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Euler pentagonal number theorem

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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 27.14 Unrestricted Partitions
Euler introduced the reciprocal of the infinite product …Euler’s pentagonal number theorem states that …where the exponents 1 , 2 , 5 , 7 , 12 , 15 , are the pentagonal numbers, defined by … Multiplying the power series for f ( x ) with that for 1 / f ( x ) and equating coefficients, we obtain the recursion formula … This is related to the function f ( x ) in (27.14.2) by …
3: 20.12 Mathematical Applications
§20.12(i) Number Theory
For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s τ ( n ) function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). … The space of complex tori / ( + τ ) (that is, the set of complex numbers z in which two of these numbers z 1 and z 2 are regarded as equivalent if there exist integers m , n such that z 1 - z 2 = m + τ n ) is mapped into the projective space P 3 via the identification z ( θ 1 ( 2 z | τ ) , θ 2 ( 2 z | τ ) , θ 3 ( 2 z | τ ) , θ 4 ( 2 z | τ ) ) . …
4: Bibliography
  • A. Abramov (1960) Tables of ln Γ ( z ) for Complex Argument. Pergamon Press, New York.
  • W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.
  • G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
  • T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.
  • H. Appel (1968) Numerical Tables for Angular Correlation Computations in α -, β - and γ -Spectroscopy: 3 j -, 6 j -, 9 j -Symbols, F- and Γ -Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.
  • 5: 24.10 Arithmetic Properties
    §24.10 Arithmetic Properties
    §24.10(i) Von Staudt–Clausen Theorem
    §24.10(ii) Kummer Congruences
    §24.10(iii) Voronoi’s Congruence
    §24.10(iv) Factors
    6: 27.2 Functions
    §27.2(i) Definitions
    (See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … This is the number of positive integers n that are relatively prime to n ; ϕ ( n ) is Euler’s totient. If ( a , n ) = 1 , then the Euler–Fermat theorem states that …The ϕ ( n ) numbers a , a 2 , , a ϕ ( n ) are relatively prime to n and distinct (mod n ). …
    7: 24.4 Basic Properties
    §24.4(ii) Symmetry
    §24.4(iii) Sums of Powers
    §24.4(iv) Finite Expansions
    Raabe’s Theorem
    Next, …
    8: 27.16 Cryptography
    §27.16 Cryptography
    Thus, y x r ( mod n ) and 1 y < n . … To do this, let s denote the reciprocal of r modulo ϕ ( n ) , so that r s = 1 + t ϕ ( n ) for some integer t . (Here ϕ ( n ) is Euler’s totient (§27.2).) By the Euler–Fermat theorem (27.2.8), x ϕ ( n ) 1 ( mod n ) ; hence x t ϕ ( n ) 1 ( mod n ) . …
    9: 27.12 Asymptotic Formulas: Primes
    §27.12 Asymptotic Formulas: Primes
    Prime Number Theorem
    where λ ( α ) depends only on α , and ϕ ( m ) is the Euler totient function (§27.2). … There are infinitely many Carmichael numbers.
    10: 24.17 Mathematical Applications
    Euler–Maclaurin Summation Formula
    Euler Splines
    are called Euler splines of degree n . …
    §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)). …