E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant 
given by
 |
(1)
|
which is good to 6 digits.
M. Hudson (pers. comm., Nov. 8, 2004) found the approximations
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
is the Euler-Mascheroni constant and
is the golden ratio, which are good to 5, 7, 7, 8,
11, and 12 digits, respectively.
A curious approximation to 
is given by
 |
(8)
|
where 
is the Euler-Mascheroni constant, which
is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).
Lima (2009) found the approximation
 |
(9)
|
where 
is Catalan's constant, which is correct to 21
digits.
