Английская Википедия:Backhouse's constant

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Use shortened footnotes Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value is approximately 1.456 074 948.

It is defined by using the power series such that the coefficients of successive terms are the prime numbers,

<math>P(x)=1+\sum_{k=1}^\infty p_k x^k=1+2x+3x^2+5x^3+7x^4+\cdots</math>

and its multiplicative inverse as a formal power series,

<math>Q(x)=\frac{1}{P(x)}=\sum_{k=0}^\infty q_k x^k.</math>

Then:

<math>\lim_{k \to \infty}\left | \frac{q_{k+1}}{q_k} \right \vert = 1.45607\ldots</math>.Шаблон:R

This limit was conjectured to exist by Backhouse,Шаблон:R and later proven by Philippe Flajolet.Шаблон:R

References

Шаблон:Reflist

Further reading


Шаблон:Numtheory-stub