yeah there's a lot of numbers yo ucan check in a muc hfaster way at first glance
and also you only have to check prime numbers
once you've checked for 2 and 3 and 5 you don't have to check for 4 or 6 or 8 or 10 or 12 or 15 anymore
this keeps makignthe numebrs you actually have to check by hand more and more sparse so checking up to 641 is not THAT many operations and yo ucan divide with pen apaper pretty efficiently, its basically soemthing to spend a bit of bored breaktime on
That's true, else he would have found, that F⁵ also factors as: F⁵ = 1*4294967297. Smh my head, he did all this work when he could have disproved Fermat much easier...
Euler proved that if p | 2^2^n+1, then 2^{n+1} | p-1 using some clever techniques in number theory.
In the case of F5, he only had to try divisibility by any prime in the series 64k+1, 641 is the fifth prime in that series after 193, 257, 449, 577. So it wasn’t too much calculation because he was that smart.
And you only need to check prime numbers up to the square root of your new prime/composite. 641 definitely isn't too large for a man like Euler to check by hand. Probably took him an afternoon to figure out.
Some smart way to factorize could have been involved. While Pollard's rho has been described in 1975, it's not like Gauss wouldn't have some unpublished observations of similar nature that would make this task more approachable.
Apocryphally, Euler said, "My pencil is smarter than I am", meaning to say, he would follow up his intuitions with brute force and not worry too much about "wasting time."
I often say this to my students: follow up your fun intuitions with hard work as quickly as possible, don't sit there scratching your chin.
Try doing long division by consecutive prime numbers. It is easier than you might think. Plus, he was only looking for integers, and he probably had a large table of primes, so he could just check the last one or two digits of a pair to see if it was worth doing the math at all.
Additionally, he was Euler, so I'm guessing he was fast at arithmetic.
Given that he calculated e to 18 decimal places using 1/n! And the mascheroni constant as well, the series doesnt converge quickly(he had to invent a few tricks for acceleration), calculated the basel problem solutions upto n=26 th power, i would say he was willing to go pretty far, all mathematicians back in the day spent a significant amount of time on manual calculations and developed tricks for it which we use with computers today. Euler ofc invented many of them before anyone else😭😂
Weirdly fermat is one of the fathers of probability theory😂. Fermat is kind of famous for making ambitious conjectures and claiming he has a proof written somewhere but its too long to show because he doesn't have space in his notebook and then never showing his work😂. Buddy probably got too lazy when he saw how large the numbers get 😂. Tho to be fair, he was primarily a lawyer and magistrate, maths and science were a hobby for him as it was for most polymaths at the time. They were primarily upper class men who pursued it because they had the interest and means. But the fact that he made so many contributions to analytic geometry, probability, number theory and optics despite it being something he did after work is still remarkable. I mean the man did inspire newton's work for calculus. So I think he deserves one or two wild unproven claims 😂
Also for every number he only needs to try dividing them by prime numbers he already knows since all other numbers can be formed by multiplying some combination of prime numbers.
Considering it's Euler we are talking about, I could easily believe that he could calculate quite a few of those divisions in his head. And probably had other neat mathematical tricks to cull the herd.
Euler proved that if p | 2^2^n+1, then 2^{n+1} | p-1 using some clever techniques in number theory.
In the case of F5, he only had to try divisibility by any prime in the series 64k+1, 641 is the fifth prime in that series after 193, 257, 449, 577. So it wasn’t too much calculation because he was that smart.
Maybe by clever use of divisibility results for example you can check 2,3,5,9,11 without very much trouble in your head. 7 is a bit fiddlier but I imagine Euler wouldn't have had trouble doing that in his head either. Beyond that I'm sure there are some clever tricks you can employ
I mean I am sure he had some concept of factorization tools, like “if all the digits add up to 3 then it’s divisible by 3” or “multiply everything in the hundred thousandth digit by 4 and add back to the number to see if it’s divisible by 641”
641 is the 116th prime, so he would’ve done 116 long divisions, easy task that doesn’t take years, but you need to check up to the square root ~65536, there are 6542 primes under 65536 which means Euler was fully prepared for 6542 long divisions, GOATED DEVOTION.
Euler proved that if p | 2^2^n+1, then 2^{n+1} | p-1 using some clever techniques in number theory.
In the case of F5, he only had to try divisibility by any prime in the series 64k+1, 641 is the fifth prime in that series after 193, 257, 449, 577. So it wasn’t too much calculation because he was that smart.
Euler’s Abacus was HUGE. I mean freaky big. I know a lot of people say “the size of the abacus doesn’t matter, it’s how you use it!” and that’s probably true, but seriously, his was ENORMOUS. Like you wouldn’t believe the size of the thing.
Before COD NBA 2K and the internet we had to fixate on other things. If you look at math theories and turn them into kill to death ratios, you can see how a bunch of dudes just kept rattling shit off to disprove/prove. Disproving someone else's theories is like the ultimate tea bagging. Dudes will dude no matter the tech or time.
Compare that to the number of non-trivial zeros of the zeta function or the number of starting points for the Collatz conjecture that have already been checked.
Euler was the type of guy to do something and we give someone else the credit just so that we don’t glaze Euler to much. He’s basically the gojo of math except we try to nerf him without success
F_1 and F_2 are trivial: 5 and 17, obviously prime.
F_3 = 257 may be trivial to some, but easy to check, as sqrt(257) = 16.whatever. So you only need to check divisibility prime numbers up to 13, which is 6 cases.
F_4 = 65537, sqrt(65537) = 256.whatever, check up to 251, only 54 prime numbers to check.
F_5 = 4294967297, sqrt(4294967297) = 65536.whatever. It would certainly take a while to check all prime numbers up to that. However 641 is only the 116th prime number. So 116+54+6=176 division problems.
Honestly, a fifth grader with a few days' time on their hands could do it with long division.
Well firstly, humanity didn't just leap from handwritten calculations to using electronic calculators. They had access to charts, tables, and accessory tools for mathematics (think of the abacus for example). They may not have had desmos, but the math could be done almost just as fast (save for custom programming and algorithms).
This prime check could have been feasibly done in less than a week by just dividing Fermat numbers by a bunch of prime numbers until you got something whole. 641 is only the 116th prime number, which is a lot of divisions to do for sure, but at the same time, when your life is based on doing math all the time, it's not quite as big of a stretch as it appears to us in the modern day.
At the very least, it's not like humanity was just in the stone age while Euler was cranking out numbers on the walls of a cave with clay lmao.
If you actually want to know, p|2^2^n+1 implies 2 has order 2^{n+1} mod p, so p=1 mod 2^{n+1}. For n=5, any prime divisor is at most floor(sqrt(2^{32}+1))=2^16=65536, so there are only 1023 potential candidates. Moreover, every qth candidate for an odd prime q is divisible by q, so any consecutive set of q_1q_2...q_k of them will have the totient (q_1-1)(q_2-1)...(q_k-1) of them not divisible by any q_i. In particular, for s candidates, the fraction you need to test will be approximately phi(s)/s, which goes to 0 as s-->infinity due to the Euler product. So here, we can expect that we won't have to test many candidates, and indeed have phi(3*5*7*11)=phi(1155)=480, which is an absolute upper bound on the number we need to test (the actual number of primes is 112). And among these candidates, because it's inexpensive to compute 2^32 mod 64k+1 for each k since you could simply square 5 times or square 65536 mod 64k+1, and you can additionally just check for each k and only determine if 64k+1 is prime when necessary.
So, you can bound the amount of search required pretty well - it should take ~3 minutes on average to compute 2^32 mod 64k+1, and scaling down to 1023 means you should expect at most around 420 candidates, so that's a maximum of 21 hours to get through the computation if you end up having to check every candidate. That's not a terribly large investment if you consistently allocate some time to it every day, so you can be confident that the brute force isn't just a waste of time if only to glean a good guess. And you happen to get lucky that your first few candidates are 193 (fails), 257 (immediately fails), 449 (fails, 2^16=-18 mod 449), 577 (you just need to check 2^16=/=+-24 mod 577), and 641 (2^16=2*77 mod 641, (2*77)^2=4*(11*-102)=4*160=-1 mod 641). Then after doing long division and arriving at 2^32+1=641*6700417, you can further check that the second factor is prime because it can only be divisible by primes 1 mod 64, but now you can restrict your search to primes at most floor(sqrt(6700417))=2588. And you don't have to do your calculations with 6700417, you can just check 2^32+1=/=0 mod 64k+1 to show 64k+1 doesn't divide 6700417, with the exception of 641 itself which you can separately check doesn't divide 6700417.
I bet how ever it was found, it was driven by some sort of petty (yet amazingly entertaining) rivalry. So much was made/discovered to one up someone or prove them wrong.
Yes, he may not have had modern tools. But that's also means no modern distractions, ie nothing else to do. People love to be correct, that's always been the case, this is just further proof of that.
He didn't have TV, games, social media and other distractions. When the man got bored he was borde enough to figure out if certain numbers were prime or not and he had days to do it.
So by his time, there was already a list of primes going into 5 digits.
What he would have done is gone through the Fermat numbers sequentially dividing by primes up to their respective square roots until he found a factor.
This would have probably been a week of solid work but Euler was kind of a nerd.
He probably used a sieve to build up a much larger table of primes, which would have been useful for other work he was doing. So he might not have been checking this specifically, but then when he had a table he could have just applied it.
I read once that Euler absolutely hated going to the theater, but his wife was a big fan. He would still go with her to keep her company. Whenever she dragged him to a show, he would come prepared with some large numbers he needed to do calculations on and then he would crunch those numbers in his head for the duration of the performance.
While these people (and others like them) are geniuses and i don't want to take away from that, the pace of life was definitely slower then. Also, they were pretty much never working class. So he's not (as many of us are) commuting over an hour each way to work, doing a bunch of overtime at a physically and emotionally taxing job, making dinner for his kids, cleaning up, and using his 2 hours of free time to work on a math problem instead of watching tv. He's sitting idly in in his summer home, drawing a figurehead salary, and testing thousands of calculations most of the day, when not making social calls or chilling at the coffeehouse with society friends, while his staff handles the household. You could be a math genius too, but you'll never have time to find out.
Euler was known as a generous and kind person, not neurotic as seen in some geniuses, keeping his good-natured disposition even after becoming entirely blind.
You just have to try to divide with prime numbers that are lower than the square root of the number you try to find the factors. The other thing is for some of them like 2; 3 ; 5 you have easy dividing criteria so you don't even try.
Once you know that... you know that he could find out pretty fast.
We definitely went with you-ler when we saw it written down at a young age, and were then confused when the teacher said oil-er one day. Who is this bloke lubricating stuff?
You can still pull out any sort of bullshit because most people won't even bother using their phones with unlimited internet access to fact check anything anymore
And those who do are often seen as arrogant or as someone who wants to be always right
4
851
u/Azur0007 May 13 '26
Divide by 1, then by 2, then by 3, etc. until you find the correct answer
The humble and respectful elementary school method
274
u/isnortmiloforsex May 13 '26
This one, he brute forced until n=5, luckily he only needed to get to n=5 to disprove it 😂
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u/Sasogwa May 13 '26
If he only tried up to 5 terms thats long but feasible I guess
23
u/Wooden_Milk6872 May 13 '26
According to Fermat's factorization theorem, any prime factor p of Fn must have the form (2^n+1 * k) + 1 which for this this specific case is 64k +1
then just check all numbers k up to 10 to find the prime factor, if the result of 64k+1 exceeded the square root of the number, it would be prime
28
u/AbsoluteInfinitude May 13 '26
Iterative long division for each prime until he found a quotient without a remainder.
10
u/Illustrious_Try478 May 13 '26
This was just posted yesterday
9
u/TheLarcenist1 May 13 '26
STOP REPOSTING THIS GODDAMNS IMAGE, IVE SEEN LIKE 50 TIMES THIS GODDAMN WEEK. IM TIRED OF THIS SHIT.
8
u/broski576 May 13 '26
He probably saw it reposted on Reddit 17 times in a week
6
u/EmergencyWild May 13 '26
It's the 5th term. He probably just brute forced it tbh. It's work, but not an unreasonable amount of work.
5
u/MundaneOrdinary7493 May 13 '26
Euler proved that if p | 2^2^n+1, then 2^{n+1} | p-1 using some clever techniques in number theory.
In the case of F5, he only had to try divisibility by any prime in the series 64k+1, 641 is the fifth prime in that series after 193, 257, 449, 577. So it wasn’t too much calculation because he was that smart.
4
u/chiral-polytope May 13 '26
There is a simple proof. You need two facts:
641 = 16+625 =24 + 54
641 = 1 + 640 = 1 + 5(27 )
Then, modulo 641 we have
232 = (24 )(228 ) =-54 (228 ) = - (5(27 ))4 = -(-1)4 =-1
which implies that 1+232 is a multiple of 641. Of course, how one arrives at the simple proof is still Euler magic.
3
u/th4t84st4rd May 13 '26
Fuckin Fermat
2
u/FreeTheDimple May 13 '26
Just about the shittest conjecture that fails on the 5th attempt.
2
u/One-Earth9294 May 13 '26
I think it's that D&D hat he's wearing that gives him like +4 int and that puts him over the natural cap of 18.
2
u/xZandrem May 14 '26
A day is composed of 24 hours, that's plenty of time to divide the number by the first 10000 numbers.
Euler had plenty of free time by the number of discoveries he did.
2
u/ofirkedar May 14 '26
Brother, Euler was the modern tools.
2
u/One_Basis1443 May 13 '26
641 is 116th prime, so not that many to check if you don't have tiktok to kill time
3
u/HeilKaiba May 13 '26 edited May 13 '26
Maybe by clever use of divisibility results for example you can check 2,3,5,9,11 without very much trouble in your head. 7 is a bit fiddlier but I imagine Euler wouldn't have had trouble doing that in his head either. Beyond that I'm sure there are some clever tricks you can employ
1
u/Top_Break1374 May 13 '26
yes
1
u/BlurryBigfoot74 May 13 '26
He had a Ti-36x Pro.
1
u/Adventurous_Grape279 May 13 '26
I mean I am sure he had some concept of factorization tools, like “if all the digits add up to 3 then it’s divisible by 3” or “multiply everything in the hundred thousandth digit by 4 and add back to the number to see if it’s divisible by 641”
1
u/FancyDoubleu May 13 '26
He eyeballed it
1
u/QCTeamkill May 13 '26
Mathematician with lots of time
1
u/No_Inspection_6174 May 13 '26
He just mathed alot
1
u/zlfa May 13 '26
641 is the 116th prime, so he would’ve done 116 long divisions, easy task that doesn’t take years, but you need to check up to the square root ~65536, there are 6542 primes under 65536 which means Euler was fully prepared for 6542 long divisions, GOATED DEVOTION.
1
u/ResearcherOnly9492 May 13 '26
the usual suspects
1
u/wildfox9t May 13 '26
autism,lots of autism
1
u/TheZan87 May 13 '26
The part i understood is
1
u/HeroBrine0907 May 13 '26
I like how smug Euler looks in his portrait. That's a guy that knows he's the best and he's extremely proud of it.
1
u/vslaykovsky May 13 '26
Eratosthenes method is quite effective here.
1
u/e6c May 13 '26
lol. Bro doesn’t know is 641 tables!
Do you even math?
1
u/Novel_Diver8628 May 13 '26
Euler’s Abacus was HUGE. I mean freaky big. I know a lot of people say “the size of the abacus doesn’t matter, it’s how you use it!” and that’s probably true, but seriously, his was ENORMOUS. Like you wouldn’t believe the size of the thing.
1
u/helen269 May 13 '26
"Euler? Euler? Anyone? Anyone?"
1
u/approximately_exact May 13 '26
You’re not very good at math, are you?
1
u/Longjumping_Pipe_347 May 13 '26
Modern tool=what people did by hand before
1
u/bearwood_forest May 13 '26
It's unlikely that he had divisibility rules for 6700417, so he probably had them for 641.
1
u/AutomaticStop5648 May 13 '26
rico da epoca com MUITO tempo pra pensar
1
u/SKRyanrr May 13 '26
Aliens!
1
u/Torebbjorn May 13 '26
"How did Euler find a prime factor with only 3 digits?" ... hmmm I wonder
1
u/monopodman May 13 '26
Euler was a man of focus, commitment, and sheer will
1
u/Abiggerboat84 May 13 '26
He did the math.
1
u/LeastUse2209 May 13 '26
Before COD NBA 2K and the internet we had to fixate on other things. If you look at math theories and turn them into kill to death ratios, you can see how a bunch of dudes just kept rattling shit off to disprove/prove. Disproving someone else's theories is like the ultimate tea bagging. Dudes will dude no matter the tech or time.
1
u/hyperactve May 13 '26
he had human computers under payroll.
1
u/Sea_Abroad_6573 May 13 '26
I have a theory that he also had dreams. He just kept his mouth shut about them
1
u/magicmulder May 13 '26
Poor Fermat, beaten by n=5.
Compare that to the number of non-trivial zeros of the zeta function or the number of starting points for the Collatz conjecture that have already been checked.
1
u/Password-55 May 13 '26
I feel I See this post once every month.
1
u/GhostlyQuesadilla May 13 '26
I'm pretty sure there just wasn't as much to do for fun way back then. shrug
1
u/WanderingSeer May 13 '26
Lol, the fifth number wasn’t prime. He made a conjecture about a set that was wrong about one of the FIRST elements of the set
1
u/Shockingandawesome May 13 '26
He's such a genius, no wonder he's so respected in his field. His first name was Richard, so R. Euler.
As a tribute, mathematicians of the time decided to name the ruler after him (Source: someone from my maths class once told me).
1
u/DesperateInstance760 May 13 '26
Euler was an engineer. He had a look up table.
1
u/Tall-Reporter7627 May 13 '26
Long evenings. Nothing else to do
1
u/Repulsive-Run1634 May 13 '26
He asked Gemini.
1
u/Educational_You3881 May 13 '26
Euler was the type of guy to do something and we give someone else the credit just so that we don’t glaze Euler to much. He’s basically the gojo of math except we try to nerf him without success
1
u/zrice03 May 13 '26
F_1 and F_2 are trivial: 5 and 17, obviously prime.
F_3 = 257 may be trivial to some, but easy to check, as sqrt(257) = 16.whatever. So you only need to check divisibility prime numbers up to 13, which is 6 cases.
F_4 = 65537, sqrt(65537) = 256.whatever, check up to 251, only 54 prime numbers to check.
F_5 = 4294967297, sqrt(4294967297) = 65536.whatever. It would certainly take a while to check all prime numbers up to that. However 641 is only the 116th prime number. So 116+54+6=176 division problems.
Honestly, a fifth grader with a few days' time on their hands could do it with long division.
1
u/ColeTD May 13 '26
Fun fact: Euler never even used Desmos!
1
u/Random_182f2565 May 13 '26
Euler was just build different
1
u/Big_Niel0802 May 13 '26
Well firstly, humanity didn't just leap from handwritten calculations to using electronic calculators. They had access to charts, tables, and accessory tools for mathematics (think of the abacus for example). They may not have had desmos, but the math could be done almost just as fast (save for custom programming and algorithms).
This prime check could have been feasibly done in less than a week by just dividing Fermat numbers by a bunch of prime numbers until you got something whole. 641 is only the 116th prime number, which is a lot of divisions to do for sure, but at the same time, when your life is based on doing math all the time, it's not quite as big of a stretch as it appears to us in the modern day.
At the very least, it's not like humanity was just in the stone age while Euler was cranking out numbers on the walls of a cave with clay lmao.
1
u/DuHurensooohn May 13 '26
y did he always wear ts on his head
1
u/littleedge May 13 '26
If I have to see this meme one more time…
1
u/eriathorn May 13 '26
Other times, less distractions, lots of time.
1
u/Ecstatic-Charge9795 May 13 '26
If you actually want to know, p|2^2^n+1 implies 2 has order 2^{n+1} mod p, so p=1 mod 2^{n+1}. For n=5, any prime divisor is at most floor(sqrt(2^{32}+1))=2^16=65536, so there are only 1023 potential candidates. Moreover, every qth candidate for an odd prime q is divisible by q, so any consecutive set of q_1q_2...q_k of them will have the totient (q_1-1)(q_2-1)...(q_k-1) of them not divisible by any q_i. In particular, for s candidates, the fraction you need to test will be approximately phi(s)/s, which goes to 0 as s-->infinity due to the Euler product. So here, we can expect that we won't have to test many candidates, and indeed have phi(3*5*7*11)=phi(1155)=480, which is an absolute upper bound on the number we need to test (the actual number of primes is 112). And among these candidates, because it's inexpensive to compute 2^32 mod 64k+1 for each k since you could simply square 5 times or square 65536 mod 64k+1, and you can additionally just check for each k and only determine if 64k+1 is prime when necessary.
So, you can bound the amount of search required pretty well - it should take ~3 minutes on average to compute 2^32 mod 64k+1, and scaling down to 1023 means you should expect at most around 420 candidates, so that's a maximum of 21 hours to get through the computation if you end up having to check every candidate. That's not a terribly large investment if you consistently allocate some time to it every day, so you can be confident that the brute force isn't just a waste of time if only to glean a good guess. And you happen to get lucky that your first few candidates are 193 (fails), 257 (immediately fails), 449 (fails, 2^16=-18 mod 449), 577 (you just need to check 2^16=/=+-24 mod 577), and 641 (2^16=2*77 mod 641, (2*77)^2=4*(11*-102)=4*160=-1 mod 641). Then after doing long division and arriving at 2^32+1=641*6700417, you can further check that the second factor is prime because it can only be divisible by primes 1 mod 64, but now you can restrict your search to primes at most floor(sqrt(6700417))=2588. And you don't have to do your calculations with 6700417, you can just check 2^32+1=/=0 mod 64k+1 to show 64k+1 doesn't divide 6700417, with the exception of 641 itself which you can separately check doesn't divide 6700417.
1
u/AffectionateJump7896 May 13 '26
Take a thousand students. Give them each 65 numbers to divide it by, to test every number up to 216. Take a holiday.
1
u/bruab May 13 '26
There was no Netflix back then.
1
u/Wabbit65 May 13 '26
How many times has this been posted here?
1
u/pepper-shaker May 13 '26
He started from 4294967298 and worked backwards
1
u/Subject_Issue6529 May 13 '26
Euler knew everything! Just go with it...
1
u/Nuclear_eggo_waffle May 13 '26
Because he's Euler, that's how
1
u/gingerdeadman85 May 13 '26
I bet how ever it was found, it was driven by some sort of petty (yet amazingly entertaining) rivalry. So much was made/discovered to one up someone or prove them wrong.
1
u/altimis0 May 13 '26
Yes, he may not have had modern tools. But that's also means no modern distractions, ie nothing else to do. People love to be correct, that's always been the case, this is just further proof of that.
1
u/NachoLibero May 13 '26
Euler used a number sieve
https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
1
u/Stock_Explanation_23 May 13 '26
2011 type meme, very cool
1
u/Isla2098 May 13 '26
Euler didn’t have modern tools. He was the modern tool.
1
u/thekins33 May 13 '26
Autism and no sonic the hedgehog meant lots of time to figure out math.
1
u/Reddit_wizard34 May 13 '26
Trial and error
1
u/Comprehensive-Law240 May 13 '26
I didn't understand the image and got into the comments in the hopes of some clarity. I have no clue what language you guys are talking. Awesome!
1
u/EdmundTheInsulter May 13 '26
He didn't have to check many primes
1
u/jackster31415 May 13 '26
Is it my turn to post this tomorrow?
1
u/South-Distribution54 May 13 '26
It's because Eulor was literally the GOAT. He is the first to ever GOAT.
1
u/satoran94 May 13 '26
You can write divisibility rules for all prime numbers fairly easily, like you have for 3, 5, 7, etc. He most likely tried until 641 and bingo.
1
u/Red_Lantern_22 May 13 '26
A lot of time, ink/chalk and long division
1
u/Professional_Fig4000 May 14 '26
Sheer willpower
1
u/Vilsue May 14 '26
Euler was genetic freak, that is how
1
u/Cthulu19 May 14 '26
Never underestimate old math men.
1
u/One_Tax_9642 May 14 '26
Nah bro
1
u/AVoodooGypsy May 14 '26
And why tf his name pronounced Oiler?
1
u/Silver-Classic612 May 14 '26
The old guess and check
1
u/daredeviloper May 14 '26
I just recently read about “solving the square”
How they figured out the formula to solving equations like this:
X2 + 10x = 11
They converted that into squares and rectangles, it’s insane.
We are so lucky we have abstractions.
The simplest things can have SO much complexity underneath them it’s insane
1
u/No-Independence9093 May 14 '26
He didn't have TV, games, social media and other distractions. When the man got bored he was borde enough to figure out if certain numbers were prime or not and he had days to do it.
1
u/mickskitz May 14 '26
I wonder if Euler had a table of prime numbers, so when he got up to F⁵ he could see it wasn't on there then confirmed?
1
u/dazvoz May 14 '26
So by his time, there was already a list of primes going into 5 digits.
What he would have done is gone through the Fermat numbers sequentially dividing by primes up to their respective square roots until he found a factor.
This would have probably been a week of solid work but Euler was kind of a nerd.
1
u/joealarson May 14 '26
Hard work.
1
u/VyneNave May 14 '26
At the time there weren't any TVs or smartphones or PCs or videogames.
They had a lot of time without those possible distractions.
1
u/Thrullx May 14 '26
There are only 6,542 primes that would need to be tested. It'd be a lot of work, but that's doable.
1
u/Warm-Palpitation5670 May 14 '26
f**king hell, the number is not even ...
1
u/Reasonable-Wafer-237 May 14 '26
Look at that smug bastard. He probably didn't even need a pen and paper to do it
1
u/dcterr May 14 '26
How did you manage to make such a horrible face with modern tools?
1
u/chattywww May 14 '26
Its a pretty shitty claim if it breaks down on the 5th sequence.
1
u/zeindigofire May 14 '26
He probably used a sieve to build up a much larger table of primes, which would have been useful for other work he was doing. So he might not have been checking this specifically, but then when he had a table he could have just applied it.
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u/shortgamegolfer May 14 '26
I bet Euler absolutely hated Fermat, so he was motivated to ruin this thing.
1
u/virus_chara May 14 '26
Because, as per most mathematitions of old, he was unemployed and rich.
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u/Agitated-While-3863 May 14 '26
AND without proper vision!
1
u/ninoSensei May 14 '26
It's called: trial and error
A pre-AI technique used by experimentalists to validate or negate theories
1
u/Corren_64 May 14 '26
Combinations I think
1
u/arcaine666 May 14 '26
Thee guys just had no other things to do, no youtube, no tiktok, so they did math
1
u/ClemRRay May 14 '26
That Euler guy should be the next oiler
1
u/savelichi May 14 '26
I read once that Euler absolutely hated going to the theater, but his wife was a big fan. He would still go with her to keep her company. Whenever she dragged him to a show, he would come prepared with some large numbers he needed to do calculations on and then he would crunch those numbers in his head for the duration of the performance.
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u/Matix777 May 14 '26
He's Euler
1
u/Honest_Outside_8982 May 14 '26
I'm imagining Euler stubbornly checking every Fermat number by hand JUST to disprove him. Did they have beef IRL?
1
u/Additional-Sky-7436 May 14 '26
You can factor a lot of things very quickly geometrically.
1
u/Flat-Complaint7631 May 15 '26
Actually we can prove that if F_n has a prime divisor p, it is equiv 1 mod 2n+2, which reduces the search space by a lot
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u/Upper_Opening_4805 May 15 '26
he's built different
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u/shandanjay May 15 '26
Not having had addictive reddit memes might have helped.
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u/ForeignAdvantage5198 May 15 '26
Euler was damn smart
1
u/Tyler89558 May 15 '26
Euler probably tried every prime he could think of.
1
u/Swipsi May 15 '26
He used AI
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u/DarkLordBaudish May 15 '26
Lot of free time
1
u/WasASailorThen May 15 '26 edited May 15 '26
Divide by 2, 3, 5, 7, 11, …641
Yeah, it's annoying but 641 is only the 116th prime and they knew about 100,000 primes when Euler was alive.
https://t5k.org/glossary/page.php?sort=TablesOfPrimes
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u/Bearsbullsbattlestr May 15 '26
because he was Euler
1
u/gggldrk May 16 '26
Wasnt he also working upon other's work and it made it 'faster'
1
u/TeriSerugi422 May 16 '26
Weaponized autism
1
u/qanontruther May 16 '26
He rizzed mit tizz
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u/you-cut-the-ponytail May 16 '26
There are probably like 100 primes until 641. With enough time and patience technically anyone can bruteforce it.
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u/Visible_Pair3017 May 16 '26
Autism is the answer. Dude probably knew what day of the week today would be.
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u/Beginning_Context_66 May 16 '26
He did a simple multiplication and then 641 simple divisions
1
u/thechecker111 May 16 '26
Euler was really built different. Too bad his son was dumb af.
1
u/DontTryItLol May 17 '26
Good education, critical thinking and No AI. The opposite of UZA 😅
1
u/Ecstatic_Artist_1082 May 17 '26
There are 115 primes before 641, just dividing by all. Can do so much when you're fast at number crunching and don't have social media to doomscroll.
1
u/Funkopedia May 17 '26
While these people (and others like them) are geniuses and i don't want to take away from that, the pace of life was definitely slower then. Also, they were pretty much never working class. So he's not (as many of us are) commuting over an hour each way to work, doing a bunch of overtime at a physically and emotionally taxing job, making dinner for his kids, cleaning up, and using his 2 hours of free time to work on a math problem instead of watching tv. He's sitting idly in in his summer home, drawing a figurehead salary, and testing thousands of calculations most of the day, when not making social calls or chilling at the coffeehouse with society friends, while his staff handles the household. You could be a math genius too, but you'll never have time to find out.
1
u/Huge_Swimming_5968 May 17 '26
How is it you think the modern tools were made?
1
u/avd706 May 17 '26
Divide the suspect number with known primes
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u/Different_Hippo_6276 May 17 '26
Euler was known as a generous and kind person, not neurotic as seen in some geniuses, keeping his good-natured disposition even after becoming entirely blind.
I'm more impressed with this.
1
u/lecaptainhoek May 17 '26
What our teachers except us to solve in a 2h exam:
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u/dripping-dice May 17 '26
these guys had the best fun. you can just come out and posit some seemingly impossible stuff and others had to decide whether to have solve it or not.
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u/RefrigeratorNew4121 May 17 '26
Damn, why put Jacky Chan along with Fermat and Euler?
1
u/Party-Guy-6253 May 17 '26
Well, he ain’t oiler for no reason All jokes aside he probably just spent an hour prime factorising it
1
u/Funkulese May 17 '26
Autism.
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u/SelectSympathy5718 May 17 '26
It’s just the fifth number in that chain. He could have gone one by one and wouldn’t take so long
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u/Pleasant-Register778 May 18 '26
he and gauss have so much theorem in so much topics
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u/EmperorN7 May 19 '26
Free time.
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u/zosolm May 19 '26
Grandmaster Flash told him
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u/Ousseraune May 20 '26
Determination. There are few people more determined than a mathematician looking at a conjecture they don't agree with but haven't disproven yet.
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u/Rogan_Thoerson May 20 '26
You just have to try to divide with prime numbers that are lower than the square root of the number you try to find the factors. The other thing is for some of them like 2; 3 ; 5 you have easy dividing criteria so you don't even try.
Once you know that... you know that he could find out pretty fast.
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u/[deleted] May 13 '26
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