Biggest Primes

This is the list of all prime numbers with more than 3 million digits:

From https://primes.utm.edu/primes/lists/all.txt

    1  2^57885161-1                    17425170 G13   2013 Mersenne 48??
    2  2^43112609-1                    12978189 G10   2008 Mersenne 47??
    3  2^42643801-1                    12837064 G12   2009 Mersenne 46??
    4  2^37156667-1                    11185272 G11   2008 Mersenne 45?
    5  2^32582657-1                     9808358 G9    2006 Mersenne 44
    6  2^30402457-1                     9152052 G9    2005 Mersenne 43
    7  2^25964951-1                     7816230 G8    2005 Mersenne 42
    8  2^24036583-1                     7235733 G7    2004 Mersenne 41
    9  2^20996011-1                     6320430 G6    2003 Mersenne 40
   10  2^13466917-1                     4053946 G5    2001 Mersenne 39
   11  19249*2^13018586+1               3918990 SB10  2007 
   12c 3*2^11895718-1                   3580969 L4159 2015 
   13f 3*2^11731850-1                   3531640 L4103 2015 
   14  3*2^11484018-1                   3457035 L3993 2014 
   15  3*2^10829346+1                   3259959 L3770 2014 

From: http://www.primenumbers.net/prptop/prptop.php

1 (2^13372531+1)/3 4025533 Ryan Propper 09/2013
2 (2^13347311+1)/3 4017941 Ryan Propper 09/2013

I don't know why the last two aren't in the main prime numbers database.

Update:  The last 2 numbers are called Wagstaff primes, and it is extremely difficult to prove that they are actually prime.  That is why the Mersenne primes are so attractive, because there is a test for primality.

The number that starts with 19249 is part of the Seventeen or Bust projects, which is an attempt to prove (or disprove) the  Sierpinski problem, which I don't understand.

The numbers that start with 3*2 are part of the 321 Prime Search project on PrimeGrid, which keeps finding new megaprimes.

What is the benefit of finding such large numbers?  Really none at all.  Except to expand the horizons of what is thought possible.