and utility modeling of economics agents based on the differential theory is fundamental to the analysis of the microeconomics models. In particular, the first and second-order derivative tests are used to specify the desired properties of the cost and utility models. Traditionally, paper-and-pencil proof methods and computer-based tools are used to investigate the mathematical properties of these models. However, these techniques do not provide an accurate analysis due to their inability to exhaustively specify and verify the mathematical properties of the cost and utility models. Additionally, these techniques cannot accurately model and analyze pure continuous behaviors of the economic agents due to the utilization of computer arithmetic. On the other hand, an accurate analysis is direly needed in many safety and cost-critical microeconomics applications, such as agriculture and smart grids. To overcome the issues pertaining to the above-mentioned techniques, in this paper, we propose a theorem proving based methodology to formally analyze and specify the mathematical properties of functions used in microeconomics modeling. The proposed methodology is primarily based on a formalization of the derivative tests and root analysis of the polynomial functions, within the sound core of the HOL-Light theorem prover. We also provide a formalization of the first-order condition, which is used to analyze the maximum of the profit function in a higher-order-logic theorem prover. We then present the formal analysis of the utility, cost and first-order condition based on the polynomial functions. To illustrate the usefulness of proposed formalization, the proposed formalization is used to formally analyze and verify the quadratic cost and utility functions, which have been used in an optimal power flow problem and demand response (DR) program, respectively.

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HILDEN, HUGH M., MARIA TERESA LOZANO, and JOSÉ MARIA MONTESINOS-AMILIBIA. "UNIVERSAL 2-BRIDGE KNOT AND LINK ORBIFOLDS." Journal of Knot Theory and Its Ramifications 02, no. 02 (June 1993): 141–48. http://dx.doi.org/10.1142/s021821659300009x.

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Let (p/q, n) be the orbifold with cyclic isotropy of order n and with singular set the 2-bridge knot or link p/q where p and q are relatively prime numbers, q is odd, q is less than p, and q is not congruent to ±1 mod p (i.e. p/q is any non toroidal 2-bridge knot or link). We show that the orbifold fundamental group π1(p/q, n) is universal for n any multiple of 12. This means that if Γ is any such group, it can be thought of as a discrete group of hyperbolic isometries of hyperbolic 3-space ℍ3, and then, given any closed, oriented 3-manifold M, there exists a subgroup of finite index G of Γ such that M is homeomorphic to G\ℍ3. Since we have shown elsewhere that the group π1(5/3, 12) is an arithmetic group, it follows that there exists an orbifold, namely (5/3, 12), whose singular set is a knot, the figure eight, and whose fundamental group is both arithmetic and universal.

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Gupal, N. A. "Methods of Numeration of Discrete Sequences." Cybernetics and Computer Technologies, no. 2 (June 30, 2021): 63–67. http://dx.doi.org/10.34229/2707-451x.21.2.6.

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Introduction. Numeration, or code, discrete sequences act fundamental part in the theory of recognition and estimation. By the code get codes or indexes of the programs and calculated functions. It is set that the universal programs are that programs which will realize all other programs. This one of basic results in the theory of estimation. On the basis of numeration of discrete sequences of Godel proved a famous theorem about incompleteness of arithmetic. Purpose of the article. To develop synonymous numerations by the natural numbers of eventual discrete sequences programs and calculable functions mutually. Results. On the basis of numerations of eventual discrete sequences numerations are built for four commands of machine with unlimited registers (MUR) in the natural numbers of type of 4u, 4u +1, 4u+2, 4u+3 accordingly. Every program consists of complete list of commands. On the basis of bijection for four commands of MUR certainly mutually synonymous numerations for all programs of MUR. Thus, on the basis of the set program it is possible effectively to find its code number, and vice versa, on the basis of the set number it is possible effectively to find the program. Conclusions. Synonymous numerations by the natural numbers of complete discrete sequences are developed mutually, programs for MUR and calculable functions. Leaning against numeration of the programs it is set in the theory of calculable functions, that the universal programs are, that programs which will realize all other programs. By application of the calculated functions and s-m-n theorem are got to operation on the calculated functions: combination φx and φy, giving work φxφy, operation of conversion of functions, effective operation of recursion. Thus, the index of function φxφy is on the indexes of x and y [2]. Keywords: numeration, Godel code number, diagonal method.

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Guglielmetti, R. "CoxIter – Computing invariants of hyperbolic Coxeter groups." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 754–73. http://dx.doi.org/10.1112/s1461157015000273.

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CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).Supplementary materials are available with this article.

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Pavlidis, Archimedes, and Dimitris Gizopoulos. "Fast quantum modular exponentiation architecture for Shor's factoring alogrithm." Quantum Information and Computation 14, no. 7&8 (May 2014): 649–82. http://dx.doi.org/10.26421/qic14.7-8-8.

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We present a novel and efficient, in terms of circuit depth, design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a quantum modular multiplier which is the fundamental block for the modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about $2000n^2$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is $1600n^3$. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.

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Journal articles on the topic 'Fundamental theorem of arithmetic'

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