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Real Number Calculations and Theorem Proving
[chapter]

2008
*
Lecture Notes in Computer Science
*

Wouldn't it be nice to be able to conveniently use ordinary

doi:10.1007/978-3-540-71067-7_19
fatcat:lccfqatnmrg5fal3zvorrlktuu
*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]

2005
*
Lecture Notes in Computer Science
*

Wouldn't it be nice to be able to conveniently use ordinary

doi:10.1007/11541868_13
fatcat:isfmzar76bh3rhuq6c6npolepm
*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

1964
*
Indagationes Mathematicae (Proceedings)
*

We arbitrarily choose a

doi:10.1016/s1385-7258(64)50030-x
fatcat:zq7g6fnztbh55fq5i2u2bm3rra
*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]

2013
*
arXiv
*
pre-print

The note is accessible for students familiar with polynomials

arXiv:0908.2029v6
fatcat:46nepmpg6rhkxblhq5shjsao2e
*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

1991
*
Journal of Number Theory
*

ACKNOWLEDGMENTS We are grateful to Anne-Marie Berg6

doi:10.1016/s0022-314x(05)80044-7
fatcat:zckzwkq4z5hdlilpvzytwntbly
*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]

2013
*
arXiv
*
pre-print

A classical

arXiv:1304.4261v1
fatcat:mxoj3heazbfjjkvapoe3jy4jji
*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

1971
*
Proceedings of the American Mathematical Society
*

It is

doi:10.2307/2037255
fatcat:qxuk5eb7izejhaabd65ys7tlpy
*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

1971
*
Proceedings of the American Mathematical Society
*

It is

doi:10.1090/s0002-9939-1971-0269603-3
fatcat:r7ormq5i2bgalb6r5sajsn2bem
*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

2006
*
Zenodo
*

For any positive integer n

doi:10.5281/zenodo.9503
fatcat:6a257ahgpvcybany4ajwznfkty
*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) ...##
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Iteration of Möbius transformations and attractors on the real line

1997
*
Computers and Mathematics with Applications
*

We

doi:10.1016/s0898-1221(96)00215-5
fatcat:ixo5n4cuwzbxtcqeydwnn3ozvy
*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*. ...##
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Lyapunov revisited

1983
*
Linear Algebra and its Applications
*

The celebrated

doi:10.1016/0024-3795(83)80033-0
fatcat:trn2loqqhbctxggrqtnlhakrqq
*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). ...##
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Products and quotients of numbers with small partial quotients

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

doi:10.5802/jtnb.364
fatcat:fu24ruzajrcffiwglxxrg4ddom
*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*...##
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Units and Periodic Jacobi-Perron Algorithms in Real Algebraic Number Fields of Degree 3

1975
*
Transactions of the American Mathematical Society
*

The main result of this paper is expressed in the following

doi:10.2307/1998626
fatcat:3emnaprxlnfbrhezqiwcyte3wq
*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 $3$

1975
*
Transactions of the American Mathematical Society
*

The main result of this paper is expressed in the following

doi:10.1090/s0002-9947-1975-0376504-7
fatcat:tkejwga7dbd43gctn7teq4qita
*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]

2019
*
arXiv
*
pre-print

With the help of this computation, we

arXiv:1806.07560v4
fatcat:j3xuu7aqkrho5onxozrkae5ffi
*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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