The Top 5000: Downloading the List
The List of Largest Known Primes
The Largest Known Primes Icon
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  The list of 5000 largest known primes will always be available at primes.utm.edu.  So you can view it and search it here!

But if you want to download a snapshot of the list to print or view at another time, there are several possibilities (this files were last modified on 30 July 2014 11:53:00 am):

all.txt
The whole list as a printable text file!  This is a large file: 429.42 KB.
all.zip
The whole list (all.txt) zipped, so it is roughly one fourth the size of all.txt: 107.38 KB.
all.pdf and all_pdf.zip
The complete list, this time as an Adobe PDF file (so it looks nicer, but is bigger)
short.txt
All of the primes with 500,000 digits or more, plus the 'interesting' smaller primes (that is, those with comments on the list). So this is a much smaller file: 132.63 KB.
short.pdf and short_pdf.zip
The same files, now as Adobe PDF's.

You might also consider one of the summary pages (these are automatically updated when new primes are accepted onto the list!)

The Top Twenty
The Top Twenty lists is a series of pages each dedicated to listing the top twenty primes of certain selected forms.
The Largest Known Primes
This is a single page which summarizes much of the information in the pages about the 5000 largest known primes from introduction through list of the top ten primes of a few selected forms.

Related lists outside of this collection:

Henri & Renaud Lifchitz's PRP Top record
A list of large probable-primes.  These are numbers that are likely to be prime, but the primality has not been proven.  (The lists above are of proven primes only.)