The number 163 is very important in number theory, since 
is the largest number such that the imaginary
quadratic field 
has class number 
.
It also satisfies the curious identities
 |  |  |
(1)
|
 |  | ![1/2[4^4+(8; 4)]](/images/equations/163/Inline9.svg) |
(2)
|
 |  | ![1/2[4^4+sum_(i=0)^(4)(4; i)^2],](/images/equations/163/Inline12.svg) |
(3)
|
where 
is a binomial coefficient.
