A Beautiful Algorithm for the Primes

What's the fastest algorithm for generating the prime sequence? Watch a frat bro hunch over a notebook and find out!
.
.
.
Walkthrough (pause, then use “,” and “.” keys to step):
• Detailed code animatio...
Computing the primes up to 40,000 (also pause and use "," and "." to step):
• The sieves of Pritchar...
Wheel generation (with inspirational quotes):
• Generating the prime n...
Wikipedia page:
en.wikipedia.org/wiki/Sieve_o...
What is a Primorial (really)?
undergroundmathematics.org/di...
0:00 Intro
0:28 Context
1:23 Prime "Fingerprints"
2:01 Relative Primes
2:27 Primorials
2:47 Bro-terlude
4:12 Primes and Primorials
5:15 Wheels
5:37 Generating wheels
6:59 Ranges of wheels
7:52 The Algorithm
8:24 Dreams of a Bro
8:38 Outro
#SoME2

[@danielcarroll3358]

Boy does this bring me back to my college days in the days beyond recall. The first task in the Fortran lab was to write a program to print all the primes that would fit in a 16-bit signed integer, i.e. up to 32,767. The grad student lab assistant's program ran faster than everyone else's because he had written an interrupt driven driver for the printer. This allowed the IBM 1130 to calculate while waiting for the printer. But mine ran faster because I knew that if a number was composite at least one of its factors was less than or equal to the square root. Pretty fundamental fact, but the look on his face was priceless as he tried to figure out how this short little program was so fast (relatively speaking).

[@CARROTMOLD]

Do you post this on every video?

[@danielcarroll3358]

@@CARROTMOLD Nope. You?

[@random6033]

which algorithm did you use?

[@m4riel]

@@random6033 Even if you take the simplest algorithm that checks if N is prime (like dividing N y everything from 2 up to N-1), if you, instead, divide by 2 up to int(sqrt(N)), it will still be a lot faster, since, for the example of 32767, needs at most about 180 checks. Now, if you do a little more work, you can, instead of checking by everything up until a number's square root, check only by the primes in this interval (there are at most 40 primes

[@danielcarroll3358]

@@random6033 @Minding My Business has summarized it well. My program printed 2 and then tested all the odd numbers starting with 3 using the sieve of Eratosthenes, "the simplest algorithm", with divisors up to int(sqrt(N)). I had also calculated that I would need less than 10 kb of space to store the prime divisors as unsigned 16 bit integers to evaluate up to 2^32. But then it was off to the next lab assignment. This was all punched into cards of course.

[@johnchessant3012]

Definitely was not expecting Pritchard to be still alive, what a twist! This feels like something that could've been discovered centuries ago but wasn't; it means even though math has gotten so specialized nowadays there are still plenty of "elementary" gems still waiting to be found!

[@oluckyman]

I also enjoyed this twist!

[@ItsMeChillTyme]

There have been older sieves, Sieve of Eratosthenes and Sieve of Sundaram to name a couple. Pritchard's is a modern optimisation on those Sieves with tradeoffs afaik.

[@JM-us3fr]

Wow I didn't know about Pritchard's sieve. This was really interesting. I thought the sieve of Atkin was the only sieve which could theoretically run faster than the sieve of Eratosthenes, but I guess I was wrong!

[@avatar098]

This is one of those videos I’ll have to rewatch over and over and really work through it 😭

[@clairecelestin8437]

I noticed this pattern independently several years ago, but couldn't explain it so clearly. Something else that's interesting about these patterns is that, unless I'm mistaken, every wheel must always contain at least some twin primes, since at each iteration we're eliminating things which have P as a factor and P must always span at least one set of twin primes. This falls a little short of proving that there's an infinite number of twin primes, because as mentioned in the video, the wheels grow very quickly and only a small section of each wheel will appear in the final prime sequence... However, I can't see a way to prevent that small visible section from periodically containing some of the twin primes that remain in the wheel.

[@kyrylosovailo1690]

Bro, I was thinking exactly about the same algorithm! However I didn't imagine it like wheels, rather repeated sequences.

[@brendanschuett]

Same!

[@fabiorota9661]

This could have been explained much more clearly honestly, when you have been working with something for a long tike all feels obvious, but here we are listening to this for the first time, tyere are many logical eductions let to the viewer, for example the reason why the wheel keeps on working after one lap, its hard if not impossible to understand things without pausing making the experience frustrating.

[@raulgalets]

he explained it actually. referentially tho, so the confusion is understandable. I think this is a more advanced video actually. try mathologer, his channel is very explanatory

[@proloycodes]

@@raulgalets advanced or not, you still need to lay out the steps before you can walk into them

[@BryanLevenson42]

Love this idea about prime DNA - never thought about it that way!

[@j.r.8176]

I love how you spend an hour explaining what prime numbers are but just zoom through the parts that actually needed explanation. Maybe skip the comedy and sketches to make up room for more explanation.

[@proloycodes]

agree

[@Martykun36]

@Larry Brin not at all! it was well-performed and entertaining. It can work a lot on the pacing but to call it cringey is just juvenile cynicism.

[@pirateskeleton7828]

There’s another fun property to this sieve method, and that is that the numbers relatively prime to the primordial are all symmetrical along the length. For example, in the case of 30, you have [1 (-29), 7 (-23), 11 (-19), 13 (-17) …] mod 30. Another fun property is that you can use larger circles of the sieve to predict what primes have the highest likelihood to being a twin prime, and definitely discard certain ones from being twin primes, because the twins will maintain their positions with respect to the ones in the circle.

[@bronzmash]

Exceptionally well delivered.

[@idontwantahandlethough]

"bro-terlude" is all I needed to subscribe. I needed more interesting math in my life anyway :) You sir... you are the Queen AND crown jewels of exhausting analogies. (but in all seriousness, super interesting video bromanguy, this was really neat!)

[@haph2087]

Neat video. I felt like it went by too fast, as I had to watch it a few times to get what's going on. Also, are you left handed? The parts where you were writing, you were writing left handed, and covering the text with your hand which made it hard to read. (I doubt this helped the "going to fast" feeling) Maybe try using more of the virtual text, or showing us the text after writing it, and just cover the parts you don't want to show with another piece of paper, revealing the next bit by moving that paper down. (I've seen teachers do this on overheads when they wanted to re-use what the wrote in previously when teaching the same course at multiple times) I guess if you want a third option, you could try using a whiteboard/chalkboard, and write bigger, so that less is covered by your had. Or trick a friend into writing it for you. Anyways, neat video, though I am curious what the proof is that shows this is the most "efficient" method of generating primes (in whatever metric it is the best).

[@hirojau9587]

I can't believe the video was so well made! It was genuinely funny and easy to understand. Also I notice you have an aops book!

[@marekmatusiak8397]

The proof of this solution is surprisingly simple. It may be interesting that, as you can see, the twin numbers in the form p # + - 1 ends at p = 11. The second remark is the vertical axis of symmetry of the circle. In my similar algorithm I use numerical sequences with cyclic differences. The use of wheels complicates and makes it difficult to understand the subject.

[@ingiford175]

I have played with this, but more as a shifting ruler then a wheel. Good to have a different thought/method of it. Especially the cancellation of removing the smaller one from the larger one.

[@KeldonA]

My favourite property is that the primorial set (that's what I called it when I first discovered this) contains numbers that are not only all coprime with each other, but are also coprime with each number (x) in the set + n * primorial_set_value. And there's more. Take any number x from the set, multiply it with each number in the set in order modulo primorial_set_value, and you will, I kid you not, get back every number in that primorial set, but each with a unique reordering. When you model the sequence as intervals, it's even more interesting. Firstly, you can clearly see that it produces an infinite seed for EVERY single prime gap from the previous primorial set (though it does not imply infinite prime gaps of any amount). This view of primes also gives you a remarkably different way of navigating and ordering potential primes. Aaand, one of my favourites ... going back to the aspect of any x,y pairing from the set modulo the primorial_set_value giving you a value from the primorial set (along with the unique shuffling behaviour) gives you a really simple method of private key encryption! I never found it faster for generating primes, but this was about 15 years ago, and written in Java in a very OO way (the JVM and most VMs are much more efficient now though).

[@mehdielaissi6486]

truly amazing!!

[@joseville]

Great video! I had used the ideas up to 5:00 before to generate prime candidates that are guaranteed not to be divisible by the first k primes. Ofc, this went further than that! 4:38 You're already using p_n to refer the nth prime, so using p_k to refer to the kth primorial is confusing. Also, it's inconsistent with your earlier notation which used a subscripted capital Pi and your later notation which used a (unsubscripted) Q. Just be mindful of consistency/notation next time. Calling one thing by one name one time and by a different name another time leads to unnecessary confusion. Calling two different things by the same name also leads to unnecessary confusion. 4:45 prime p divides the primorial p_k if only if p

[@ConnoisseurOfExistence]

Awesome! I've always been fascinated by primes! Need to watch this few more times.

[@violetasuklevska9074]

0:49 I find it too much of a coincidence that both you and Graenolf (don't ask me why I'd remember this) both picked a random number that ends in 86, 286 and 7186 respectively. Now the number 7 is statistically the most commonly chosen number between 1 and 10 (Numberphile among others have stated this). I don't know why 86 should be the most commonly picked 2 digit number, but I'll say a few things: Given what we know about 7 there are clearly preferences. I believe when picking a two digit number you wouldn't want to choose extremes like 0, 1 and 9 or even the middleground 5, as your two digit number would be imbalanced. That leaves two sets {2, 3, 4} and {6, 7, 8} which you'd intuitevuly construct as 'small' and 'big'. If you pick a number from both sets you are likely to go for the middleground of both to get nice representation, so something like 73. If you pick a single set you probably prefer the 'big' set to make your number stand out (this is why I said 73 and not 37), maybe the 7 sticks out too much as the middle of the pack, maybe you decide to go for a bigger number and start with an 8 but not the biggest, somehow you end up at 86. To reiterate 0, 1, 5 and 9 shouldn't appear, you want a bigger number to be distinct, you pick 73 or 86. I swear this is not a coincidence.

[@dalmationblack]

The bit around 4:12 felt a little fast at first, so I had to watch it twice, but the animations afterwards made it more clear. Good video nonetheless.

[@HenokJackson]

Great video !, Don't stop making vids lik these..

[@stefanalecu9532]

I really enjoyed this video

[@Max-yb1mx]

That's cool. I wrote some code years ago to do this. Didn't know anyone else who had done this.

[@lexinwonderland5741]

...the tissues on the table are a nice touch. (great video btw!!)

[@udayrajd]

Beautiful !

[@leyasep5919]

Also note that if you have a wheel built for factorial of p, you have p symmetries but they are not "compatible" with each other so you have to choose one to reduce the actual size of the wheel.

[@TymexComputing]

nice one - thanks! And thank You Pritchard - in Poland the surname: Pritchard is mainly known by some S-F books :) but it wheel soon change :)

[@leyasep5919]

Not enough views yet for this meaningful algorithm... Which reminds me of my own exploration and the very interesting things I found in this domain 😀

[@aleksandervadla9881]

Queen(Queen(Queen(natural science)))=prime numbers

[@tejasvishrivastava5888]

Natural Sciences ->Mathematics -> Number Theory -> Prime Numbers

[@skilz8098]

@@tejasvishrivastava5888 you for got PI

[@file4318]

Amazing!

[@egohicsum]

great video

[@user-jc2lz6jb2e]

Nice video, Egon Spengler.

[@Isignorancebliss]

Unrelated question to the video topic. But how do you like the Art of Problem Solving books? I've never seen one in real life but have bought them before for nieces/nephews. Do they deliver as promised at least for those who work them? I'm not certain my money was well invested but don't know if that was down to the child, the parent or the book.

[@rome8432]

Beautiful books bro, I recognized his copy too, they are beyond insightful, and have given me an incredible amount of fun problem solving and learning moments. I would absolutely suggest reading them, but I’m a highschooler and idk if an adult would really find much use in them, I use them for primarily fun but also to study for contests

[@Brotcrunsher]

This is going to explode.

[@Astro_retired]

Please make the preview more appealing, more people would know about this great video

[@DerekSpeareDSD]

Wouldn't Number Theory be more like the "Princess" of mathematics?

[@secret6365]

Art of problem solving vol 1 is one of the books lying on the floor

[@lincolnschoolpreservation3574]

Thanks for posting, i love numbers n math. Let me toss in a couple things. 1st get a 70s kids toy, called a Spirograph, 2nd lol look closer at the number 2. Now all things considered switch to metric..............have fun

[@danielgautreau161]

Ghandi's Recursive Formula for the next prime: Let p be an odd prime and let Q be the product of the primes less than p. In MathJax/LaTex let S=(-1/2)+\sum_{d|Q}\mu(d)/(2^d-1) where \mu is the Mobius function. Then 1

[@joseville]

What do you mean? The summation has cardinality of Q terms.

[@danielgautreau161]

@@joseville No. The summation is over all divisors of Q, not over the prime divisors of Q. For each divisor d of Q there is a term \mu(d)/(2^d-1). If Q is the product of the first 46 primes then Q has 2^{46) divisors .

[@joseville]

@@danielgautreau161 Ah I see. Thanks for explaining!

[@danielgautreau161]

@@joseville To prove Ghandi's result, write each (2^d-1)=(2^{-d}+2^{-2d}+2^{-3d}+...). (power series). Now collect "like powers" of 2 in the expression for S. Now use the Mobius function property \sum_{d|n}\mu (d)=0 when n>1, and \mu (1)=1, to show S=\sum_{n\in K} 2^{-n}, where K is the set of all n>1 that are co-prime to Q. The smallest member of K is p, and all members of K are odd, so 2^{-p}

[@nischalkc1141]

One ring to rule them all.😅 that was amazing and remind me again to do lotr marathon 😂😂. So see you soon 😂

[@samwehner8808]

This is coprime patterns that eventually filter out the primes as you add primes to filter. Current filters are very symmetric, even mirrored halfway betweeen periods. It's beautiful but still not efficient.

[@SillySussySally]

liked just for the "you can go back to the EXTREEEEMELY productive thing you were doing before" XDDD

[@ChristofApolinario]

Can you use Prime DNA using sets? It could be a good idea to tell if they're alike.

[@yashprajapati8857]

The greeting in the beginning sounded like Agadmator

[@yairkaz]

Thank you for making my head hurt

[@TrimutiusToo]

Oh well... It is pretty obvious... Discovered this algorithm myself before youtube was a thing (saying it that way makes me feel old though) Though making wheel out of wheel is slightly more efficient than what i was doing

[@harsh8899]

Gold ✨😂

[@amaratvak6998]

My dear boy, your fascination for the primes and your good intention to explore them in this video are praiseworthy...but sorry to say, your presentation is too technical and hard to comprehend. You'd do well to use more of graphics and animations along with a somewhat slower paced narration, to make your math videos more appealing to lay audience like me (who are also intrigued by these weird creatures called prime numbers!😂). God bless🙌

[@DaneBrooke]

Please redo with higher volume.

[@F00Lsmack]

I felt personally attacked by prime bro

[@a0z9]

Eso es como el círculo de quintas de la música.

[@leesweets4110]

Egon Spengler, in his younger years.

[@sanferrera]

Honest question...I am not the smartest guy...at 4:52, the reminder of 37/5 is 2, not 1, right? What am I getting wrong?

[@MichaelRothwell1]

The division of 7 & 37 is by 2, not 5, so remainder 1 is correct.

[@sanferrera]

@@MichaelRothwell1 Oh, I think I get it now, the test for 5 would be 7%5=2 and 37%5=2 also. Thank you very much, Michael.

[@MichaelRothwell1]

@@sanferrera Exactly.

[@sanferrera]

@@MichaelRothwell1 🙏

[@GlitchiPitch]

oh my gosh, you're cool

[@Inception1338]

OK, my main app is stuck in a loop...

[@phitsf5475]

286 may be number theory but is it numberwang?

[@whoknowsnubby]

Imagine the factors on the wheel were actually functions instead of numbers lol.

[@proloycodes]

fyi that's lambda calculus

[@yashmandaviya1356]

Doesn't this proves that there are actually infinite primes and there are also infinite twin/cousin primes(prime pair with diff 2)?

[@alejorabirog1679]

Wilson's Theorem

[@joseville]

How does it play into this? Just curious.

[@de_michael1222]

Isn't the sieve of Atkin faster?

[@vaels5682]

I'm impressed with myself i actually got 49 almost immediately :D

[@maddoxlane5040]

There are no negative whole numbers so saying the study of POSITIVE whole numbers was unneeded

[@markwrede8878]

The abundance of composites is 70.6%.

[@humanrightsadvocate]

Your definition of prime numbers is inaccurate. A prime number is a number that has 2 and only 2 factors. This is the correct definition. Number 1 is not a prime number.

[@bobitsmagic4961]

Very cute

[@GlitchiPitch]

where did you find this algorithm?

[@jdvon]

3:10 this is what i did when i was in sixth grade😅😅

[@General12th]

Hi Aaron!

[@MgtowRubicon]

You expect me to believe left hand proof?

[@lukephilbrecht3876]

7:40 ans: 49 time: 3 seconds

[@vitorsaramago104]

It would be much easier to understand if you put past info from the video in the current explanation.

[@skilz8098]

Not that anyone cares, but here's the first 256 prime values in a decimal, hexadecimal and binary representation followed by a truth table entry. This is using an 8 bit unsigned integer scheme! // ------------------------------------------------------------- // first true case (2) is the only even prime! // -------------------------------------------------------------- 0: 0x00: 0000 0000 false 1: 0x01: 0000 0001 false 2: 0x02: 0000 0010 true // Every prime from here out is odd! 3: 0x03: 0000 0011 true 4: 0x04: 0000 0100 false 5: 0x05: 0000 0101 true 6: 0x06: 0000 0110 false 7: 0x07: 0000 0111 true 8: 0x08: 0000 1000 false 9: 0x09: 0000 1001 false 10: 0x0A: 0000 1010 false 11: 0x0B: 0000 1011 true 12: 0x0C: 0000 1100 false 13: 0x0D: 0000 1101 true 14: 0x0E: 0000 1110 false 15: 0x0F: 0000 1111 false 16: 0x10: 0001 0000 false 17: 0x11: 0001 0001 true 18: 0x12: 0001 0010 false 19: 0x13 0001 0011 true 20: 0x14: 0001 0100 false 21: 0x15: 0001 0101 false 22: 0x16: 0001 0110 false 23: 0x17: 0001 0111 true 24: 0x18: 0001 1000 false 25: 0x19: 0001 1001 false 26: 0x1A: 0001 1010 false 27: 0x1B: 0001 1011 false 28: 0x1C: 0001 1100 false 29: 0x1D: 0001 1101 true 30: 0x1E: 0001 1110 false 31: 0x1F: 0001 1111 true 32: 0x20: 0010 0000 false 33: 0x21: 0010 0001 false 34: 0x22: 0010 0010 false 35: 0x23: 0010 0011 false 36: 0x24: 0010 0100 false 37: 0x25: 0010 0101 true 38: 0x26: 0010 0110 false 39: 0x27: 0010 0111 false 40: 0x28: 0010 1000 false 41: 0x29: 0010 1001 true 42: 0x2A: 0010 1010 false 43: 0x2B: 0010 1011 true 44: 0x2C: 0010 1100 false 45: 0x2D: 0010 1101 false 46: 0x2E: 0010 1110 false 47: 0x2F: 0010 1111 true 48: 0x30: 0011 0000 false 49: 0x31: 0011 0001 false 50: 0x32: 0011 0010 false 51: 0x33: 0011 0011 false 52: 0x34: 0011 0100 false 53: 0x35: 0011 0101 true 54: 0x36: 0011 0110 false 55: 0x37: 0011 0111 false 56: 0x38: 0011 1000 false 57: 0x39: 0011 1001 false 58: 0x3A: 0011 1010 false 59: 0x3B: 0011 1011 true 60: 0x3C: 0011 1100 false 61: 0x3D: 0011 1101 true 62: 0x3E: 0011 1110 false 63: 0x3F: 0011 1111 false 64: 0x40: 0100 0000 false 65: 0x41: 0100 0001 false 66: 0x42: 0100 0010 false 67: 0x43: 0100 0011 true 68: 0x44: 0100 0100 false 69: 0x45: 0100 0101 false 70: 0x46: 0100 0110 false 71: 0x47: 0100 0111 true 72: 0x48: 0100 1000 false 73: 0x49: 0100 1001 true 74: 0x4A: 0100 1010 false 75: 0x4B: 0100 1011 false 76: 0x4C: 0100 1100 false 77: 0x4D: 0100 1101 false 78: 0x4E: 0100 1110 false 79: 0x4F: 0100 1111 true 80: 0x50: 0101 0000 false 81: 0x51: 0101 0001 false 82: 0x52: 0101 0010 false 83: 0x53: 0101 0011 true 84: 0x54: 0101 0100 false 85: 0x55: 0101 0101 false 86: 0x56: 0101 0110 false 87: 0x57: 0101 0111 false 88: 0x58: 0101 1000 false 89: 0x59: 0101 1001 true 90: 0x5A: 0101 1010 false 91: 0x5B: 0101 1011 false 92: 0x5C: 0101 1100 false 93: 0x5D: 0101 1101 false 94: 0x5E: 0101 1110 false 95: 0x5F: 0101 1111 false 96: 0x60: 0110 0000 false 97: 0x61: 0110 0001 true 98: 0x62: 0110 0010 false 99: 0x63: 0110 0011 false 100: 0x64: 0110 0100 false 101: 0x65: 0110 0101 true 102: 0x66: 0110 0110 false 103: 0x67: 0110 0111 true 104: 0x68: 0110 1000 false 105: 0x69: 0110 1001 false 106: 0x6A: 0110 1010 false 107: 0x6B: 0110 1011 true 108: 0x6C: 0110 1100 false 109: 0x6D: 0110 1101 true 110: 0x6E: 0110 1110 false 111: 0x6F: 0110 1111 false 112: 0x70: 0111 0000 false 113: 0x71: 0111 0001 true 114: 0x72: 0111 0010 false 115: 0x73: 0111 0011 false 116: 0x74: 0111 0100 false 117: 0x75: 0111 0101 false 118: 0x76: 0111 0110 false 119: 0x77: 0111 0111 false 120: 0x78: 0111 1000 false 121: 0x79: 0111 1001 false 122: 0x7A: 0111 1010 false 123: 0x7B: 0111 1011 false 124: 0x7C: 0111 1100 false 125: 0x7D: 0111 1101 false 126: 0x7E: 0111 1110 false 127: 0x7F: 0111 1111 true 128: 0x80: 1000 0000 false 129: 0x81: 1000 0001 false 130: 0x82: 1000 0010 false 131: 0x83: 1000 0011 true 132: 0x84: 1000 0100 false 133: 0x85: 1000 0101 false 134: 0x86: 1000 0110 false 135: 0x87: 1000 0111 false 136: 0x88: 1000 1000 false 137: 0x89: 1000 1001 true 138: 0x8A: 1000 1010 false 139: 0x8B: 1000 1011 true 140: 0x8C: 1000 1100 false 141: 0x8D: 1000 1101 false 142: 0x8E: 1000 1110 false 143: 0x8F: 1000 1111 false 144: 0x90: 1001 0000 false 145: 0x91: 1001 0001 false 146: 0x92: 1001 0010 false 147: 0x93: 1001 0011 false 148: 0x94: 1001 0100 false 149: 0x95: 1001 0101 true 150: 0x96: 1001 0110 false 151: 0x97: 1001 0111 true 152: 0x98: 1001 1000 false 153: 0x99: 1001 1001 false 154: 0x9A: 1001 1010 false 155: 0x9B: 1001 1011 false 156: 0x9C: 1001 1100 false 157: 0x9D: 1001 1101 true 158: 0x9E: 1001 1110 false 159: 0x9F: 1001 1111 false 160: 0xA0: 1010 0000 false 161: 0xA1: 1010 0001 false 162: 0xA2: 1010 0010 false 163: 0xA3: 1010 0011 true 164: 0xA4: 1010 0100 false 165: 0xA5: 1010 0101 false 166: 0xA6: 1010 0110 false 167: 0xA7: 1010 0111 true 168: 0xA8: 1010 1000 false 169: 0xA9: 1010 1001 false 170: 0xAA: 1010 1010 false 171: 0xAB: 1010 1011 false 172: 0xAC: 1010 1100 false 173: 0xAD: 1010 1101 true 174: 0xAE: 1010 1110 false 175: 0xAF: 1010 1111 false 176: 0xB0: 1011 0000 false 177: 0xB1: 1011 0001 false 178: 0xB2: 1011 0010 false 179: 0xB3: 1011 0011 true 180: 0xB4: 1011 0100 false 181: 0xB5: 1011 0101 true 182: 0xB6: 1011 0110 false 183: 0xB7: 1011 0111 false 184: 0xB8: 1011 1000 false 185: 0xB9: 1011 1001 false 186: 0xBA: 1011 1010 false 187: 0xBB: 1011 1011 false 188: 0xBC: 1011 1100 false 189: 0xBD: 1011 1101 false 190: 0xBE: 1011 1110 false 191: 0xBF: 1011 1111 true 192: 0xC0: 1100 0000 false 193: 0xC1: 1100 0001 true 194: 0xC2: 1100 0010 false 195: 0xC3: 1100 0011 false 196: 0xC4: 1100 0100 false 197: 0xC5: 1100 0101 true 198: 0xC6: 1100 0110 false 199: 0xC7: 1100 0111 true 200: 0xC8: 1100 1000 false 201: 0xC9: 1100 1001 false 202: 0xCA: 1100 1010 false 203: 0xCB: 1100 1011 false 204: 0xCC: 1100 1100 false 205: 0xCD: 1100 1101 false 206: 0xCE: 1100 1110 false 207: 0xCF: 1100 1111 false 208: 0xD0: 1101 0000 false 209: 0xD1: 1101 0001 false 210: 0xD2: 1101 0010 false 211: 0xD3: 1101 0011 true 212: 0xD4: 1101 0100 false 213: 0xD5: 1101 0101 false 214: 0xD6: 1101 0110 false 215: 0xD7: 1101 0111 false 216: 0xD8: 1101 1000 false 217: 0xD9: 1101 1001 false 218: 0xDA: 1101 1010 false 219: 0xDB: 1101 1011 false 220: 0xDC: 1101 1100 false 221: 0xDD: 1101 1101 false 222: 0xDE: 1101 1110 false 223: 0xDF: 1101 1111 true 224: 0xE0: 1110 0000 false 225: 0xE1: 1110 0001 false 226: 0xE2: 1110 0010 false 227: 0xE3: 1110 0011 true 228: 0xE4: 1110 0100 false 229: 0xE5: 1110 0101 true 230: 0xE6: 1110 0110 false 231: 0xE7: 1110 0111 false 232: 0xE8: 1110 1000 false 233: 0xE9: 1110 1001 true 234: 0xEA: 1110 1010 false 235: 0xEB: 1110 1011 false 236: 0xEC: 1110 1100 false 237: 0xED: 1110 1101 false 238: 0xEE: 1110 1110 false 239: 0xEF: 1110 1111 true 240: 0xF0: 1111 0000 false 241: 0xF1: 1111 0001 true 242: 0xF2: 1111 0010 false 243: 0xF3: 1111 0011 false 244: 0xF4: 1111 0100 false 245: 0xF5: 1111 0101 false 246: 0xF6: 1111 0110 false 247: 0xF7: 1111 0111 false 248: 0xF8: 1111 1000 false 249: 0xF9: 1111 1001 false 250: 0xFA: 1111 1010 false 251: 0xFB: 1111 1011 true 252: 0xFC: 1111 1100 false 253: 0xFD: 1111 1101 false 254: 0xFE: 1111 1110 false 255: 0xFF: 1111 1111 false // --------------------------------------------------------- // post 8 bit unsigned first two cases: // --------------------------------------------------------- 256: 0x100: 0001 0000 0000 false occurrences for those groups in order: {6, 5, 4, 3, 4, 2, 5, 2, 3, 3, 3, 3, 3, 2, 4, 2} remember, this pattern is based on log 2 and log 16.

[@Ald3r_]

I've never seen your channel before, but saw this video in my recommended, and since I occasionally enjoy watching math videos, I clicked it. But man, you come off as extremely pretentious in this video. I really hope it is done in sarcasm.

[@jessasto947]

In the beginning was the Word, and the Word was with God, and the Word was God. 2He was with God in the beginning. 3Through him all things were made; without him nothing was made that has been made. (John 1:1-3)

[@volbla]

Hmm.

[@DanielPflager]

At 4:25 you say "first k primes" and write p_n. Did you mean to write p_k?

[@fabiorota9661]

Your voice tone is should be more friendly, calm and exited, so that people tend to feel what you appear like feeling when talking

[@Andres-qo2zg]

247

[@PaulFisher]

ok but who plays the bro

[@mauijttewaal]

LOL

[@FerdinandCoding]

You sound like Sheldon Cooper

[@tolkienfan1972]

"If you can count you can understand number theory" 🤔 I'm pretty sure that doesn't logically follow

[@gplgomes]

Aaron, as yoy like primes numbers study, than see my "Advanced Eratosthenes Sieve" in my channel about primes list.

[@joseville]

Can you provide a link?

[@gplgomes]

@@joseville In kzbin.info/www/bejne/bZLRlmSbf5Z_itU

[@haleshs66]

Queen?? Really??

[@GlitchiPitch]

sorry, after my previous comment you told about algorithm)

[@rounitkamal2832]

Difficult to understand

[@plugplagiate1564]

WHO IS Paul Pritchard ?

[@oluckyman]

I am.

[@Relkond]

If primes are defined by what they are not, then Mersenne primes are the defined by…

[@jakeaustria5445]

This is too fast. You are introducing something like a modulo without slowing down. It's not like we can prove everything you say immediately. How can I even understand the circle thing in a video format without pausing the vid? Yeah, I will like it if you started with proofs that is already true and build upon them.

[@bigmonster_RB]

totalement inutile, cela revient juste à dire qu'on invente toutes les suites ax+b, avec une suite par nombre premier, et de regarder tout ce qui n'a pas a été pris par le passage de toutes les suites

[@bawbak8800]

I didn't understand anything from this video :|

[@tunafllsh]

Has it been proven to be theoretically fastest?

[@swagnegrocongucci134]

It's not the fastest. It's on par with the sieve of Atkin asymptotically(which pritchhard basically ripped off btw, he only made a slight change to reduce the space complexity), if anything it has higher constant factors usually speaking.

[@tunafllsh]

@@swagnegrocongucci134 but is it asymptomatically optimal?

[@swagnegrocongucci134]

​@@tunafllsh Might be. There is n/logn as a trivial lower bound but there is not a tighter one known yet

[@disoriented3971]

@@swagnegrocongucci134 Atkin was introduced over 20 years later.

[@swagnegrocongucci134]

@@disoriented3971 Right, sorry. I got the names mixed up

[@Ruktiet]

Fun topic, but horrible explanation starting from 4:58 up to 7:16; the most important part…