Porter's constant

Template:Short description In mathematics, Porter's constant C arises in the study of the efficiency of the Euclidean algorithm. It is named after J. W. Porter of University College, Cardiff.

Euclid's algorithm finds the greatest common divisor of two positive integers Template:Mvar and Template:Mvar. Hans Heilbronn proved that the average number of iterations of Euclid's algorithm, for fixed Template:Mvar and averaged over all choices of relatively prime integers Template:Math, is

$\frac{12\ln 2}{{\pi }^2}\ln n+o(\ln n).$

Porter showed that the error term in this estimate is a constant, plus a polynomially-small correction, and Donald Knuth evaluated this constant to high accuracy. It is:

$\begin{array}{cc}C& =\frac{6\ln 2}{{\pi }^2}[3\ln 2+4\gamma -\frac{24}{{\pi }^2}{\zeta }^{\prime }(2)-2]-\frac{1}{2}\\ & =\frac{6\ln 2((48\ln A)-(\ln 2)-(4\ln \pi )-2)}{{\pi }^2}-\frac{1}{2}\\ & =1.4670780794\dots \end{array}$

where

$\gamma$ is the Euler–Mascheroni constant

$\zeta$ is the Riemann zeta function

$A$ is the Glaisher–Kinkelin constant

$-{\zeta }^{\prime }(2)=\frac{{\pi }^2}{6}[12\ln A-\gamma -\ln (2\pi )]=\sum _{k=2}^{\mathrm{\infty }}\frac{\ln k}{k^2}$

${\displaystyle }$

Approximations

The simple approximations of the Porter's constant accurate up to 3 digits can be found as

$C=2^{21/38}=\mathrm{1.466758730139474...},$

accurate up to 4 digits as

$C=2^{26/47}=\mathrm{1.467328090156686...},$

and up to 6 digits as

$C=2^{47/85}=\mathrm{1.467073525425601....}$

See also

• Lochs' theorem
• Lévy's constant

References

Template:Numtheory-stub