## Largest Prime Number Discovered – With More Than 23m Digits (mersenne.org) 117

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.*
## I'll fine one right now (Score:1, Insightful)

2^98,435,672 -- 1

This is easy. Where's my fookin medal?

## Re: (Score:3)

Exercise for the interested maths nerd: Prove that if q is any even number, then 2^q - 1 is divisible by 3.

## Primes with prime ordinals (Score:2)

One is prime, and it is the first prime, which is also prime.

Two is prime, and its ordinal is 2, which is prime.

3 is prime, its ordinal is 3, also prime

5 is prime, its ordinal is 4, which is not prime

7 is prime, its ordinal is 5, which is prime

11 is prime, its ordinal is 6, not prime

etc

So is there a rule that would answer whether any given prime's ordinal in the list of primes is also prime?

Extra points for a calculator trick to answer this.

Super extra bonus points: is there a largest prime number

## Re: (Score:2)

Also, every number has a unique prime factorization. Consider 1 prime and you get the prime factorizations of 10 being 2*5, 1*2*5, 1*1*2*5, etc. Also, if 1 is prime, then 1 = 1 * 1, so 1 is the multiple of two primes. It's much easier to consider 1 neither prime nor composite, just the multiplicative identity.

## Re: (Score:2)

The Sieve of Eratosthenes is perhaps the oldest method of identifying prime numbers, BUT it is not the definition of a prime number. It is an algorithm ---not a definition--- that was not developed until a long time after the Greeks (and probably other ancient peoples) had identified the nature of prime numbers.

The definition of a prime number is any number that is only divisible by itself and by one. Do not mistake an early algorithm for identifying prime numbers with the definition of prime. For one thin

## Re: (Score:2)

I studied mathematics for some time. I know what I'm talking about. Under your definition, one of the fundamental theorems of arithmetic (that each number has a unique prime factorization) is false. This has nothing to do with the Sieve, which as you point out is an algorithm. It has to do with making useful mathematical definitions.

Mathematics doesn't depend on the physical world. It's a collection of deductions from axioms and definitions, and we tailor the axioms and definitions to be useful. De

## Re: (Score:2)

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. More concisely, a prime number p is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. [see Wolfram MathWorld [wolfram.com]]

Note that when p is set to 1, the test is also true: its only positive integer divisors are itself, p, and 1. (1/1=1 which is both 1 and the current value of p). Wolfram simply ignores this edge case, since for all but esoteric levels of advanced mathematics it is not necessary to go there.

From a historical perspective this makes sense. When at the end of the day's harvest it was time to evenly divide up the bags of grain that had been gathered in, it turned out that it was always the case that if there we

## Re: (Score:1)

Fails if q = -2.

## Re: (Score:2)

+1, appropriately pedantic

## Re:I'll fine one right now (Score:5, Interesting)

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)

## Re: (Score:2)

Nice argument!

That was clearly too easy. Show that 2^98,435,672 - 1 is divisible by 5.

## Re: (Score:2)

Very good. To lay this out formally, you should say up front that you're using proof by truthiness.

## Re: (Score:2)

Note that the same argument can be used to show that 2^(4n)-1 is divisible by 15.

In general, 2^(mn)-1 is divisible by 2^m - 1 (and without loss of generality 2^n - 1). It follows that if p is not prime, 2^p-1 isn't prime either. And

thatis how I instantly knew that 2^98,435,672-1 couldn't possibly be prime.## Re: (Score:2)

Nice argument!

That was clearly too easy. Show that 2^98,435,672 - 1 is divisible by 5.

It won't calculate on Excel, therefore its truth is unverifiable.

## Re: (Score:2)

## Re: (Score:2)

2^98,435,672 - 1 is equal to (2^49217836 + 1)(2^49217836 - 1); it's a difference of squares.

But we're discussing primes. what's 4^2 - 3^2 ?

## Re: (Score:3, Interesting)

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.

## Re: (Score:2)

Good stuff!

## Re: (Score:2)

As an extension... every prime greater than 3 is of the form 6n+1 or

6n -1 (wish I knew how to get the plus or minus sign into Slash comments)

6n is obviously not prime (divisible by 6)

6n+2, 6n+4 are divisible by 2

6n+3 is divisible by 3

which just leaves 6n+1 and 6n+5

6n+5 is equivalent to 6m-1 (where m=n+1)

so 6n+1 6n-1 are necessary but not sufficient conditions but do provide a quick & dirty test for primality for small - medium sized numbers [divisibility by 2 is easily checked; divisibility by 3

## Re: (Score:2)

All primes p are odd, p+1 and p-1 are always even.

Except for 2, which is prime and even.

## Re: (Score:2)

wow you were able to check that pretty fast. it took a Radeon Vega GPU over 34 hours to verify the number in the OP.

## If GIMPS Can Find Such a Huge Prime (Score:5, Funny)

Just think how big a prime PHOTOSHOPS could find!

## Re:If GIMPS Can Find Such a Huge Prime (Score:5, Informative)

Serious reply in response to a decent joke: GIMPS is the Great Internet Mersenne Prime Search [wikipedia.org], which is more or less like SETI@home or Folding@home, but for Mersenne primes. I wasn't aware what it was, so I figured I'd share. Also, I had forgotten that Prime95, which is oftentimes used to stress cooling solutions in PCs, was actually created for use in finding large prime numbers, and was apparently developed by GIMPS.

## Application (Score:1)

## Re:Application (Score:5, Informative)

"At present there are few practical uses for this new large prime, prompting some to ask "why search for these large primes"? Those same doubts existed a few decades ago until important cryptography algorithms were developed based on prime numbers. For seven more good reasons to search for large prime numbers, see here [utm.edu].

## Re:Application (Score:4, Interesting)

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.

## Re:Application (Score:5, Interesting)

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.

## Re: (Score:3)

A typical practical usage is PRNGs of girgantual periods (the period is typically the MP itself, or multiple of thereof) for HPC number crunching. The perfect number property indeed is a nice bonus in there, as it often leads to better k-distr

## Re: (Score:3, Informative)

The more prime numbers that are discovered, the more likely we are to be able to discover a pattern within an arbitrary base number set. The larger numbers are useful because we also want to make sure that the entire range is consistent, or in other words that any pattern, or lack of pattern, is the same across the entire set of numbers. There is always a benefit to trying to find patterns in number theory -- it's one of the coolest and most interesting fields in pure mathematics.

## Re: (Score:2)

I have to wonder if looking for just Mersenne primes will reveal anything interesting about the primes in general. It seems unlikely to me.

Having said that, finding a pattern in the Mersenne exponents would be very interesting indeed.

## Re: (Score:2)

An entity which could encode a message in the distribution of Mersenne exponents would be very powerful indeed. It'd be even harder than encoding a message into physical constants which a hyper advanced civilisation may be able to do by creating new universes inside black holes.

## Re: (Score:3)

think of all the dogecoin they could have mined!

## Re: (Score:3, Informative)

## Re: (Score:2)

It's interesting for pure mathematics people. There are some minor applications for this although it's mostly theoretical.

Once we get bigger computers that can hold these numbers, they may be used to prime a PRNG or a cryptographic constant, especially once quantum computing starts threatening the constants in the lower ranges. Quantum computers can't break cryptography, they just do it faster and for larger primes you still need more q-bits.

Highly theoretical there may be some constant to the prime numbers

## Headline Color? (Score:1)

## Posted by FirehoseFavorites? (Score:2)

Is this an indication that we're going to be getting more placed content and less user-voted content?

I'm not saying that's a good or bad thing - I'm just wondering what this is... or maybe it's already been around a while and I just am not observant.

## Re: (Score:2)

Note that the story now has changed to say "Posted by msmash", and the banner color from dark blue to the familiar green. Perhaps someone jumped the gun..

## Re:Posted by FirehoseFavorites? (Score:5, Informative)

## Re: (Score:1)

Wouldn't /. or is improving it not the aim?

moreeditor input be the way to improve## Re: (Score:2)

## Re: (Score:2)

Wouldn't /. or is improving it not the aim?

moreeditor input be the way to improveYou must be new round here ^_^

## Re: (Score:2)

Wouldn't /. or is improving it not the aim?

moreeditor input be the way to improveI think that depends on your opinion of the editors.

## Re: (Score:3)

FirehoseFavorites is purely user voted content. Something new we are testing. Requires zero editor input to make it to the front page, just user votes from the firehose.

Ah, that makes sense - thank you for the explanation.

## Re: (Score:2, Insightful)

Unleash the bots!

## Fireho5hin (Score:2)

Is this the second coming of Kuro5hin?

## Headline writer is a boob (Score:4, Insightful)

Subject says it all.

## Re: (Score:2)

## Re: (Score:2)

So 6 extra characters (known_) is apparently a bridge too far? Its not that the posted headline can be interpreted in different ways; it is outright wrong.

## Re: (Score:2)

The phrase "largest prime number discovered" refers to the largest prime that has been discovered. I know English is a hard language, but it's not

thathard.Having said that, pity the Slashdotter who needs "raising a number to the power of two" explained to them.

## Re: (Score:2)

The phrase "largest prime number discovered" refers to the largest prime that has been discovered. I know English is a hard language, but it's not that hard.

The problem is that the phrase is ambiguous.

It could mean:

The (largest prime number) has been discovered. -- i.e. there is a largest prime number and we just found it.

or it could mean:

The largest prime number [that we have so far] discovered [is two-to-the-whatever-minus-one].

Presumably the headline writer mean to communicate the latter, but ended up mostly communicating the former. It's true that people familiar with the properties of prime number can probably

## Off by two (Score:2)

discovered the 50th known Mersenne prime, 2^77,232,917 -- 1 on December 26, 2017.

I've done the math and 2^77,232,917 -- 1 is not prime. Although decrementing it by 2 to get 2^77,232,917 - 1 is indeed a prime.

## Re: (Score:2)

Please write down all numbers from 0 to your prime for proof.

I've done what you suggested, but the result was slightly too large to include in the margins of this web page.

## Thats (Score:5, Funny)

## Re: (Score:2)

Never mind, the TSA staff have cut your lock off. They're going through your underwear as we speak.

## Re: (Score:2)

## Found a bigger one (Score:2)

2^77,232,917 + 1

## Re: (Score:2)

Number Theory 101 exercise: why is 3 the only Mersenne number to be the smaller half of a prime pair?

## Re: (Score:2)

## Re: (Score:2)

There's a simpler pigeonhole answer: one of (2^n - 1), 2^n, (2^n + 1) is divisible by 3. Clearly it's not 2^n. Therefore if (2^n - 1) is a prime other than 3, (2^n + 1) is the one which is divisible by 3.

## Re: (Score:1)

That IS divisible by 3. (M=2^p-1 isn’t, M+1=2^p isn’t, so M+2 is.)

## Re: (Score:2)

That's not how prime numbers work.

## Re: (Score:2)

Standard? This is my Bitcoin wallet key. I'm ruined!

## Re: (Score:2)

## Might be a provable prime, but... (Score:1)

Now for the important questions: Is it executable? And is it illegal?

http://fatphil.org/maths/illeg... [fatphil.org]

## Here's part of it... (Score:2)

I evaluated the full number using the mpmath Python library. It starts with:

46733318335923109998833558556111...

and ends with: ...1136582730618069762179071

It took over an hour, but there are likely better ways to do it than I did, even with mpmath.

## Re: (Score:2)

Did they check it on Intel or AMD CPU?Both.

https://www.mersenne.org/prime... [mersenne.org]

The results were checked with several different pieces of software on multiple platforms.