Fermat%20last%20theorem

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1: 27.2 Functions

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§27.2(i) Definitions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►This is the number of positive integers $\le n$ that are relatively prime to $n$; $\varphi \left(n\right)$ is Euler’s totient. ►If $\left(a,n\right)=1$, then the Euler–Fermat theorem states that …

2: 27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. …

3: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

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§35.9.

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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13 Lectures on Fermat’s Last Theorem. Springer-Verlag, New York.

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ISBN 0-387-90432-8, MathReview (T. Metsänkylä), ZentralBlatt

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§24.10(iv), §24.17(iii).

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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§7.25(vi).

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4: 24.17 Mathematical Applications

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§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …

5: 20 Theta Functions

Chapter 20 Theta Functions

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6: Bibliography C

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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MathReview (K. Goldberg), ZentralBlatt

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§24.6.

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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Document

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5th item.

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

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7: 27.18 Methods of Computation: Primes

… ►These algorithms are used for testing primality of Mersenne numbers, $2^n-1$, and Fermat numbers, $2^{2^n}+1$. …

8: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product $m$ has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

9: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

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10: 27.8 Dirichlet Characters

… ►For any character $\chi (modk)$, $\chi \left(n\right)\ne 0$ if and only if $\left(n,k\right)=1$, in which case the Euler–Fermat theorem (27.2.8) implies ${\left(\chi \left(n\right)\right)}^{\varphi \left(k\right)}=1$. …