In mathematics, Cahen's constant is defined as the value of an infinite series of unit fractions with alternating signs: Here
denotes Sylvester's sequence, which is defined recursively by Combining these fractions in pairs leads to an alternative expansion of Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence.
This series for Cahen's constant forms its greedy Egyptian expansion: This constant is named after Eugène Cahen [fr] (also known for the Cahen–Mellin integral), who was the first to introduce it and prove its irrationality.
[1] The majority of naturally occurring[2] mathematical constants have no known simple patterns in their continued fraction expansions.
[3] Nevertheless, the complete continued fraction expansion of Cahen's constant
where the sequence of coefficients is defined by the recurrence relation
All the partial quotients of this expansion are squares of integers.
Davison and Shallit made use of the continued fraction expansion to prove that
is transcendental.
[4] Alternatively, one may express the partial quotients in the continued fraction expansion of Cahen's constant through the terms of Sylvester's sequence: To see this, we prove by induction on
in the first step respectively the recursion for
in the final step.
As a consequence,
, from which it is easy to conclude that
Cahen's constant
has best approximation order
That means, there exist constants
has infinitely many solutions
has at most finitely many solutions
This implies (but is not equivalent to) the fact that
has irrationality measure 3, which was first observed by Duverney & Shiokawa (2020).
To give a proof, denote by
the sequence of convergents to Cahen's constant (that means,
As a consequence, the limits (recall that
) both exist by basic properties of infinite products, which is due to the absolute convergence of
Numerically, one can check that
0 < α < 1 < β < 2
Thus the well-known inequality yields for all sufficiently large
has best approximation order 3 (with
to is necessarily a convergent to Cahen's constant.