Largest prime numbers ever printed on the computer has 24 million digits with over $100,000 prize.

A portrait of Marin Mersenne and his prime formula


Prime numbers are positive numbers other than $1$ that have no divisors other than $1$ and itself e.g $2,3,5,7,11,13,17,19,23,29,...$. 
The search for very large prime numbers called Mersenne primes was recorded as early as the 14th century but was revolutionized in the 1940s when electronic digital computers first emerged, during this period the first mathematician to search digitally was Alan Turing who seached for them using the Manchester Mark 1 computer in 1949 but was unsuccessful. The first successful attempt was in 1952 in the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles, this first attempt produced the $M_{521}$ Mersenne prime. Other successful attempts have been launched since then and in 2020 the 24 million digits Mersenne prime that was printed in 2018 still remains the largest prime ever printed on a computer.
The Mersenne prime is a prime number that is one less than a power of two i.e $M_n=2^n-1$ for some $n$ prime, is used to evaluate a value for the largest known primes, named after Marin Mersenne, a French mathematician and scientist who introduced the Mersenne prime formula. The first terms of the numbers are $3,7,31,127,8191$. 
The longest record holder known was $M_{19}=524,287$ which was the largest known prime number for over 144 years, this record was set by Pietro Cataldi. As at 2018, Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) found $M_{82589933}$ Mersenne prime which was estimated at 24,862,048 digits and is $M_{82589933}$:
$148894445742041325$
$5478064584723979166$
$0302627399279532418$
$5271289425213239361$
$0644753103099711321$
$80337174752834401423587560...$
(24,861,808 digits omitted)
$...0621075579479582$
$9753159520880719269$
$3676521782184472526$
$6400769121143553083$
$1196948763376645782$
$3695074037951210325$
$217902591$
Which shows the first and last 120 digits only.  

Since then no new record has been set to beat this. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search (GIMPS) a distributed computing project. One question about the Mersenne prime which is yet to be solved is:
Are there infinitely many Mersenne primes? Because It is not even known whether the set of Mersenne primes is finite or infinite.  
Researchers intending to go into Mersenne primes can take this up. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite. The Mersenne primes have been studied to have a close connection to the perfect numbers, Euclid proposed this in the 14th century where he proved that if $2^p-1$ is a prime number then $2^{p-1}(2^p-1)$ is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. This is known as the Euclid–Euler theorem. Many theorems on Mersenne primes have been presented by number theorist and other interested mathematicians. 

The Great Internet Mersenne Prime Search (GIMPS) currently offers a US$3,000 research discovery award for participants who download and run their free software and whose computer discovers a new Mersenne prime having fewer than 100 million digits.
In 1999, the first one million digit was produced and won US$50,000 
price while in 2008 the record passed ten million digit and won US$100,000 price and a Cooperative Computing Award from the Electronic Frontier Foundation.
GIMPS is currently on the search for a 100 million digits which attracts a whooping US$150,000 price in collaboration with the Electronic Frontier Foundation.