The IMEI (International Mobile Equipment Identity) number is a unique set of 15 digits used on GSM phones
to identify them. Because the SIM card is associated with the user and
can be swapped from phone to phone, a method is needed to keep track of
the hardware itself, and that's why the IMEI was developed. Math nerds
will enjoy the way they are calculated -- the first 14 digits are
decided by the GSM association, and the final check digit is computed
using what's called the Luhn Formula --
crazy base-8 math that mere mortals like most of us don't understand,
nor want to understand. You can see the IMEI of your Android phone by
looking in settings > About phone if you're curious. Or under the battery or on the phone itself.
What is this number used for? That's the real question, isn't it. Like the MEID number on phones
(think Sprint and Verizon), the IMEI is used for network control. It's
not very common, but your mobile operator can block a phone based on
it's IMEI in cases where it's been reported stolen or someone didn't pay
the bill. Because it's not easy to change the IMEI of your phone (and
maybe even illegal -- check your local laws) it's also used to keep
track of phones that were involved in criminal activity, and the UK in
particular has a handy database of phones used for these purposes. The
IMEI number is also used to specify a phone for wiretapping by federal
governments worldwide.
On a lighter note, Android apps can also use your IMEI number.
and the IMEI can be used to keep track of the device in a remote
database. This sounds pretty scary, but it's an easy way (though not
necessarily the best way) for app developers to keep a settings database
online for your phone, in their app. Let's say you mark a bunch of
favorites in a wallpaper app. Those favorites are kept in a small
database file online, and when you reconnect to the app it reads your
IMEI number to find your preferences. Not an ideal method, but it's
easy.
One last thing -- now that we know a little
more about IMEI numbers, they will soon be changing to IMEISV numbers.
The use-case scenarios are the same, but the data structure and method
of calculation is different. An IMEISV gets rid of the check digit (and
its complicated Luhn formulated calculation) in favor of two digits used
for software version numbering. Like everything else in the mobile
space, network identification changes rapidly.

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