# Foias constant

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

If x1 > 0 and

${\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots ,}$

then the Foias constant is the unique real number α such that if x1 = α then the sequence diverges to ∞.[1] Numerically, it is

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

No closed form 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 usual natural logarithm.

A fortuitous observation between the prime number theorem and this constant goes as follows,

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

where π is the prime-counting function.[2]

## Notes and references

1. Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse and G. Păun). London: Springer-Verlag, pp. 119–126, 2000.
2. Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse and G. Păun). London: Springer-Verlag, pp. 119–126, 2000.
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