@@numberphile2 It is believed that there are infinitely many Mersenne Primes and heuristics support it, but it has NOT been proven. I think it is a bit like the twin prime conjecture, they are pretty confident it is true.
While the largest prime is certainly interesting, what this has got me thinking about is: "what is the largest prime that we know where we know all the primes that are less than it?". Like, we presume that there are primes less than this new Mersenne Prime that we haven't discovered, so how far up have we gone wherein we're sure that there are none that we have missed? Is anyone working on driving that sequence forward?
We know for sure there are primes we don't know less than this new Mersenne prime because for every prime p, there is another prime less than 2p. Bertrand's postulate
17:50 it's not known whether there are infinitely many Mersenne primes, but a heuristic argument shows that you would guess that there are infinitely many (as with all such arguments about primes, it's based on there not being some sort of "conspiracy" that makes Mersenne numbers particularly less likely to be prime than would otherwise be assumed). However, they should be quite sparse of course (fairly clear given that Mersenne numbers themselves increase exponentially fast, combined with the 1/log n density of primes), and this can be roughly seen in the distribution of known Mersenne primes. Edit: I should add that such a heuristic argument shouldn't necessarily be fully trusted, because indeed there are trivial conspiracies for such numbers that indeed change how "likely" a Mersenne number is to be prime. For instance, a simple factorisation shows that the Mersenne number's exponent must be prime for it to be prime itself (well known), but on the contrary, a simple argument about the multiplicative order of 2 modulo an assumed prime factor shows that any prime factor of a Mersenne number 2^p - 1 for p prime must be a prime of the form 2kp + 1 (so factors are already somewhat rarer than for an "average number" of the same size).
The number of digits is a multiple of 80 so it prints out all nice and tidy in a default 80-column console. What a coincidence and yet the number itself has no factors, so wild.
There are about 2.75 million prime numbers between 82,589,933, the previous power of 2 and 136,279,841, 53¾ million powers later... not one of which yields a Mersenne prime! After nearly 6 years I was beginning to think, "Maybe there is only a finite number of Mersenne primes!"
@@Jupiterninja95, I believe they did but they are now double-checking all the numbers in between. This is what I got from Luke’s interview when he mentioned that now he’s switching his cluster to verify the backlog.
‘I’ve shifted the computer to work on the double checking backlog […], which is their experiments that they did Lucas-Lehmer testing on but they only got one failed result […]. Currently Mersenne prime number 48 through 52, there may be other Mersenne primes mixed in […] but I’m helping to steam-roll through that a little bit.’ So everything has been verified, but to make extra sure, he’s double checking Mersenne numbers between Mersenne prime 48 and 52.
The plank length which is roughly `1e-35 m` is the smallest length that exists (like an IRL pixel) Turn that into volume and you have roughly `1e-105 m^3`. The Volume of the observable universe is roughly `1e80 m^3` This means that the amount of smallest possible things (i.e. the amount of "pixels") in our observable universe is roughly `1e80 m^3 / 1e-105 m^3 =1e185` This number has 185 digits. This is the amount of smallest possible things that could ever exist in our whole (observable) universe and it's ONLY 185 digits long. Now how big is that prime number?
And yet, as much as the volume of the observable universe in planck volumes is essentially zero compared to this newly discovered prime, this prime is essentially zero compared to the likes of Graham's number or Tree(3) and it's incomprehensibly bigger numbers all the way up.
In Space Engine, accelerate to 100m light years per second. It takes 15 minutes to cross the Universe, Galaxy clusters passing like nighttime snowflakes in headlights. Try it. You're welcome.
18:00 No proof that there's infinite Mersenne primes, but heuristics suggest that the probability 2^p-1 is prime is on the order 1/p and the sum of 1/p diverges (extremely slowly) so I'm quite certain there's infinitely many mersenne primes but only the first ~50-100 are even accessible
Generally I think we can only safely say that there's a materialistic limit when it would probabilistically take a non-trivial amount of time to reach the next mersenne prime, in which we can safely say that is the limit for our modern computing hardware and lifetimes As Matt mentioned, that "number" is no longer in the region of "countable numbers" in the traditional sense of counting pebbles, so in terms of infinity, its no longer about the specific yes/no, it is about the technical capabilities in that point in time Theoretically, stronger computing would give you more and more digits
I remember the mile long prime. I forgot how awe inspiring it was thinking about it watching the video until matt referenced it was a religious experience then it all came back. Its crazy that we have numbers so big that its hard to comprehend the actual length of the number let alone its value
What's even more wild to me is that even though we can't comprehend the size of this number, we _can_ comprehend a proof for why there are infinitely many more primes larger than it. We can even figure out an estimate for how frequently primes occur at this magnitude (roughly 1 in 94 million numbers of this size are prime). Math is so crazy.
You could print it out in base 2. You could come up with a special font for printing it out in base 2. Base 2 with kerning. A one is 2x4, eight cells, three with dots. A zero is 4x4, 16 cells, eight with dots. Very compact. 20:18
The observable universe has around 10^80 particles and a total volume of 10^108 cubic nanometers. A number with this many digits is just impossible to think about in physical (counting) terms. It's a testament to the power of our mathematics that we can imagine and reason about such things.
the Tech Model Railroad Club at MIT birthed hacker culture. early open-source vibes, pioneered interactive computing, and developed one of the first video games, "Spacewar!" Their tinkering laid groundwork for AI advancements and modern computing.
I wonder if anyone has searched for interesting patterns in the decimal version of this number like fifty 9s in a row or the first hundred digits of pi.
A Mersenne number is any number of the form N = M(n) = 2ⁿ - 1, where n = any positive integer. A Mersenne prime is any Mersenne number that's prime; a necessary, but not sufficient condition for which, is that n be prime. The first four primes, p = {2, 3, 5, 7}, all produce Mersenne primes - M(p) = 2ᵖ - 1 = {3, 7, 31, 127}. The fifth prime, p = 11, is the first to fail to produce a Mersenne prime: M(11) = 2047 = 23·89. You needn't go far down the list of Mersenne primes before merely testing the "Mersenne seed," p, for primality, by hand, becomes daunting. The procedure used for testing M(p) for primality is something of a miracle of mathematics. Without it, Mersenne primes would never be famous. Fred
From the OEISWiki: It is not known whether the set of Mersenne primes is finite or infinite. The Lenstra-Pomerance-Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth.
As a Mathematician, what I think of as BIG is the Number of Bridge Hands, 52!, or Games of GO, 361! But those only look like 200 or 1000 Digits Max. Millions..... You can Save Time and Paper if you Print the Results in Hexadecimal.
Considering this took 6 years to find, how long do y’all think it’ll take to find the 53rd Mersenne Prime/Perfect Number? What are your thoughts, predictions, and methods you think could speed up the process?
I feel, I don't know, disappointed? It's like someone told me "hey we found the digit 11,548,211,798,954,357 of pi" and I say "Really, do we know all the digits op pi up to that one?" - "Oh no no... we don't know most of the digits below that one, but we do know that one!". Almost every prime number smaller than the new biggest prime are not known. I could not even find what is the largest prime in the longest uninterrupted (no gaps) list of primes, that is, what is the biggest prime for which all smaller primes are known. But I know it will be ridiculously small compared to this one because the largest non-Mersenne prime known is already ridiculously smaller and most primes smaller that that one are not know either.
13:53 I think the answer here is that Matt Parker, like many other mathematicians who have spent a substantial part of their career contemplating these large numbers, appreciates better than most humans exactly how bad we are at comprehending how large these numbers are. You think you can't, but you have no idea how badly you can't - not even close. Only mathematicians approach knowing exactly how badly we have no idea how badly we can't comprehend them.
Thinking on the largest discovered prime number... I'm wondering what the smallest number is that nobody knows whether or not it's a prime? and alternatively the smallest composite number which we don't know what factors it has
I wonder if there are (m)any primes between this largest one and the previously largest one.. What is the largest number below which we know for sure every prime number?
I did some quick back of the envelope calculations during the video (so I don't guarantee my values are correct even assuming the data is): Given that a million digits was a mile, 41 million digits would be 41 miles, which, at 3 miles per hour, would take almost 14 hours to walk - which is basically an entire day with some time at either end to get up and go to bed, and maybe some breaks for little things like eating. Very roughly, printing a hundred million digit number would take on the order of a week (so more than a couple of days, but probably less than a month) - and require some well-planned paper management. If you can fit 5000 digits on a single sheet (which is probably a slight overestimate) then you'd need 20,000 sheets to fit a hundred million digits, which would stack about the height of a tall person (a ream of 500 sheets is around 5cm or 2 inches, so 40 reams is 2m or 80 inches - the difference between the two is less than the errors in approximation)
The Online Encyclopedia of Integer Sequences (OEIS) has a page that converts sequences to MIDI files. Neil Sloane has done some Numberphile videos with that music. See the OEIS website with link "Music" at the bottom of the page.
If every particle in the Universe contained an entire universe of that many particles, and each of those... for 400,000 levels, you still wouldn't be more than this number.
So this is the highest prime number, but. Being a mersenne prime there are a lot of prime numbers between this one and the previous record.... Now my question is, which prime number discovered is the biggest one with no gaps, without skipping any prime number in between?
If you plant grass you might be able to imagine 41,000,000 pieces of grass in groups of 1,000s but that's just the digits definitely a few orders of magnitude off from imagining these numbers
The way I conceptualized it was reading out 3 digits per second 24 hours aa day would take 158 days. But again, like all the other methods, that's still only conceptualizing 41 million, and not the number itself. Lol. Mind boggling
Does anybody know the largest value, for which the prime counting function is not known EXACTLY. Or in other words, what order of magnitude is the smallest unknown prime.
This is more of a philosophical question of what it means to be "known". To have it stored? You can easily generate all 64-bit primes (i.e. all primes below 2^64) on the fly - and that's actually done by a lot of software that is involved in prime-finding (sieving stuff mostly). But there's about 4.15829×10^17 of them. If you want to store their decimal representation with a single space character separating them it's gonna take (I made some calculations through wolfram alpha) 7127 petabytes. Of course, much more efficient to store them in binary. 2955 petabytes. Still way too much for any reasonable person to store. But that doesn't make any of them "unknown".
I only vaguely understand the process by which we find these numbers. From a theory perspective, what're the odds there are 0 other primes between this and the previous largest known prime?
There are many primes between the current and previous record holder. if you meant other "Mersenne primes", we will need to wait for the backlog re-verification to be sure that no Mersenne primes exist between 2^82589933 -1 and 2^136279841 -1.
And if Matt wants to make this printing installation for the next Mersenne prime, he can print it in binary and start even before it's found. Just make it print 1s. Then, when the prime is found, just update it with when to stop.
In what sense are Mersenne Primes the easiest to find, as Matt claims at the beginning. We’ve only got 52 verified, right? Sounds like they are the hardest to find. What hidden qualifiers in his claim am I missing?
We have proven that we know 100% of the Mersenne primes less than 10²⁰⁰⁰⁰⁰⁰⁰. We will only ever know 0% of all primes less than this limit, to the nearest 1%.
(There are roughly 10¹⁹⁹⁹⁹⁹⁹³ primes less than this limit whereas the age of the universe in Planck time units multiplied by the volume of the observable universe in Planck volume units is about 10⁶⁰ . 10¹⁸⁶ < 10³⁰⁰.)
@@backwashjoe7864 Yes those are (roughly speaking) the hidden qualifiers that you asked for. A different set of qualifiers could give a different answer. For instance, we are absolutely sure we know the first 52 primes, yet we are guessing that we know the first 52 Mersenne primes, since there may be some between the ones discovered so far. A more specific interpretation of “easy to find” is the process rather than the outcome - we have no practical way of proving whether an arbitrary prime with 41 million digits is actually prime. It is only the special structure of Mersenne primes that allows us to determine whether Mersenne numbers of that size are prime. The one nuance hidden in my “roughly speaking” statement is that we can’t speak of a “percentage of the ones that exist” since there are infinitely many primes, and there are probabilistic (non-rigorous) indications that there are infinitely many Mersenne primes. So we do need some way of making it a percentage of a finite quantity.
The test for primality is simpler to calculate for Mersenne primes than for ordinary primes. Thus the largest known Mersenne prime is (much) bigger than the largest known non-Mersenne prime.
'The less incorrect' is grammatically more correct than 'the least', given that there are only 2 folk involved - and also less incorrect, as it happens.
Not possible. 2^p isn't divisible by 3, so either 2^p+2 or 2^p-2 must be divisible by 3. 2^p+2 can't be divisible by 3, because then 2^p-1 would also be divisible by 3. Therefore, 2^p-2 must be divisible by 3, which means 2^p+1 is also divisible by 3. p=2 is the only exception, giving 3 and 5 as the only pair of twin primes consisting of a Mersenne prime and a Fermat prime generated from the same power of 2. And even without the above, if there are any more Fermat primes, they require the exponent to be a power of two, where as Mersenne primes require the exponent to be a prime. it's 2^p-3 we need to look at if we want to find twin primes to Mersenne primes.... Though I've checked the first few Mersenne primes and the only such pairs below a billion are 5 and 7 and 29 and 31, but that leaves 40-some Mersenne primes to check for a little twin.
Since the probability of n being prime is 1/log(n), the probability of (2^n-1) goes like 1/n, so we "know" there are infinitely many, barring an improbable conspiracy. That's not a mathematical proof, but it's a scientific certainty.
I don't think you can really use "scientific certainty" in this kind of context. And what qualifies as an "improbable conspiracy"? Numbers have structure, they are not purely probabilistic.
In other videos he said it cost him (?) less than 2 million in server rent to calculate. Either he's crazy obsessed by primes or had some generous help from others to get all the money to fund this.
A Googol is 10^100 2^10 = 1024 is approximately 1000 = 10^3 10^99 is approximately 2^330 The exponent on this prime is in the hundreds of millions, so flipping huge compared to a Googol. 2^100 million is about 10^30 million, though considering this prime is 41 million or so digits, those 24s in our rough conversion between powers of 2 and powers of 10 are adding up
In perhaps 100,000 years, mathematicians might discover new principles or structures that enhance our understanding of prime numbers, potentially even leading to a function that generates them.
When you print the Mersenne Prime #52 in a book, it takes 10500 pages where each contains 4000 digits (80 x 50) , you will realize that you wont be able to spot a printing mistake in 1000th page 🤣.
Why is this titled and done like Matt isnt a regular appearance 😂 I'm guessing its because he's technically still on vacation, not to mention something tells me this is technically vacation material for him to talk about this new mersenne prime, even better when its for numberphile (i'd imagine) For me, this seems like it can be pretty useful as well towards the question of "Is there a true limit?" Because there will come a point when it would probabilistically take a non-trivial amount of time to reach the next mersenne prime, in which we can safely say that is the limit for our modern computing hardware and lifetimes Also, might break encryption/cipher algorithms requiring the use of prime numbers (i.e. RSA), but I'd presume that would speed up the motivation to improve computing, instead of whatever the heck Nvidia is doing just messing around with money
@@numberphile2 How long do you think you could sustain 1 digit a second? The point of giving the time for 5 digits per second is that that's about the fastest most people can rattle off digits (doing some quick research, there are world records up in the 8 per second range - though the record for reciting most memorised digits of pi is around 1 per second for about 17 hours), and that's already taking significantly more than "a few hours". When you're talking about more than a couple of days, you start having to schedule eating, drinking and sleeping (the essential activities you can't read out digits during), and it stops being a simple calculation...
More like a few months. Average human speech rate is about 180 words per minute or about 10,800 words per hour. 41 million divided by 10,800 is about 3800 hours or about 160 days if he recited digits 24/7. Granted, except for 7 and 0, all of the digits are monosyllabic and zero has it's monosyllabic synonyns, so even at a normal speech rate, digits might be faster than most speech... still, for comparison, Harry Potter and the Deathly Hallows is about 200k words and its audiobook has a total runtime of about 24 hours and we're talking reciting 41 million digits.
Something about this is very against what it means to be interested in the nuances of mathematics and number theory. You think a new biggest prime is a big deal, and then you realize that if AWS put a chunk of servers on this, they’d have several new biggest primes by lunch tomorrow. I mean, yay! but where is the compelling story?
@@david-melekh-ysroel but that's wrong. It isn't defined because it has no consistent meaning. Quantum have probabilistic values until it decoheres than it has a definite value.
On the topic of why spend time and resources on this. I imagine that a lot of constants or ratios or physics things were discoverd/proven/invented entirely in theoretical contexts and only found a practical usecase years/centuries/millenia later. RSA encryption uses prime numbers, IIRC. With increased computing power, breaking the RSA encryption becomes less impossible, so you'd need to use larger primes. If we stopped finding bigger primes, that list would eventually run out. I doubt that usecase was on the minds of Mersenne, Euler or Aristotle, when they messed around with maths.
Disagree. Finding the Next Prime yields just another Arbitrary Piece of Data. It's just More of the Same. Perhaps the Ancients knew when Enough was Enough.
If we've had Mersenne Primes for thousands of years, why are they named after a 17th century French mathematician? Shouldn't they instead be named after a mathematician from antiquity?
Well, certain bits of mathematics were discovered a long time ago in China or India yet they’re still associated with someone much later who rediscovered them…
Isn't their a law, observation, or something that states things named after people are rarely named after the first person to discover them? Also, I'm willing to bet at least four of the 5 known Fermat primes were known in antiquity even if their relevance to the problem of constructible polygons wasn't, and even 65537 seems like it might be small enough to confirm primality by hand if one cared to do so.
Matt is the kind of guy who supports anything 'green', yet he doesn't boycott the huge waste of resources involved in finding such large prime numbers.
The people searching for these numbers develop software that can be used elsewhere. You can't really optimize software without running it for some time. I'm pretty sure that most things you've done today were less useful than searching for the next biggest Mersenne prime.
New prime has Brady uploading like rent is due
I'll admit I am pretty excited
@@numberphile2 It is believed that there are infinitely many Mersenne Primes and heuristics support it, but it has NOT been proven.
I think it is a bit like the twin prime conjecture, they are pretty confident it is true.
@@LocallyConstantDuck New plan for when rent is due: spend millions of dollars running a computer farm to find a prime. 🙂
New born gotta eat!
Bradys rent must be a huge prime number lol😂
$2M to find a "new largest" prime, and only $1M prize to solve a problem which settles primes for all.
''if you're not keeping nerds entertained and having fun, its gonna slow down the entire nerd industry. You know, math as a whole''
Classic
Quite true for most areas as well, because if you dont have bits and pieces of recreation, you'll burn out pretty quickly
While the largest prime is certainly interesting, what this has got me thinking about is: "what is the largest prime that we know where we know all the primes that are less than it?". Like, we presume that there are primes less than this new Mersenne Prime that we haven't discovered, so how far up have we gone wherein we're sure that there are none that we have missed? Is anyone working on driving that sequence forward?
According to Wikipedia, The Goldbach conjecture verification project supposedly calculated every prime smaller than 4 x 10^18
Great question!
We know for sure there are primes we don't know less than this new Mersenne prime because for every prime p, there is another prime less than 2p. Bertrand's postulate
17:50 it's not known whether there are infinitely many Mersenne primes, but a heuristic argument shows that you would guess that there are infinitely many (as with all such arguments about primes, it's based on there not being some sort of "conspiracy" that makes Mersenne numbers particularly less likely to be prime than would otherwise be assumed).
However, they should be quite sparse of course (fairly clear given that Mersenne numbers themselves increase exponentially fast, combined with the 1/log n density of primes), and this can be roughly seen in the distribution of known Mersenne primes.
Edit: I should add that such a heuristic argument shouldn't necessarily be fully trusted, because indeed there are trivial conspiracies for such numbers that indeed change how "likely" a Mersenne number is to be prime. For instance, a simple factorisation shows that the Mersenne number's exponent must be prime for it to be prime itself (well known), but on the contrary, a simple argument about the multiplicative order of 2 modulo an assumed prime factor shows that any prime factor of a Mersenne number 2^p - 1 for p prime must be a prime of the form 2kp + 1 (so factors are already somewhat rarer than for an "average number" of the same size).
+
The number of digits is a multiple of 80 so it prints out all nice and tidy in a default 80-column console. What a coincidence and yet the number itself has no factors, so wild.
There are about 2.75 million prime numbers between 82,589,933, the previous power of 2 and 136,279,841, 53¾ million powers later... not one of which yields a Mersenne prime! After nearly 6 years I was beginning to think, "Maybe there is only a finite number of Mersenne primes!"
Have they said they verified all the primes in between? I didn't think they had yet
@@Jupiterninja95 Depends on how much you require to "verify"
Ongoing I believe - so while the new one is the 52nd found, it *might* not be the 52nd in sequence... We'll see.
@@Jupiterninja95, I believe they did but they are now double-checking all the numbers in between. This is what I got from Luke’s interview when he mentioned that now he’s switching his cluster to verify the backlog.
‘I’ve shifted the computer to work on the double checking backlog […], which is their experiments that they did Lucas-Lehmer testing on but they only got one failed result […]. Currently Mersenne prime number 48 through 52, there may be other Mersenne primes mixed in […] but I’m helping to steam-roll through that a little bit.’
So everything has been verified, but to make extra sure, he’s double checking Mersenne numbers between Mersenne prime 48 and 52.
The plank length which is roughly `1e-35 m` is the smallest length that exists (like an IRL pixel)
Turn that into volume and you have roughly `1e-105 m^3`.
The Volume of the observable universe is roughly `1e80 m^3`
This means that the amount of smallest possible things (i.e. the amount of "pixels") in our observable universe is roughly `1e80 m^3 / 1e-105 m^3 =1e185`
This number has 185 digits.
This is the amount of smallest possible things that could ever exist in our whole (observable) universe and it's ONLY 185 digits long.
Now how big is that prime number?
That is astonishing
And yet, as much as the volume of the observable universe in planck volumes is essentially zero compared to this newly discovered prime, this prime is essentially zero compared to the likes of Graham's number or Tree(3) and it's incomprehensibly bigger numbers all the way up.
non infinite enormous numbers should be called 'Big'. They are essentially identical to each other.
In Space Engine, accelerate to 100m light years per second. It takes 15 minutes to cross the Universe, Galaxy clusters passing like nighttime snowflakes in headlights. Try it. You're welcome.
18:00
No proof that there's infinite Mersenne primes, but heuristics suggest that the probability 2^p-1 is prime is on the order 1/p
and the sum of 1/p diverges (extremely slowly)
so I'm quite certain there's infinitely many mersenne primes but only the first ~50-100 are even accessible
Generally I think we can only safely say that there's a materialistic limit when it would probabilistically take a non-trivial amount of time to reach the next mersenne prime, in which we can safely say that is the limit for our modern computing hardware and lifetimes
As Matt mentioned, that "number" is no longer in the region of "countable numbers" in the traditional sense of counting pebbles, so in terms of infinity, its no longer about the specific yes/no, it is about the technical capabilities in that point in time
Theoretically, stronger computing would give you more and more digits
"I'm now obsessed with {subject}" *defines* a nerd!
And long may it continue!
1) Matt - have a shave already
2) Well done, posting your first (?) Rush video!
I remember the mile long prime. I forgot how awe inspiring it was thinking about it watching the video until matt referenced it was a religious experience then it all came back. Its crazy that we have numbers so big that its hard to comprehend the actual length of the number let alone its value
What's even more wild to me is that even though we can't comprehend the size of this number, we _can_ comprehend a proof for why there are infinitely many more primes larger than it. We can even figure out an estimate for how frequently primes occur at this magnitude (roughly 1 in 94 million numbers of this size are prime). Math is so crazy.
I watched the 6 or so minutes of the largest prime. At first, I was like - oh no, but at the end I was happy. Well done!
You can probably solve NHS funding by cancelling football, and the "what's the point" brigade would go silent in a heartbeat.
Matt’s new beard is awesome.
Usually the beard size correlates with his process in writing a book, but I think this is 'simply' his vacation beard
7:46 very very well put
You could print it out in base 2. You could come up with a special font for printing it out in base 2. Base 2 with kerning. A one is 2x4, eight cells, three with dots. A zero is 4x4, 16 cells, eight with dots. Very compact. 20:18
You do realise that mersenne primes by nature are all ones right?
The observable universe has around 10^80 particles and a total volume of 10^108 cubic nanometers. A number with this many digits is just impossible to think about in physical (counting) terms. It's a testament to the power of our mathematics that we can imagine and reason about such things.
I would go visit an art display that was just a mile of pi, just the numbers running along a walkable pathway or something. It sounds incredible.
the Tech Model Railroad Club at MIT birthed hacker culture. early open-source vibes, pioneered interactive computing, and developed one of the first video games, "Spacewar!" Their tinkering laid groundwork for AI advancements and modern computing.
Yep. It's a fallacy to think that only the things which are known to be productive are, in fact, productive. It's just bean-counting myopia.
I wonder if anyone has searched for interesting patterns in the decimal version of this number like fifty 9s in a row or the first hundred digits of pi.
A Mersenne number is any number of the form N = M(n) = 2ⁿ - 1, where n = any positive integer.
A Mersenne prime is any Mersenne number that's prime; a necessary, but not sufficient condition for which, is that n be prime.
The first four primes, p = {2, 3, 5, 7}, all produce Mersenne primes - M(p) = 2ᵖ - 1 = {3, 7, 31, 127}.
The fifth prime, p = 11, is the first to fail to produce a Mersenne prime: M(11) = 2047 = 23·89.
You needn't go far down the list of Mersenne primes before merely testing the "Mersenne seed," p, for primality, by hand, becomes daunting.
The procedure used for testing M(p) for primality is something of a miracle of mathematics. Without it, Mersenne primes would never be famous.
Fred
From the OEISWiki: It is not known whether the set of Mersenne primes is finite or infinite. The Lenstra-Pomerance-Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth.
As a Mathematician, what I think of as BIG is the Number of Bridge Hands, 52!, or Games of GO, 361! But those only look like 200 or 1000 Digits Max. Millions.....
You can Save Time and Paper if you Print the Results in Hexadecimal.
For the next largest prime, Matt Parker should go all in and read it out on camera.
All of it.
1977, the fearsome roar of UCLA's IBM line printer at full tilt usually meant infinite loop.🥰
3:57 bit funny how nerdy these hobbies are that matt parker comes up with there :D
In base 2 these numbers are easy to calculate the digits - they are all ones. That exponent number of them.
Considering this took 6 years to find, how long do y’all think it’ll take to find the 53rd Mersenne Prime/Perfect Number? What are your thoughts, predictions, and methods you think could speed up the process?
I feel, I don't know, disappointed? It's like someone told me "hey we found the digit 11,548,211,798,954,357 of pi" and I say "Really, do we know all the digits op pi up to that one?" - "Oh no no... we don't know most of the digits below that one, but we do know that one!". Almost every prime number smaller than the new biggest prime are not known.
I could not even find what is the largest prime in the longest uninterrupted (no gaps) list of primes, that is, what is the biggest prime for which all smaller primes are known. But I know it will be ridiculously small compared to this one because the largest non-Mersenne prime known is already ridiculously smaller and most primes smaller that that one are not know either.
13:53 I think the answer here is that Matt Parker, like many other mathematicians who have spent a substantial part of their career contemplating these large numbers, appreciates better than most humans exactly how bad we are at comprehending how large these numbers are. You think you can't, but you have no idea how badly you can't - not even close. Only mathematicians approach knowing exactly how badly we have no idea how badly we can't comprehend them.
Matt's beard is in it's prime.
"It's no longer in their prime." I see what you did there...
Thinking on the largest discovered prime number... I'm wondering what the smallest number is that nobody knows whether or not it's a prime? and alternatively the smallest composite number which we don't know what factors it has
What is the longest string of the initial digits of Pi that has been found in a prime number?
No replies yet! Come on, we wanna know
I wonder if there are (m)any primes between this largest one and the previously largest one..
What is the largest number below which we know for sure every prime number?
I did some quick back of the envelope calculations during the video (so I don't guarantee my values are correct even assuming the data is):
Given that a million digits was a mile, 41 million digits would be 41 miles, which, at 3 miles per hour, would take almost 14 hours to walk - which is basically an entire day with some time at either end to get up and go to bed, and maybe some breaks for little things like eating.
Very roughly, printing a hundred million digit number would take on the order of a week (so more than a couple of days, but probably less than a month) - and require some well-planned paper management. If you can fit 5000 digits on a single sheet (which is probably a slight overestimate) then you'd need 20,000 sheets to fit a hundred million digits, which would stack about the height of a tall person (a ream of 500 sheets is around 5cm or 2 inches, so 40 reams is 2m or 80 inches - the difference between the two is less than the errors in approximation)
We need a QR code of this number
+
It's all 1s, so it might be a bit boring.
That printing operation should have a percentage, & zn understanding of how long it takes for each %age point to change.
Could be cool to have an algorithm that converts each prime to a song based on its digits
The Online Encyclopedia of Integer Sequences (OEIS) has a page that converts sequences to MIDI files. Neil Sloane has done some Numberphile videos with that music. See the OEIS website with link "Music" at the bottom of the page.
@@omaro77 use Morse code on the binary representation 😉
I think I'm right in saying that the number of atoms in the observable universe is basically 0 compared to this number?
If every particle in the Universe contained an entire universe of that many particles, and each of those... for 400,000 levels, you still wouldn't be more than this number.
So this is the highest prime number, but. Being a mersenne prime there are a lot of prime numbers between this one and the previous record.... Now my question is, which prime number discovered is the biggest one with no gaps, without skipping any prime number in between?
the parker prime is undefined
The Parker prime is even and > 3.
The Parker prime is negative.
luckily, the Parker prime can be found by writing a piece of horrible Python code.
At $0.05 a page, how much would it cost to print?
When we pursue what appear to be ephemeral goals, we develop practical, applicable tools along the way.
Is it possible this is one of a pair?
If you plant grass you might be able to imagine 41,000,000 pieces of grass in groups of 1,000s but that's just the digits definitely a few orders of magnitude off from imagining these numbers
The way I conceptualized it was reading out 3 digits per second 24 hours aa day would take 158 days.
But again, like all the other methods, that's still only conceptualizing 41 million, and not the number itself. Lol. Mind boggling
why do people go to all the trouble to climb Mt Everest!? 6:17
famously answered already by first climbers: "Because it is there"
Does anybody know the largest value, for which the prime counting function is not known EXACTLY.
Or in other words, what order of magnitude is the smallest unknown prime.
This is more of a philosophical question of what it means to be "known".
To have it stored?
You can easily generate all 64-bit primes (i.e. all primes below 2^64) on the fly - and that's actually done by a lot of software that is involved in prime-finding (sieving stuff mostly). But there's about 4.15829×10^17 of them. If you want to store their decimal representation with a single space character separating them it's gonna take (I made some calculations through wolfram alpha) 7127 petabytes. Of course, much more efficient to store them in binary. 2955 petabytes. Still way too much for any reasonable person to store.
But that doesn't make any of them "unknown".
I only vaguely understand the process by which we find these numbers. From a theory perspective, what're the odds there are 0 other primes between this and the previous largest known prime?
There are many primes between the current and previous record holder.
if you meant other "Mersenne primes", we will need to wait for the backlog re-verification to be sure that no Mersenne primes exist between 2^82589933 -1 and 2^136279841 -1.
One of the most interesting properties of Mersenne primes is that in binary representation they are all 1s and no 0s.
And if Matt wants to make this printing installation for the next Mersenne prime, he can print it in binary and start even before it's found. Just make it print 1s. Then, when the prime is found, just update it with when to stop.
The thing about Mersenne primes is that everyone already knows all of the digits, in base 2 anyway. :)
In what sense are Mersenne Primes the easiest to find, as Matt claims at the beginning. We’ve only got 52 verified, right? Sounds like they are the hardest to find. What hidden qualifiers in his claim am I missing?
We have proven that we know 100% of the Mersenne primes less than 10²⁰⁰⁰⁰⁰⁰⁰. We will only ever know 0% of all primes less than this limit, to the nearest 1%.
(There are roughly 10¹⁹⁹⁹⁹⁹⁹³ primes less than this limit whereas the age of the universe in Planck time units multiplied by the volume of the observable universe in Planck volume units is about 10⁶⁰ . 10¹⁸⁶ < 10³⁰⁰.)
@@andrewkepert923 So, "easiest to find" by the metric of "found the highest % of the ones that exist"?
@@backwashjoe7864 Yes those are (roughly speaking) the hidden qualifiers that you asked for.
A different set of qualifiers could give a different answer.
For instance, we are absolutely sure we know the first 52 primes, yet we are guessing that we know the first 52 Mersenne primes, since there may be some between the ones discovered so far.
A more specific interpretation of “easy to find” is the process rather than the outcome - we have no practical way of proving whether an arbitrary prime with 41 million digits is actually prime. It is only the special structure of Mersenne primes that allows us to determine whether Mersenne numbers of that size are prime.
The one nuance hidden in my “roughly speaking” statement is that we can’t speak of a “percentage of the ones that exist” since there are infinitely many primes, and there are probabilistic (non-rigorous) indications that there are infinitely many Mersenne primes. So we do need some way of making it a percentage of a finite quantity.
The test for primality is simpler to calculate for Mersenne primes than for ordinary primes. Thus the largest known Mersenne prime is (much) bigger than the largest known non-Mersenne prime.
that would be 5000 or so sheets of paper that would need two cartons of paper. You would have to interrupt the print out half way through
Some of those old IBM printers would print continuously through many boxes of paper.
'The less incorrect' is grammatically more correct than 'the least', given that there are only 2 folk involved - and also less incorrect, as it happens.
I hope someone has tested adding two to this newest biggest prime and see if we’ve happened upon an amazing twin prime!
Not possible.
2^p isn't divisible by 3, so either 2^p+2 or 2^p-2 must be divisible by 3. 2^p+2 can't be divisible by 3, because then 2^p-1 would also be divisible by 3. Therefore, 2^p-2 must be divisible by 3, which means 2^p+1 is also divisible by 3.
p=2 is the only exception, giving 3 and 5 as the only pair of twin primes consisting of a Mersenne prime and a Fermat prime generated from the same power of 2. And even without the above, if there are any more Fermat primes, they require the exponent to be a power of two, where as Mersenne primes require the exponent to be a prime.
it's 2^p-3 we need to look at if we want to find twin primes to Mersenne primes.... Though I've checked the first few Mersenne primes and the only such pairs below a billion are 5 and 7 and 29 and 31, but that leaves 40-some Mersenne primes to check for a little twin.
Since the probability of n being prime is 1/log(n), the probability of (2^n-1) goes like 1/n, so we "know" there are infinitely many, barring an improbable conspiracy. That's not a mathematical proof, but it's a scientific certainty.
I don't think you can really use "scientific certainty" in this kind of context. And what qualifies as an "improbable conspiracy"? Numbers have structure, they are not purely probabilistic.
You would ideally need to set the printer up on the top of a tall building to watch the printout "pile up" far below Matt!!!
it is not proven that there are an infinite number of Mersenne primes.... 19:38
Would the world be a better place if Luke just spent his $2million on a yacht? I don't think so
It's an extraordinary achievement for a private individual. Kudos.
In other videos he said it cost him (?) less than 2 million in server rent to calculate. Either he's crazy obsessed by primes or had some generous help from others to get all the money to fund this.
@@cyrilio I think he had a successful career in IT beforehand
Would be interesting to set this prime into perspective to a googol or some hyperoparations. Can we grasp either?
A Googol is 10^100
2^10 = 1024 is approximately 1000 = 10^3
10^99 is approximately 2^330
The exponent on this prime is in the hundreds of millions, so flipping huge compared to a Googol.
2^100 million is about 10^30 million, though considering this prime is 41 million or so digits, those 24s in our rough conversion between powers of 2 and powers of 10 are adding up
Matt could be wrong? Impossible
3:45
Ohhhh this is me with all my hobbies. Numbers and data go brrrrr and I just get dopamine for free, no drugs.
In perhaps 100,000 years, mathematicians might discover new principles or structures that enhance our understanding of prime numbers, potentially even leading to a function that generates them.
Who will be the first to remember all the digits ? Forth & back.
My favourite primes are the 2^n + 1 primes.
Like: 257
I like big primes and I cannot lie.
Like talk to me I have proven the Goldbach conjecture.
How do you not edit in the answer like Matt said? Big miss.
"It's so outrageously big"
Erm ... Tree(g64)?
Yeah, but you don't know that number. You just have a process by which you can't compute it.
Just o(4) is way bigger than this outrageously big number.
And א(4) is way bigger than TREE(G64)
11:19 STPC™ - some terrible python code
It would have been a bit easier to show it in binary.
Now you've created a new problem. What is the HARDEST prime with 2745418 digits to find?
When you print the Mersenne Prime #52 in a book, it takes 10500 pages where each contains 4000 digits (80 x 50) , you will realize that you wont be able to spot a printing mistake in 1000th page 🤣.
Just imagine errata to such book...
Unless you print the number in binary, in which case any mistake will be blindingly obvious.
@@MarsAnonymous "We're terribly sorry, we realized we failed to print two more 'ones' in this book. Please correct this error yourself." 😉
@@MarsAnonymous please check again, it is indeed in decimal system only , in binary it takes approx 4 times more
@@adammarkiewicz3375 it needs to be precise enough 🤣
Put it in the back of box truck lol
you are freaking out the youtube compression algorithm
Why is this titled and done like Matt isnt a regular appearance 😂
I'm guessing its because he's technically still on vacation, not to mention something tells me this is technically vacation material for him to talk about this new mersenne prime, even better when its for numberphile (i'd imagine)
For me, this seems like it can be pretty useful as well towards the question of "Is there a true limit?" Because there will come a point when it would probabilistically take a non-trivial amount of time to reach the next mersenne prime, in which we can safely say that is the limit for our modern computing hardware and lifetimes
Also, might break encryption/cipher algorithms requiring the use of prime numbers (i.e. RSA), but I'd presume that would speed up the motivation to improve computing, instead of whatever the heck Nvidia is doing just messing around with money
I'm feeling like such a major nerd right now, even more so now that I've graduated university
Crash must be horrible
I'd pay to see Matt Parker stand at a podium and read the number out loud. Should only take a few hours...
At 5 digits per second, 41 million digits would take nearly three months...
And for long do you think you could sustain 5 digits a second? 1 digit a second is getting towards a year and a half.
@@numberphile2 how about we crowd source it? Everyone reads like 400 digits. Only a hundred thousand people needed
@@numberphile2 How long do you think you could sustain 1 digit a second?
The point of giving the time for 5 digits per second is that that's about the fastest most people can rattle off digits (doing some quick research, there are world records up in the 8 per second range - though the record for reciting most memorised digits of pi is around 1 per second for about 17 hours), and that's already taking significantly more than "a few hours".
When you're talking about more than a couple of days, you start having to schedule eating, drinking and sleeping (the essential activities you can't read out digits during), and it stops being a simple calculation...
More like a few months.
Average human speech rate is about 180 words per minute or about 10,800 words per hour. 41 million divided by 10,800 is about 3800 hours or about 160 days if he recited digits 24/7.
Granted, except for 7 and 0, all of the digits are monosyllabic and zero has it's monosyllabic synonyns, so even at a normal speech rate, digits might be faster than most speech... still, for comparison, Harry Potter and the Deathly Hallows is about 200k words and its audiobook has a total runtime of about 24 hours and we're talking reciting 41 million digits.
It will never be as famous as Optimus Prime
Something about this is very against what it means to be interested in the nuances of mathematics and number theory.
You think a new biggest prime is a big deal, and then you realize that if AWS put a chunk of servers on this, they’d have several new biggest primes by lunch tomorrow. I mean, yay! but where is the compelling story?
How big is this number? It is near to zero if you can imagine infinity. 😅
Even compared to o(4)
2 points off of Matt's nerd card for not generating the decimal digits file himself.
He's on vacation. Cut him some slack. 😅
HUMANS
There may come a time when alien invaders will only let you live based on what primes you have found
This is not a waste of resource
I wonder how many FLOPS it took to find this prime?
Is 0/0 a prime?
Maths is simple beautiful
No. 0/0 is not defined.
@@CorwynGC0/0 is the Quantum version of Mathematics
@@david-melekh-ysroel What does that even mean?
@@CorwynGC it means that it depends on how you view it.
Some cases you'll get 1, otherwise 0, or any other number.
@@david-melekh-ysroel but that's wrong. It isn't defined because it has no consistent meaning. Quantum have probabilistic values until it decoheres than it has a definite value.
On the topic of why spend time and resources on this. I imagine that a lot of constants or ratios or physics things were discoverd/proven/invented entirely in theoretical contexts and only found a practical usecase years/centuries/millenia later.
RSA encryption uses prime numbers, IIRC. With increased computing power, breaking the RSA encryption becomes less impossible, so you'd need to use larger primes. If we stopped finding bigger primes, that list would eventually run out. I doubt that usecase was on the minds of Mersenne, Euler or Aristotle, when they messed around with maths.
RSA encryption FINDS prime numbers when constructing a key, it doesn't take from a list. (details omitted)
Disagree. Finding the Next Prime yields just another Arbitrary Piece of Data. It's just More of the Same.
Perhaps the Ancients knew when Enough was Enough.
GPU/CPU power vs prn science lol
I didn’t know Tom segura likes primes so much
Largest prime ? Still about zero compared to the rest of the primes to be found. So not that special.
Highest jump? Nothing compared to the distance from Earth to Sun. Your point was?
If we've had Mersenne Primes for thousands of years, why are they named after a 17th century French mathematician? Shouldn't they instead be named after a mathematician from antiquity?
Well, certain bits of mathematics were discovered a long time ago in China or India yet they’re still associated with someone much later who rediscovered them…
Isn't their a law, observation, or something that states things named after people are rarely named after the first person to discover them?
Also, I'm willing to bet at least four of the 5 known Fermat primes were known in antiquity even if their relevance to the problem of constructible polygons wasn't, and even 65537 seems like it might be small enough to confirm primality by hand if one cared to do so.
Maths horrible approximation of physics
Fun? Most of human history has not been fun for most people.
I wonder about hobbies like these, and the carbon footprint associated with them. Like, how do you justify stuff like this...I dunno.
Parker has nothing to do with the discovery
Why interview him instead of the actual mathematician
Why not view the other videos on this channel and see for yourself?
Matt is the kind of guy who supports anything 'green', yet he doesn't boycott the huge waste of resources involved in finding such large prime numbers.
The people searching for these numbers develop software that can be used elsewhere. You can't really optimize software without running it for some time. I'm pretty sure that most things you've done today were less useful than searching for the next biggest Mersenne prime.