Euler%E2%80%93Fermat%20theorem
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1: 24.1 Special Notation
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Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 27.2 Functions
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►This is the number of positive integers that are relatively prime to ; is Euler’s totient.
►If , then the Euler–Fermat theorem states that
…The numbers are relatively prime to and distinct (mod ).
…Note that .
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►Table 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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3: 27.16 Cryptography
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►Thus, and .
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►To do this, let denote the reciprocal of modulo , so that for some integer .
(Here is Euler’s totient (§27.2).)
By the Euler–Fermat theorem (27.2.8), ; hence .
But , so is the same as modulo .
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4: 24.17 Mathematical Applications
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Euler–Maclaurin Summation Formula
… ►Euler Splines
… ►are called Euler splines of degree . … ►§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 -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)); -adic analysis (Koblitz (1984, Chapter 2)). …5: Bibliography
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Tables of for Complex Argument.
Pergamon Press, New York.
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Some determinants of Bernoulli, Euler and related numbers.
Portugal. Math. 18, pp. 91–99.
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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument.
ACM Trans. Math. Software 16 (2), pp. 178–182.
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A proof that Euler missed: Evaluating the easy way.
Math. Intelligencer 5 (3), pp. 59–60.
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Numerical Tables for Angular Correlation Computations in -, - and -Spectroscopy: -, -, -Symbols, F- and -Coefficients.
Landolt-Börnstein Numerical Data and Functional Relationships
in Science and Technology, Springer-Verlag.
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6: 24.4 Basic Properties
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§24.4(i) Difference Equations
… ►§24.4(ii) Symmetry
… ►§24.4(iii) Sums of Powers
… ►§24.4(iv) Finite Expansions
… ►Next, …7: 5.22 Tables
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►Abramowitz and Stegun (1964, Chapter 6) tabulates , , , and for to 10D; and for to 10D; , , , , , , , and for to 8–11S; for to 20S.
Zhang and Jin (1996, pp. 67–69 and 72) tabulates , , , , , , , and for to 8D or 8S; for to 51S.
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►Abramov (1960) tabulates for () , () to 6D.
Abramowitz and Stegun (1964, Chapter 6) tabulates for () , () to 12D.
…Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of , , and for , to 8S.
8: 24.2 Definitions and Generating Functions
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§24.2(ii) Euler Numbers and Polynomials
… ►§24.2(iii) Periodic Bernoulli and Euler Functions
… ► ► …9: Bibliography D
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On the real roots of Euler polynomials.
Monatsh. Math. 106 (2), pp. 115–138.
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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials.
J. Approx. Theory 49 (4), pp. 321–330.
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials.
Mem. Amer. Math. Soc. 73 (386), pp. iv+94.
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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