Backhouse's constant

Template:Short description Template:Use shortened footnotes Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value is approximately 1.456074948.

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

$P(x)=1+\sum _{k=1}^{\mathrm{\infty }}{p}_k{x}^k=1+2x+3x^2+5x^3+7x^4+\cdots$

and its multiplicative inverse as a formal power series,

$Q(x)=\frac{1}{P(x)}=\sum _{k=0}^{\mathrm{\infty }}{q}_k{x}^k.$

Then:

$\underset{k\to \mathrm{\infty }}{lim}\left|\frac{q_{k+1}}{q_k}\right|=1.45607\dots$.Template:R

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

Further reading

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