# Foias constant

In mathematical analysis, the Foias constant is a real number named after Ciprian Foias.

It is defined in the following way: for any real number x1 > 0, there is a sequence defined by the recurrence relation

${\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}}$

for n = 1, 2, 3, .... The Foias constant is the unique choice α such that if x1 = α then the sequence diverges to infinity.[1] Numerically, it is

${\displaystyle \alpha =1.187452351126501\ldots }$ .

No closed form for the constant is known.

When x1 = α then we have the limit:

${\displaystyle \lim _{n\to \infty }x_{n}{\frac {\log n}{n}}=1,}$

where "log" denotes the natural logarithm. Consequently, one has by the prime number theorem that in this case

${\displaystyle \lim _{n\to \infty }{\frac {x_{n}}{\pi (n)}}=1,}$

where π is the prime-counting function.[1]