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1: 27.19 Methods of Computation: Factorization

§27.19 Methods of Computation: Factorization

►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. ►Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time. … ►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. … ►As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …

2: 27.22 Software

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Cunningham Project. This includes updates of factorization records.

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ECMNET Project. Links to software for elliptic curve methods of factorization and primality testing.

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Number Theory Web. References and links to software for factorization and primality testing.

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Prime Pages. Information on primes, primality testing, and factorization including links to programs and lists of primes.

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Wolfram’s Mathworld. Descriptions, references, and Mathematica algorithms for factorization and primality testing.

3: 33.4 Recurrence Relations and Derivatives

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33.4.2 $$R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-T_{\mathrm{\ell }}{X}_{\mathrm{\ell }}+R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1}=0,$$ $\mathrm{\ell }\ge 1$,

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Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $T_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.3

Permalink:

http://dlmf.nist.gov/33.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

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33.4.3 $$X_{\mathrm{\ell }}^{\prime }=R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-S_{\mathrm{\ell }}{X}_{\mathrm{\ell }},$$ $\mathrm{\ell }\ge 1$,

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.1

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

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33.4.4 $$X_{\mathrm{\ell }}^{\prime }=S_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }}-R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1},$$ $\mathrm{\ell }\ge 0$.

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.2

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

4: 28.22 Connection Formulas

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28.22.3 $${\mathrm{Mc}}_m^{(2)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{e,m}(h){\mathrm{ce}}_m(0,h^2)}\left(-f_{e,m}(h){\mathrm{Ce}}_m(z,h^2)+\frac{2}{\pi C_m(h^2)}{\mathrm{Fe}}_m(z,h^2)\right),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Fe}}_n(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors, $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors and $C_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.4 $${\mathrm{Ms}}_m^{(2)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{o,m}(h){\mathrm{se}}_m^{\prime }(0,h^2)}\left(-f_{o,m}(h){\mathrm{Se}}_m(z,h^2)-\frac{2}{\pi S_m(h^2)}{\mathrm{Ge}}_m(z,h^2)\right).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ge}}_n(z,q)$: modified Mathieu function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, ${\mathrm{Se}}_{\nu }(z,q)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors, $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors and $S_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

►The joining factors in the above formulas are given by … ►

28.22.9 $$f_{e,m}(h)=-\sqrt{\pi /2}{g}_{e,m}(h){\mathrm{Mc}}_m^{(2)}(0,h),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors

A&S Ref:

20.8.12 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.22.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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$${\mathrm{ge}}_m(0,h^2)=\frac{1}{2}\pi S_m(h^2){\left(g_{o,m}(h)\right)}^2{\mathrm{se}}_m^{\prime }(0,h^2).$$

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5: 21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

6: 27.16 Cryptography

… ►Applications to cryptography rely on the disparity in computer time required to find large primes and to factor large integers. … ►But to decode, both factors $p$ and $q$ must be known. With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers. …

7: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. In Wagstaff (2002) these results are extended to $n=60(2)152$ and $n=40(2)88$, respectively, with further complete and partial factorizations listed up to $n=300$ and $n=200$, respectively. …

8: David M. Bressoud

… ► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

9: 24.10 Arithmetic Properties

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§24.10(iv) Factors

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10: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. …7 of Abramowitz and Stegun (1964) also lists the factorizations in Glaisher’s Table I(a); Table 24. …