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Largest Prime Number Discovered – With More Than 23m Digits (mersenne.org)
chalsall writes: Persistence pays off. Jonathan Pace, a GIMPS volunteer for over 14 years, discovered the 50th known Mersenne prime, 2^77,232,917 -- 1 on December 26, 2017. The prime number is calculated by multiplying together 77,232,917 twos, and then subtracting one. It weighs in at 23,249,425 digits, becoming the largest prime number known to mankind. It bests the previous record prime, also discovered by GIMPS, by 910,807 digits. You can read a little more in the press release.
Not a rigorous proof, but here's my favorite explanation:
for any positive integer k, the binary representation of 2^k-1 consists of k 1's. If k is even, this is an even number of 1's lined up together. Since 3 is 11 in binary, you can divide 2^k-1 by 3 and get a quotient of the form 10101..01.
e.g. 2^10 = 1111111111=11(101010101)
To clarify, these types of primes aren't useful for cryptography as they are much too large and not 'typical' primes.
From a theoretical perspective they are quite interesting: they are in bijective correspondence with perfect numbers and no one knows whether there are infinitely many. For all we know, this one could theoretically be the last.
"What lies behind us, and what lies before us are tiny matters compared to what lies within us." Ralph Waldo Emerson
Exercise for the interested maths nerd: Prove that if q is any even number, then 2^q - 1 is divisible by 3.
Because q is even, we can write 2^q - 1 = [2^r -+1] * [2^r - 1], where 2r = q and r is a (positive) integer.
Now consider the numbers 2^r - 1, 2^r, and 2^r + 1. One of these numbers must be divisible by 3. Clearly it is not 2^r because it only has factors that are powers of 2. Therefore either 2^r - 1 or 2^r + 1 must be divisible by 3. QED.
If it weren't for deadlines, nothing would be late.
OK people, we're done here! We found the last prime! Time to shut it down! You don't have to go home, but you can't stay here!
OK, that was a joke but we can still be clear. He was talking about the last perfect number. There is an infinite number of primes. That proof is pretty simple.
1. Assume there is a limited number of primes. Given the list of all the prime numbers
2. Multiply them all together and add 1.
The new number you get is can not divisible by any of the prime numbers in your list (e.g. if you divide the number by 2, you have a reminder of 1, if you divide the number by 3, you have a remainder of 1, if you divide the number by 5, you have a remainder of 1...)
So there must be at least one number not on your list which invalidates the given statement.