# fundamental theorem of arithmetic

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###### §19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
As $n\to\infty$, $a_{n}$ and $g_{n}$ converge to a common limit $M\left(a_{0},g_{0}\right)$ called the AGM (Arithmetic-Geometric Mean) of $a_{0}$ and $g_{0}$. …showing that the convergence of $c_{n}$ to 0 and of $a_{n}$ and $g_{n}$ to $M\left(a_{0},g_{0}\right)$ is quadratic in each case. …
19.8.5 $K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)},$ $-\infty.
Again, $p_{n}$ and $\varepsilon_{n}$ converge quadratically to $M\left(a_{0},g_{0}\right)$ and 0, respectively, and $Q_{n}$ converges to 0 faster than quadratically. …
##### 2: 27.2 Functions
###### §27.2(i) Definitions
Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, … (See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … If $\left(a,n\right)=1$, then the Euler–Fermat theorem states that …
##### 3: 28.29 Definitions and Basic Properties
###### §28.29(ii) Floquet’s Theorem and the Characteristic Exponent
28.29.4 $w_{\mbox{\tiny I}}(z+\pi,\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda),$
28.29.5 $w_{\mbox{\tiny II}}(z+\pi,\lambda)=w_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda).$
If $\nu$ $(\neq 0,1)$ is a solution of (28.29.9), then $F_{\nu}(z)$, $F_{-\nu}(z)$ comprise a fundamental pair of solutions of Hill’s equation. …
28.29.15 $\bigtriangleup(\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda)$
##### 4: 27.15 Chinese Remainder Theorem
###### §27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences $x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …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 $\pmod{m}$, which is correct to 20 digits. …
##### 5: 27.4 Euler Products and Dirichlet Series
The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …
##### 6: 28.2 Definitions and Basic Properties
(28.2.1) possesses a fundamental pair of solutions $w_{\mbox{\tiny I}}(z;a,q),w_{\mbox{\tiny II}}(z;a,q)$ called basic solutions with …
28.2.6 $\mathscr{W}\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}\right\}=1,$
###### §28.2(iii) Floquet’s Theorem and the Characteristic Exponents
If $q\neq 0$, then for a given value of $\nu$ the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
##### 8: Bibliography C
• L. Carlitz (1961b) The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.
• B. C. Carlson (1978) Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.
• H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.
• D. A. Cox (1984) The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.
• D. A. Cox (1985) Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.
• ##### 9: Bibliography K
• R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.
• Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series ${}_{r+3}F_{r+2}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
• B. J. King and A. L. Van Buren (1973) A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.
• D. E. Knuth (1968) The Art of Computer Programming. Vol. 1: Fundamental Algorithms. 1st edition, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont.
• T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.