Euler pentagonal number theorem

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1: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, 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 $E_n$, $E_n\left(x\right)$, 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 $1$, $2$, $5$, $7$, $12$, $15$, $\mathrm{\dots }$ are the pentagonal numbers, defined by … ►Multiplying the power series for $f\left(x\right)$ with that for $1/f\left(x\right)$ and equating coefficients, we obtain the recursion formula … ►This is related to the function $f\left(x\right)$ in (27.14.2) by …

3: 20.12 Mathematical Applications

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§20.12(i) Number Theory

… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ 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 $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (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 $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …

4: Bibliography

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Tables of $\ln \mathrm{\Gamma }(z)$ for Complex Argument. Pergamon Press, New York.

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External Links:

ZentralBlatt

Referenced by:

§5.22(iii).

Encodings:

BibTeX

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Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

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Notes:

In memory of Gian-Carlo Rota

External Links:

ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt

Referenced by:

§17.12.

Encodings:

BibTeX

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A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

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External Links:

MathReview (Giovanni Coppola), ZentralBlatt

Referenced by:

(25.16.2), (25.16.4), §25.16(i), §25.16.

Encodings:

BibTeX

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Numerical Tables for Angular Correlation Computations in $\alpha$-, $\beta$- and $\gamma$-Spectroscopy: $3j$-, $6j$-, $9j$-Symbols, F- and $\mathrm{\Gamma }$-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:

ISBN 978-3-540-04218-1

Referenced by:

§34.14.

Encodings:

BibTeX

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5: 27.16 Cryptography

§27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . … ►To do this, let $s$ denote the reciprocal of $r$ modulo $\varphi \left(n\right)$, so that $rs=1+t\varphi \left(n\right)$ for some integer $t$. (Here $\varphi \left(n\right)$ is Euler’s totient (§27.2).) By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. …

6: 27.2 Functions

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§27.2(i) Definitions

… ►(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 …The $\varphi \left(n\right)$ numbers $a,a^2,\mathrm{\dots },a^{\varphi \left(n\right)}$ are relatively prime to $n$ and distinct (mod $n$). …

7: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

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§24.10(i) Von Staudt–Clausen Theorem

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§24.10(ii) Kummer Congruences

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§24.10(iii) Voronoi’s Congruence

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§24.10(iv) Factors

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8: 24.4 Basic Properties

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§24.4(ii) Symmetry

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§24.4(iii) Sums of Powers

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§24.4(iv) Finite Expansions

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Raabe’s Theorem

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9: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

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Prime Number Theorem

… ►where $\lambda (\alpha )$ depends only on $\alpha$, and $\varphi \left(m\right)$ is the Euler totient function (§27.2). … ►There are infinitely many Carmichael numbers.

10: 24.17 Mathematical Applications

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Euler–Maclaurin Summation Formula

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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)). …