If anyone is curious, 2^67 - 1 has the factors 193,707,721 × 761,838,257,287 and 2^257 - 1 has the factors 535,006,138,814,359 × 1,155,685,395,246,619,182,673,033 × 374,550,598,501,810,936,581,776,630,096,313,181,393
I find the story about 2^67 - 1 and Frank Nelson Cole quite interesting and amusing. American Mathematical Society, New York 1903, Cole going on stage without saying a word, writing the arithmetic series 2^0 + 2^1 + 2^2 ...+2^65 + 2^66 = 2^67 -1 and writing the whole number on the board. Then he wrote the multiplication on the board and showed that same result. The whole time Cole did not say a word, went back to his seat and the audience erupted in applause. Frank Nelson Cole said checking all possible divisors took him every Sunday afternoon for three years.
@@Kirkaje I can't believe he calculated (2^67)-1 as the sum of that series. Having calculated 2^66, the final term of the series, surely he'd just double it and subtract one, rather than add it to another 65 numbers, each of which has up to 20 digits? And, it's probably more efficient to calculate 2^66 = (2^33)^2 = ((2^11)^3))^2 = (2048^3)^2, rather than 2*2*2*...*2. That requires multiplying two four-digit numbers, multiplying the seven-digit answer by a four-digit number, then multiplying the resulting ten-digit number by itself. Finally, double the answer and subtract one. That has to be faster than repeated multiplications by two.
I've watched Dianna's video (standing up for the first time in 2 years) yesterday. Still makes me sad seeing those old shots of her, but good to see her make some progress.
Most comments are about him missing on 2 of them, meanwhile Im flabbergasted that he got number 127 correct!! How? Other then a pure guess, how could he have figured that out before computers?
Actually, 2^127 was proven correct by Edouard Lucas before computers (the first 'L' in the Lucas-Lehmer test). By hand. It took 19 years of testing. Oh, and he figured out the method and started the test for 2^127 when he was only 15 years old... Quite an achievement compared to Mersenne's guesses
Even by Mersenne's time, it was known that the possible prime factors of 2^p-1 are all of the form 2kp + 1. He could have tried a bunch and concluded that there probably weren't any, without having a proof.
Even in Mersenne's day, the prime factors of 2^p - 1 were known to satisfy a strong constraint: by Fermat's little theorem (proven 1640) they must be congruent to 1 (mod 2p). Many of these candidates will themselves have small prime factors, ruling them out. Shortcuts like these are what make Mersenne primes easier to find. Of course, Mersenne still made mistakes. I imagine for p = 127 he checked a bunch of candidate factors, didn't find any, and called it a day. For the primes he missed, perhaps he miscalculated and found a fake factor.
Is there any chance that a method exists for generating large primes that can be verified with less computation than Mersenne primes, but has not yet been discovered? Or is there an aspect of 2^p-1 that ensures primes generated by it are almost certainly (perhaps provably) less computationally-expensive to verify than any other conceivable method of generating primes at the same scale?
I always get a weird feeling when people touch such old books. I'm like "dude stop touching it, you'll break it!" but if no-one ever read them, what would be the point of keeping them around?
Not square (2n-1)? (1 side and length) 1/3, 1/5, 1/7 ( 3 sides and 105 length). 2 x 1 -1= 1 side (base 105). 35 =1. 2 x 2-1= 3 sides (base 105). 3 =35. 2x3-1= 5. 5=21. 2x4-1=7 sides. (Base 105) 7=15. Its always finds a way too loop back. Singular loops. Binary doesn't.
There's a way of generalizing Mersenne primes which, recognizing that mᵖ-1 is always divisible by m-1, goes M(m,p) = (mᵖ-1)/(m-1) in which m,p>1 are integers, with p = a prime. When m=2, M(2,p) is just the Mersenne number = M(p) = 2ᵖ-1 Some primes that have this form are M(3,3) = 13 M(3,7) = 1093 M(5,3) = 31, coincidentally = M(2,5) = M(5) M(6,3) = 43 M(7,5) = 2801 M(8,3) = 73 etc. Fred
What do you mean by "not being able to compute them"? You can quickly calculate large powers (large as in hundreds or thousands) by repeatedly squaring a value. Calculating 2^257 with pen and paper takes a few minutes. Or were you talking about factoring them?
What AHBelt said; plus, mⁿ - 1 is always divisible by m - 1, which, when m=2 is just 1, so Mersenne numbers can still be prime, but for m,n>2, mⁿ - 1 is never prime. Also, when n is composite, say n = st, where s,t > 1, then mⁿ - 1 = mˢᵗ - 1 = (mˢ)ᵗ - 1 = (mᵗ)ˢ - 1, so it's divisible by mˢ - 1 and mᵗ - 1. Now if you restrict n to be prime, n = p, and divide by the automatic factor m - 1, then you CAN get some primes of the form M(m,p) = (mᵖ - 1)/(m - 1) Note that when m=2, this just boils down to the Mersenne numbers, 2ᵖ - 1; and like those, M(m,p) is sometimes composite. Fred
@@ffggddss When M,N are both greater than 2, how to proof M^N-1 is never prime? If M is of the form of (2*S)+1, then M^N-1 is allways multiple of 2 so not prime, but how to proof when M is of the form of (2*S)? When M=2 there are primes that are 2^N-1, like N=2, N=3, N=5, but not N=4, ...
@@androidlogin3065 Pretty simple, really. [BTW, I should maybe have said, m>2 & n>1.] mⁿ - 1 is always divisible by m - 1. This is key to the formula for the finite geometric series. mⁿ - 1 = (m - 1)(mⁿ⁻¹ + mⁿ⁻² +...+ m + 1) If m > 2, (m - 1) > 1, and with n > 1, the other factor is also > 1, so this factorization demonstrates that mⁿ - 1 is composite.
2n-1 is illogical. Its not even 2n+1. Frequency in frequency. We have 3 break down 1of 1= 1/2 ( 2 parallelogram) (1/2,1/4, 1/6, 1/8, etc). 1/2, 1/4, 1/6= 3(1+1+1) and 2 as the differential (parallelogram). How many times does 1 pop up for prime in 3? 3 (2+1), 9(8+1), 15(14+1), etc. so frequency of 2 with another frequency resonates at 3. So 1, 6, 12, 18, etc.. why aren't you using your base function? (Parallelograms, or not squared). What is your math based off? Units of what?
It seems like Mersenne basicaly got thing more wrong then right. 7 Mersenne primes known before him. He correctly identified 2 more. He incorrectly suggest 2 more. He incorrectly missed 3 more within the range of exponents he had explored. He totalled 5 mistakes to 2 right numbers.
This is an odd way of framing it. If you really think he searched that entire range of prime exponents (primes greater than the previously known 19 and up to 257), that's 47 candidate prime numbers. If you think he actually "explored" all of these, that would mean: He apparently correctly determined that 40 of them were NOT prime. Correctly determined that 2 of them were prime. Incorrectly suggested 2 were prime. Incorrectly suggested 3 were not prime when they were. That's 5 mistakes to 42 right numbers. A LOT more "right" than wrong. However, I don't think that's an accurate way of considering this either, as we really don't know the search method employed. Mersenne didn't really clarify how he narrowed down his list. It's very doubtful he considered all 47 candidate prime numbers in this range. Someone can correct me if I'm wrong, but I'm not sure Mersenne claimed this list was exhaustive either for that range. That is, he may have been working off a pattern he thought might be a sufficient condition for being a prime, but not a necessary one, in which case the three he "missed" shouldn't necessarily be counted against him. Really, however, without knowing exactly what criteria he was using to narrow them down (and historians of math have identified some ideas about what that may have been, without coming up with anything precisely consistent with his list that I know of), we can't really determine exactly how many "mistakes" he made among numbers he didn't discuss.
I'm surprised they let you touch such an old book with your bare hands. Shouldn't you wear gloves to not damage the paper? I always thought this would cause damage to the pages over time
That used to be the case, but now common practice is to use your bare hands, because when you're wearing gloves your fingers are less sensitive and dexterous and you're more likely to tear a page accidentally, so it's generally thought that the oil from your hands is less of a risk than that.
@@LofferLogge: Very well put! You can also find the answer in 'Objectivity' videos or their comments when old books or manuscripts from the Royal Society archives (and elsewhere) are featured - and they often are.
Mersenne did not, in fact, start the conversation about Mersenne primes. As Stigler's law puts it, nothing is named after its original discoverer. Hopefully, actual history books will care about more than who manages to get famous for other people's work, even if the general public doesn't.
@@RobinDSaunders You raise a great point. Not always, but it does happen. Do we revere the silent geniuses that changed the future, or the mouths that gave them voice (And just rhetorically of course, which would you choose) ? 😉
@@onecupofconsciousnessplease Elon Musk is a Fellow of the Royal Society. This video promotes the Royal Society; Federico has explained that he objects to that.
@ The Royal Society has an enormous endowment so they're not at all reliant on Musk and his fans. For example, the UK government granted them a quarter of a billion pounds a couple of years ago, to be invested to fund fellowships and research grants.
See Objectivity (293 videos and more to come) - ua-cam.com/users/objectivityvideos
Please do consider subscribing too! 🔔
cogitate-ta both a's are pronounced differently. Right Hermione? It's levi-oh-sar. I Before E Except After C or T.
If anyone is curious, 2^67 - 1 has the factors 193,707,721 × 761,838,257,287 and 2^257 - 1 has the factors 535,006,138,814,359 × 1,155,685,395,246,619,182,673,033 × 374,550,598,501,810,936,581,776,630,096,313,181,393
Thank you! :)
It's ridiculous that Mersenne gets all the fame when he didn't even know his 193,707,721 times table.
I find the story about 2^67 - 1 and Frank Nelson Cole quite interesting and amusing. American Mathematical Society, New York 1903, Cole going on stage without saying a word, writing the arithmetic series 2^0 + 2^1 + 2^2 ...+2^65 + 2^66 = 2^67 -1 and writing the whole number on the board. Then he wrote the multiplication on the board and showed that same result. The whole time Cole did not say a word, went back to his seat and the audience erupted in applause. Frank Nelson Cole said checking all possible divisors took him every Sunday afternoon for three years.
@@Kirkaje why the geometric series ? did he have to calculate every previous power of 2 ?
@@Kirkaje I can't believe he calculated (2^67)-1 as the sum of that series. Having calculated 2^66, the final term of the series, surely he'd just double it and subtract one, rather than add it to another 65 numbers, each of which has up to 20 digits?
And, it's probably more efficient to calculate 2^66 = (2^33)^2 = ((2^11)^3))^2 = (2048^3)^2, rather than 2*2*2*...*2. That requires multiplying two four-digit numbers, multiplying the seven-digit answer by a four-digit number, then multiplying the resulting ten-digit number by itself. Finally, double the answer and subtract one. That has to be faster than repeated multiplications by two.
2^67-1 and 2^257-1, the parker primes.
2^57 - 1 is the Grothendieck-Mersenne prime
Almost 13 years since your top video and you haven’t aged a day
A Numberphile Objectivity crossover in the Bradyverse -Nice
"See the lonely boy out on the weekend trying to make it pay" 🎶
My guess as to why he included those larger numbers, despite being unable to confirm them, is that he believed he had some rule for generating them.
Brady’s finally playing the game 📺 🧠 👏🏻
I've watched Dianna's video (standing up for the first time in 2 years) yesterday. Still makes me sad seeing those old shots of her, but good to see her make some progress.
Yay!
I was surprised when you mentioned it, as I'm subscribed. But YT doesn't tend to put shorts in my notifications.
@@jursamaj Have you seen "Hello from Dianna!", from back in November?
301 تحية من المملكة المغربيه ❤🇲🇦🇲🇦
Most comments are about him missing on 2 of them, meanwhile Im flabbergasted that he got number 127 correct!! How? Other then a pure guess, how could he have figured that out before computers?
Actually, 2^127 was proven correct by Edouard Lucas before computers (the first 'L' in the Lucas-Lehmer test). By hand. It took 19 years of testing.
Oh, and he figured out the method and started the test for 2^127 when he was only 15 years old... Quite an achievement compared to Mersenne's guesses
Simple. You check whether it's divisible by 2, 3, 5, 7, 11, 13 and so on all the way up to roughly 2^63. And you make sure to not make any mistakes.
@@dev_ceb We knew whether each of the first 258 Mersenne numbers was prime by 1947 -- all before computers were able to contribute much.
Even by Mersenne's time, it was known that the possible prime factors of 2^p-1 are all of the form 2kp + 1. He could have tried a bunch and concluded that there probably weren't any, without having a proof.
Even in Mersenne's day, the prime factors of 2^p - 1 were known to satisfy a strong constraint: by Fermat's little theorem (proven 1640) they must be congruent to 1 (mod 2p). Many of these candidates will themselves have small prime factors, ruling them out. Shortcuts like these are what make Mersenne primes easier to find.
Of course, Mersenne still made mistakes. I imagine for p = 127 he checked a bunch of candidate factors, didn't find any, and called it a day. For the primes he missed, perhaps he miscalculated and found a fake factor.
Who came here from the "301 view freeze" video?
Very Parker-esque of Mersenne, but he gave it a try! Thanks Brady
So 2^67 - 1 is a Parker Mersenne prime?
Parker Prime!
The Mersenne prime for n = 61 is also known as the Pervushin's number.
I thought at first that the origin story had been slightly bungled, but I guess it's the list of primes that was slightly bungled.
Is there any chance that a method exists for generating large primes that can be verified with less computation than Mersenne primes, but has not yet been discovered? Or is there an aspect of 2^p-1 that ensures primes generated by it are almost certainly (perhaps provably) less computationally-expensive to verify than any other conceivable method of generating primes at the same scale?
please do a video on the number 274
GPT-4o was able to guess the author and the book from a screenshot of the text.
(2^67)-1 ... The Parker Prime... 😂💜
0:39 Notwithstanding my lack of Latin skills, I only recognize perfect numbers here.
Take a look a bit lower on the page
I was expecting mat parker to do this one.
Please do a video about how to solve Maxi Nerdles. Thanks
I always get a weird feeling when people touch such old books. I'm like "dude stop touching it, you'll break it!" but if no-one ever read them, what would be the point of keeping them around?
Not square (2n-1)? (1 side and length) 1/3, 1/5, 1/7 ( 3 sides and 105 length). 2 x 1 -1= 1 side (base 105). 35 =1. 2 x 2-1= 3 sides (base 105). 3 =35. 2x3-1= 5. 5=21. 2x4-1=7 sides. (Base 105) 7=15. Its always finds a way too loop back. Singular loops. Binary doesn't.
Great
For me the view count seems to be stuck at 301, why?
Nice reference, dude!
(at least I hope it's a reference?)
I hereby declare "Muderpillow Primes" primes that follow the form (3^n)-1
Theres only one of them •~•
You may like the story of "Legendre's constant" :)
There's a way of generalizing Mersenne primes which, recognizing that mᵖ-1 is always divisible by m-1, goes
M(m,p) = (mᵖ-1)/(m-1)
in which m,p>1 are integers, with p = a prime.
When m=2, M(2,p) is just the Mersenne number = M(p) = 2ᵖ-1
Some primes that have this form are
M(3,3) = 13
M(3,7) = 1093
M(5,3) = 31, coincidentally = M(2,5) = M(5)
M(6,3) = 43
M(7,5) = 2801
M(8,3) = 73
etc.
Fred
What do you mean by "not being able to compute them"? You can quickly calculate large powers (large as in hundreds or thousands) by repeatedly squaring a value. Calculating 2^257 with pen and paper takes a few minutes.
Or were you talking about factoring them?
What about 3^N-1 be a prime? How are they called? ... At lesat one exist: 3^1-1=2
And in general, M^N-1 be a prime, how is that called?
3^N, N an integer, is odd, then 3^N-1 even, only prime if 2 itself.
What AHBelt said; plus, mⁿ - 1 is always divisible by m - 1, which, when m=2 is just 1,
so Mersenne numbers can still be prime, but for m,n>2, mⁿ - 1 is never prime.
Also, when n is composite, say n = st, where s,t > 1, then mⁿ - 1 = mˢᵗ - 1 = (mˢ)ᵗ - 1 = (mᵗ)ˢ - 1, so it's divisible by mˢ - 1 and mᵗ - 1.
Now if you restrict n to be prime, n = p, and divide by the automatic factor m - 1, then you CAN get some primes of the form
M(m,p) = (mᵖ - 1)/(m - 1)
Note that when m=2, this just boils down to the Mersenne numbers, 2ᵖ - 1; and like those, M(m,p) is sometimes composite.
Fred
@@ffggddss When M,N are both greater than 2, how to proof M^N-1 is never prime?
If M is of the form of (2*S)+1, then M^N-1 is allways multiple of 2 so not prime, but how to proof when M is of the form of (2*S)?
When M=2 there are primes that are 2^N-1, like N=2, N=3, N=5, but not N=4, ...
@@androidlogin3065 Pretty simple, really. [BTW, I should maybe have said, m>2 & n>1.]
mⁿ - 1 is always divisible by m - 1. This is key to the formula for the finite geometric series.
mⁿ - 1 = (m - 1)(mⁿ⁻¹ + mⁿ⁻² +...+ m + 1)
If m > 2, (m - 1) > 1, and with n > 1, the other factor is also > 1, so this factorization demonstrates that mⁿ - 1 is composite.
The viowers not are in 301
aging very gracefully Mr Haran
2n-1 is illogical. Its not even 2n+1. Frequency in frequency. We have 3 break down 1of 1= 1/2 ( 2 parallelogram) (1/2,1/4, 1/6, 1/8, etc). 1/2, 1/4, 1/6= 3(1+1+1) and 2 as the differential (parallelogram). How many times does 1 pop up for prime in 3? 3 (2+1), 9(8+1), 15(14+1), etc.
so frequency of 2 with another frequency resonates at 3. So 1, 6, 12, 18, etc.. why aren't you using your base function? (Parallelograms, or not squared). What is your math based off? Units of what?
Because he listed them in that book it’s called… ?
Mersenne Prime Suspect?
Omega Speedmaster Professional Moonwatch? Noice!
It seems like Mersenne basicaly got thing more wrong then right.
7 Mersenne primes known before him.
He correctly identified 2 more.
He incorrectly suggest 2 more.
He incorrectly missed 3 more within the range of exponents he had explored.
He totalled 5 mistakes to 2 right numbers.
And yet those numbers will be forever given his name…
This is an odd way of framing it. If you really think he searched that entire range of prime exponents (primes greater than the previously known 19 and up to 257), that's 47 candidate prime numbers. If you think he actually "explored" all of these, that would mean:
He apparently correctly determined that 40 of them were NOT prime.
Correctly determined that 2 of them were prime.
Incorrectly suggested 2 were prime.
Incorrectly suggested 3 were not prime when they were.
That's 5 mistakes to 42 right numbers. A LOT more "right" than wrong.
However, I don't think that's an accurate way of considering this either, as we really don't know the search method employed. Mersenne didn't really clarify how he narrowed down his list. It's very doubtful he considered all 47 candidate prime numbers in this range. Someone can correct me if I'm wrong, but I'm not sure Mersenne claimed this list was exhaustive either for that range. That is, he may have been working off a pattern he thought might be a sufficient condition for being a prime, but not a necessary one, in which case the three he "missed" shouldn't necessarily be counted against him. Really, however, without knowing exactly what criteria he was using to narrow them down (and historians of math have identified some ideas about what that may have been, without coming up with anything precisely consistent with his list that I know of), we can't really determine exactly how many "mistakes" he made among numbers he didn't discuss.
If Mersenne was basically a coun toss on listing correct numbers after the relatively simple ones, isn't he a bit overrated?
Salve Brady, hodie est formosus.
Wow, UA-cam's auto-trainslate handles Latin.
Okay but hear me out. Neo got my back culture things tech tech on my mind
Why would he "not be able to compute" these numbers?
This could be shorthand for "test by computation".
@RobinDSaunders Possibly, if so then it is pretty clumsy. I watched it back and have a hard time knowing exactly what he meant.
Whaaaaaat, no white gloves?
Beautiful Speedmaster.
Why no gloves on when handling an old book like that?….
2n-1 or 2n+1 are or have to be coordinates in a system. Not a value.
Neat. :)
Looking good Brady
Parker primes
I'm surprised they let you touch such an old book with your bare hands. Shouldn't you wear gloves to not damage the paper? I always thought this would cause damage to the pages over time
That used to be the case, but now common practice is to use your bare hands, because when you're wearing gloves your fingers are less sensitive and dexterous and you're more likely to tear a page accidentally, so it's generally thought that the oil from your hands is less of a risk than that.
@@LofferLogge: Very well put! You can also find the answer in 'Objectivity' videos or their comments when old books or manuscripts from the Royal Society archives (and elsewhere) are featured - and they often are.
hi
A blursed math artifact!
2 of the quickest ways to get into the history books are to either start or end a conversation.
🖖😎👍
Mersenne did not, in fact, start the conversation about Mersenne primes. As Stigler's law puts it, nothing is named after its original discoverer.
Hopefully, actual history books will care about more than who manages to get famous for other people's work, even if the general public doesn't.
@@RobinDSaunders You raise a great point. Not always, but it does happen. Do we revere the silent geniuses that changed the future, or the mouths that gave them voice (And just rhetorically of course, which would you choose) ?
😉
please remove the auto dubbing from your videos
301views💀
Fermat conjectured that Mersenne primes existed where n = 2^k, but it only works up to k = 4.
Fermat numbers are of the form 2^2^k + 1 and not 2^2^k - 1, which would only be prime for k=1 since for k>=1 it is always divisible by 3
Brady interviewing Brady?!
Did you lose weight? You look so much better than in my memories.
Keep up if you actually did something :D
Mersenne's Square
First?
Yup.😂
2^n-1-st, n=1 to be precise.
No.
10th comment 🕶️
Nope...Sorry, but i will watch a video even remotly related to the royal society when they will expel Elon Musk form their fellows list.
Sir this video has nothing to do with Elon Musk.
This does nothing but cede ground and would make them *more* dependent on Musk and his fans.
@@onecupofconsciousnessplease Maybe, in the future, everything will have to do with Elon Musk. What a dystopic perspective!
@@onecupofconsciousnessplease Elon Musk is a Fellow of the Royal Society. This video promotes the Royal Society; Federico has explained that he objects to that.
@ The Royal Society has an enormous endowment so they're not at all reliant on Musk and his fans. For example, the UK government granted them a quarter of a billion pounds a couple of years ago, to be invested to fund fellowships and research grants.
Hello from our new republic🫡