Should define what is "Arabic Numerals"

Nice kata! A couple of minor things you could improve:

• The yes, the leading zero not pass rule should be part of the main description.
• As it stands, any code can be run--not just regular expressions. You could require a simple variable as a RegExp object, and run the tests yourself.

Two things:
You say a) between 0 - 65536 (compare from a mathematical point) and also when Input is '0' Output should be false I think you should write between 1 - 65535 there or add a note stating the limits aren't included.
The second thing is in the tests labels, should return false when input bigger than 1 shouldn't it be should return false when input bigger than 65535?
I've already done both changes, see if that's ok with you.

input between 1 - 65535

So, I started writing

string > "1" && string < "65535"

Imagine my surprise when that didn't work! :P

Seriously, please specify more accurately we should compare numerically, and the bounds are inclusive ( "between" is used in the wild with inclusive and with exclusive bounds ).