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### Real Number Calculations and Theorem Proving [chapter]

David R Lester
2008 Lecture Notes in Computer Science
Wouldn't it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework.  ...  The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs.  ...  This work was supported by the National Aeronautics and Space Administration under NASA Cooperative Agreement NCC-1-02043.  ...

### Real Number Calculations and Theorem Proving [chapter]

César Muñoz, David Lester
2005 Lecture Notes in Computer Science
Wouldn't it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework.  ...  The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs.  ...  This work was supported by the National Aeronautics and Space Administration under NASA Cooperative Agreement NCC-1-02043.  ...

### Markov Chains and Intuitionism. III Note on Continuous Functions with an Application to Markov Chains

J.G. Dijkman
1964 Indagationes Mathematicae (Proceedings)
We arbitrarily choose a real number 81 > 0 and we calculate a corresponding real number b1 according to (1), then from (1) follows t1, t2 E (0, b1) =* I f(t1)-f(t2) I< 81. Theorem.  ...  Now we have proved that for every real number x <)::: 0, for which {en} is a defining sequence, and for every real number 8 > 0 a natural number N can be calculated such that l!ln(x) -!  ...  In particular the function g(x) is continuous at x =a, hence: Now we choose a natural number k1 and we calculate a natural number l1 such that However, the function /( · ), which is defined on (oo, a),  ...

### Yet another proof from the Book: the Gauss theorem on regular polygons [article]

A. Skopenkov
2013 arXiv   pre-print
The note is accessible for students familiar with polynomials and complex numbers, and could be an interesting easy reading for professional mathematicians.  ...  The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known.  ...  Prasolov and M. N. Vyalyi for useful discussions. A reduction to complex numbers. A real number is called (real)-constructible, if we can calculate this number using our calculator.  ...

### Ratios of regulators in totally real extensions of number fields

Antone Costa, Eduardo Friedman
1991 Journal of Number Theory
ACKNOWLEDGMENTS We are grateful to Anne-Marie Berg6 and Jacques Martinet for their numerous and inspired letters. We are also indebted to M. Pohst, whose work [8] is essential to this paper.  ...  To prove (2.13) we use (2.11) and the known values of 7r to calculate Table I We now aim for Theorem 3, which bounds Reg(ELm) in terms of the discriminant of L.  ...  Let K S L and let L be a totally real number field. Then >( [L ! Q](log(!  ...

### A short proof of d'Alemberts theorem [article]

Tord Sjödin
2013 arXiv   pre-print
A classical theorem of d'Alembert states that if a polynomial P(x) with real coefficients has a non-real root x=a+ib, then it also has a root x=a-ib.  ...  We give a short and elementary inductive proof that avoids any properties of the complex conjugation operator.  ...  Fix any natural number k and assume that P k holds. Then A k is a real number and B k is a purely imaginary number and hence A k+1 is a real number by the recursion formula.  ...

### Sums and Products of Continued Fractions

T. W. Cusick
1971 Proceedings of the American Mathematical Society
It is proved that every real number is representable as a sum of two real numbers each of which has a fractional part whose continued fraction expansion contains no partial quotient less than 2, and that  ...  [l] proved that every real number is representable as a sum of two real numbers each of which has a fractional part whose continued fraction expansion contains no partial quotient greater than 4, and that  ...  It is proved that every real number is representable as a sum of two real numbers each of which has a fractional part whose continued fraction expansion contains no partial quotient less than 2, and that  ...

### Sums and products of continued fractions

T. W. Cusick
1971 Proceedings of the American Mathematical Society
It is proved that every real number is representable as a sum of two real numbers each of which has a fractional part whose continued fraction expansion contains no partial quotient less than 2, and that  ...  [l] proved that every real number is representable as a sum of two real numbers each of which has a fractional part whose continued fraction expansion contains no partial quotient greater than 4, and that  ...  It is proved that every real number is representable as a sum of two real numbers each of which has a fractional part whose continued fraction expansion contains no partial quotient less than 2, and that  ...

### On The Primitive Numbers Of Power P

Zhu Weiyi
2006 Zenodo
For any positive integer n and prime p, let Sp(n) denotes the smallest positive integer m such that m! is divisible by pn.  ...  And, let p be a fixed prime, then for any real number x ≥ 1, ∞ n=1 Sp(n)≤x 1 S p (n) = 1 p − 1 ln x + γ + p ln p p − 1 + O(x − 1 2 +ε ), where γ is the Euler constant, ε denotes any fixed positive number  ...  Yi Yuan [4] had studied the asymptotic property of S p (n) in the form 1 p n≤x |S p (n + 1) − S p (n)| , and obtained the following conclusion: For any real number x ≥ 2, we have 1 p n≤x |S p (n + 1)  ...

### Iteration of Möbius transformations and attractors on the real line

A. Barrlund, H. Wallin, J. Karlsson
1997 Computers and Mathematics with Applications
We prove that with probability one the orbit is attracted to the real axis. In the proof, we have to do some calculations on a computer.  ...  In Section 2, we state our main result as a theorem. The theorem is proved in Section 3 except for a lemma which is proved in Section 4 by computer calculations.  ...  By calculating the right-hand side of this relation we get the lemma. | When we apply this lemma to prove Theorem 1 we will have a < -1/4, which implies that ZA, B are complex conjugate numbers.  ...

### Lyapunov revisited

Morris Newman
1983 Linear Algebra and its Applications
The celebrated theorem of Lyapunov (see [2] ) states that a real n x n matrix A is stable (that is, has all its eigenvalues in the left half plane) if and only if a real symmetric positive definite matrix  ...  This theorem has been the subject of much study, and has been  ...  THEOREM 2 . 2 Let H, K be n X n hermitian matrices such that the eigenvalues of HK have positive real parts. Then H and K have the same number of positive eigenualues (and hence the same signature).  ...

### Products and quotients of numbers with small partial quotients

Stephen Astels
2002 Journal de Théorie des Nombres de Bordeaux
Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 387-Products and quotients of numbers with small partial quotients par STEPHEN ASTELS  ...  In this paper we characterize most products and quotients of sets of the form F(m).  ...  In 1947 Marshall Hall, Jr. proved [6] that where for two sets A and B of real numbers we denote by A ~ B the set In fact, we shall show in Theorem 1.2 that More generally we will examine products and  ...

### Units and Periodic Jacobi-Perron Algorithms in Real Algebraic Number Fields of Degree 3

Leon Bernstein
1975 Transactions of the American Mathematical Society
The main result of this paper is expressed in the following theorem: 3 There are infinitely many real cubic fields Q(w), w cubefree, a and T natural 2 numbers, such that the Jacobi-Perron Algorithm of  ...  Periodic Jacobi-Perron Algorithms are important, because they can be applied, inter alia, to calculate units in the corresponding algebraic number fields.  ...  Another important application is the calculation of units, as was mentioned in the. previous chapter. In [3(b) ] Hasse and the author proved the Basic Theorem. Let the J. P.  ...

### Units and periodic Jacobi-Perron algorithms in real algebraic number fields of degree \$3\$

Leon Bernstein
1975 Transactions of the American Mathematical Society
The main result of this paper is expressed in the following theorem: 3 There are infinitely many real cubic fields Q(w), w cubefree, a and T natural 2 numbers, such that the Jacobi-Perron Algorithm of  ...  Periodic Jacobi-Perron Algorithms are important, because they can be applied, inter alia, to calculate units in the corresponding algebraic number fields.  ...  Another important application is the calculation of units, as was mentioned in the. previous chapter. In [3(b) ] Hasse and the author proved the Basic Theorem. Let the J. P.  ...

### A computational criterion for the irrationality of some real numbers [article]

Peyman Nasehpour
2019 arXiv   pre-print
With the help of this computation, we prove that if a real number cannot be represented as a finite decimal and the asymptotic average of its decimals is zero, then it is irrational.  ...  In this paper, we compute the asymptotic average of the decimals of some real numbers.  ...  In Section 2, by The Chebychev's Estimate Theorem [7, Theorem 4.2.1] and calculating the asymptotic average of the decimals of some real numbers, we give an alternative proof for Hardy and Wright's theorem  ...
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