Jan van Leeuwen, S. Barry Cooper's Alan Turing: His Work and Impact PDF

By Jan van Leeuwen, S. Barry Cooper

ISBN-10: 0123869803

ISBN-13: 9780123869807

In this obtainable new collection of writings by means of info Age pioneer Alan Turing, readers will locate some of the most important contributions from the four-volume set of the Collected Works of A. M. Turing.
These contributions, including commentaries from present specialists in a large spectrum of fields and backgrounds, supply perception at the importance and modern impression of A.M. Turing's paintings.

Offering a extra sleek standpoint than whatever presently on hand, Alan Turing: His paintings and Impact supplies large insurance of the various ways that Turing's clinical endeavors have impacted present examine and realizing of the realm. His pivotal writings on topics together with computing, synthetic intelligence, cryptography, morphogenesis, and extra reveal endured relevance and perception into today's medical and technological landscape.

This assortment offers a good provider to researchers, yet can be an approachable access aspect for readers with constrained education within the technological know-how, yet an urge to benefit extra concerning the information of Turing's work.

• cheap, key choice of the main major papers through A.M. Turing.
• statement explaining the importance of every seminal paper by means of preeminent leaders within the box.
• extra assets on hand online.

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Merten’s theorem (see for example [BS96]) states that n 1 pi = eγ , n→∞ log n p − 1 i i=1 lim where pi is the ith prime number. Thus this probability is asymptotically equal to (eγ log ap)/p log 2. The first two claims easily follow from this observation. 2 Let E be a prime-generating elliptic curve defined over the field Fq . Then the following statements can be made about the distribution of the primes generated by E: (i) The number of primes of the form Ek /E1 less than or equal to x, is approximately (eγ / log q) log log x.

It is unfortunately not known whether or not there exist prime-generating elliptic curves, but some good candidates will be given later. First, the possible values of e for which Ee /E1 can be prime will be narrowed down. 3 Let E be an elliptic curve defined over a finite field Fq . Suppose that Ee /E1 is a prime. Then one of the following statements holds: (i) e is prime, (ii) q = 2 and e ∈ {4, 6, 9}1 , (iii) q = 3 and e = 4. Proof: Suppose that e is not a prime and let f be a non-trivial divisor of e.

Then the Frobenius eigenvalues are given by 1 ± i. Furthermore  e 2 − 2e/2+1 + 1    e   2 − 2(e+1)/2 + 1 Ee = 2e + 1   2e + 2(e+1)/2 + 1    e 2 + 2e/2+1 + 1 if if if if if e≡0 e ≡ 1, 7 e ≡ 2, 6 e ≡ 3, 5 e≡4 (mod (mod (mod (mod (mod 8), 8), 8), 8), 8). Let E be the elliptic curve defined over F3 with Weierstrass equation Y 2 = X 3 −X−1. √ Then the Frobenius eigenvalues are given by (3 ± i 3)/2 and  e 3 − 2 · 3e/2 + 1     3e − 3(e+1)/2 + 1     3e − 3e/2 + 1  Ee = 3e + 1   3e + 3e/2 + 1      3e + 3(e+1)/2 + 1   e 3 + 2 · 3e/2 + 1 if if if if if if if e≡0 e ≡ 1, 11 e ≡ 2, 10 e ≡ 3, 9 e ≡ 4, 8 e ≡ 5, 7 e≡6 (mod (mod (mod (mod (mod (mod (mod 12), 12), 12), 12), 12), 12), 12).

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Alan Turing: His Work and Impact by Jan van Leeuwen, S. Barry Cooper

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