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Interesting Prime Pattern

Hello!

I have a really crappy English class. We're doing a unit on public speaking, and we get to choose a topic and give a "lecture" to the class. Some people are doing instructional talks, like how to tie shoelaces; and other people are giving persuasive talks.

To annoy the crap out of my teacher, I'm doing a 45 minute long presentation on the Green-Tao Theorem. If you have any suggestions or links to good papers, I'd appreciate it, but that's not what this note is about.

I have discovered a truly marvelous property that may help with a compact proof. I observed (in my science class) that given an arithmetic sequence, there tended (see the last paragraph for more info) to be prime numbers in prime positions! For example, consider the sequence:

\[1, 4, \boxed{7}, 10, \boxed{13}, 16, \boxed{19}, 22, 25, 28, \boxed{31}, 34, \boxed{37}, \dots\]

Notice how all boxed numbers are primes. Here is a list of the positions at which the prime numbers appear:

\[3, 5, 7, 11, 13\]

These are all primes! In fact, these are all CONSECUTIVE primes! Wow! Why is this? Can we generalize this for any arithmetic progression \(a_1, a_1+d, a_1+2d, \dots\)? Is it a coincidence that the difference in this sequence (\(3\)) is prime, since the same property obviously wouldn't hold for \(d=6\)? And is this an application of the GT Theorem or part of it's proof? Isn't it just the difference \(d\) in the APs that matter because \(a_1\) can be shifted back or forward by \(d\) to align the primes into prime positions?

I've taken steps toward a proof that may or may not be really awesome. I'll publish it when (or if) I finish it. What do you think?


As another example, consider \(a_1=-1\), \(d=3\). The sequence is:

\[-1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29\]

which has primes at positions:

\[2, 3, 5, 7, 11\]

which seems pretty odd (get it?).


As pointed out by Daniel Liu, each "good" sequence such as the first one (\(a_1=1, d=3\)) can be shifted by changing \(a_1\). This will throw the "prime pattern". But what needs to be shown is that there exists an optimal \(a_1\) for ALL \(d\). In this case, at \(d=3\) the optimal solution is \(a_1=1\). What about \(d=10\)? What can we say about \(d\)? Must it be prime? What else can be observed?

Note by Finn Hulse
2 years, 11 months ago

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In the fist arithmetic sequence that you have written, we have 79 (a prime) on 27 (not a prime) th position. Abhishek Sinha · 2 years, 11 months ago

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@Abhishek Sinha Oh I'm not saying it will work for ALL primes because otherwise we'd have a prime generator more powerful than any Riemann Hypothesis or similar algorithm. Finn Hulse · 2 years, 11 months ago

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@Finn Hulse If anyone could do that, the world would literally have to bow down to their knees. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa There's a NUMB3RS episode about this. Have you seen that show? Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Nope. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa Yes they will... :D Finn Hulse · 2 years, 11 months ago

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@Finn Hulse MWAHAHAHAHAHA! Finn Hulse · 2 years, 11 months ago

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@Finn Hulse I had seen this this morning, for me, just before I had to go to school. I told my maths teacher about this and I wanted to see where it failed. my maths teacher wanted to see some sort of pattern in how it worked. He has a PhD in math so he's the right person to talk to. BTW, how the hell are you supposed to do 45 minute presentations?! For us, we need to do a max of 5 minutes. Green-Tao theorem, one way to annoy the crap out of most teenagers. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa Actually the cap is at 5-6 minutes. I'm just having a little fun. :D Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Good, for a second I was worried. Sharky Kesa · 2 years, 11 months ago

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@Finn Hulse How about you use the common difference as a prime which isn't a factor of \(10^n\)? 7, 13 17, etc. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa ?? Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Like, you can try 1, 8, 15, 22, etc. It doesn't work as good as 3 though. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa Mmm. Finn Hulse · 2 years, 11 months ago

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@Finn Hulse How about you use 210? 1, 211, 421, etc.? It works to over 1000. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa Wait no because 841, the 5th number, isn't prime. Finn Hulse · 1 year, 8 months ago

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@Finn Hulse Where have you been? Also, yeah, it isn't. I think I was referring to some other pattern. Sharky Kesa · 1 year, 8 months ago

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@Sharky Kesa Hehe, I've been just living life. Finn Hulse · 1 year, 8 months ago

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@Finn Hulse Nothing special? Sharky Kesa · 1 year, 8 months ago

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@Sharky Kesa Started high school, started running, did swimming over the summer... Played way too much League of Legends. xD Finn Hulse · 1 year, 8 months ago

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@Finn Hulse LOL, I did nothing too special. I started high school last year. Got $1000 cheque from school because I did good academic stuff, came 12th in regional cross country, came 3rd in 50m butterfly against 5 people, 2 of which couldn't do butterfly, and I did AMC, AIMO and AMO with varying successes. Finally top 20 in my state for chess. Sharky Kesa · 1 year, 8 months ago

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@Sharky Kesa OMG!!!!!! Tell me more about all of those things!! What is your 50m butterfly time? What is your 5k XC time? What scores did you get on AMC? Finn Hulse · 1 year, 8 months ago

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@Finn Hulse 50m butterfly time = 1 minute 12 seconds. (Think about what I just said in the previous comment "2 of which cannot ..."). 5km cross country 13 minutes 09 seconds (I think could've been 15). AMC (Australian Mathematics Competition, not American) scores not released but I know I won a prize because I did AIMO for free. Sharky Kesa · 1 year, 8 months ago

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@Sharky Kesa Hmm... Butterfly time isn't so impressive... Sorry.

13 minute 5k however is extremely impressive. If you moved to USA you could be #1! Finn Hulse · 1 year, 8 months ago

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@Finn Hulse I'm really bad at swimming. There was a massive 10m difference between 2nd and me. Sharky Kesa · 1 year, 8 months ago

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@Sharky Kesa Wait but you actually went 13:09 for 5K? Finn Hulse · 1 year, 8 months ago

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@Finn Hulse Yes. Sharky Kesa · 1 year, 8 months ago

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@Sharky Kesa I came 7th. Sharky Kesa · 1 year, 8 months ago

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@Sharky Kesa Isn't the world record like 12:30? Finn Hulse · 1 year, 8 months ago

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@Sharky Kesa Holy crap it does. :O Finn Hulse · 2 years, 11 months ago

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@Finn Hulse I just found it. Sharky Kesa · 2 years, 11 months ago

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I just wrote a computer program that checks for this pattern. For the case \(a=1, d=3\), with upper bound the highest prime under \(100000\), the number of prime indices were \(921\) and the number of composites were \(3862\). This gives a \(19.3\%\) prime yield.

I'm not sure if this is higher than what we should expect or lower than what we expect. Daniel Liu · 1 year, 8 months ago

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@Daniel Liu OMG thank you so much Daniel! I literally was just learning Python so I could do it myself, but I'll take your word for it!

As far as seeing if this is abnormally high, take the same amount of numbers but randomize the called "prime" numbers but with the same frequency. Or, vice versa it could look at all the numbers in the prime spots and see what percent of those were prime. It's not really important.

How general can you go? Here is the ideal program to solve this problem:

First, it finds the ideal \­(a\­) for EACH \­(d\­) (within some preferably massive limit).

It does exactly what you have described above to each ideal pattern, finding the % prime yield.

But it also adds the step I have mentioned above, where it calculates if that % was abnormally high or low.

After considering all these factors, hopefully the program can give a nice simple answer as to the nature of my so-called pattern.

Dude thanks so much though, I totally appreciate it cause if I can show this to be true then I might win Breakthrough Junior this year. Finn Hulse · 1 year, 8 months ago

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@Finn Hulse Ideal \(a\) is probably small, because the larger we go the less frequent that the pattern holds.

My program can calculate percentage for any given input \(a,d\), but right now I'm a little busy to change it. It isn't hard to change it to what you said, but runtime will be a real pain. Daniel Liu · 1 year, 8 months ago

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@Finn Hulse Tell us how the "talk" went on , in the future! This sounds like a great speech Elliott Macneil · 2 years, 10 months ago

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@Elliott Macneil I only got to talk for 10 minutes and then my teacher made me sit down. D; Finn Hulse · 2 years, 10 months ago

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@Finn Hulse Teachers are sooooooo boring. Can't they have let you say the rest of your talk? I would have liked to have you say it to me. Such is life. :( Sharky Kesa · 2 years, 10 months ago

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@Sharky Kesa The thing is, it was the last day of school and all 30 of my classmates also needed to present... :O Finn Hulse · 2 years, 10 months ago

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hey, but in d second AP, you havent considered 23!!!! Aviroop Pal · 2 years, 10 months ago

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@Aviroop Pal Yes, I've overlooked it only because it doesn't help prove the point I'm making. Finn Hulse · 2 years, 10 months ago

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Finn , can I know how did you come to know about these theorems ? It's your wish to answer this question . I would truly say that you are a young inspiration to many of them , including me !!!! you know that i don't know this theorem at all . In fact , i heard the word Calculus only after coming to brilliant and became interested in maths and theoretical physics after coming to brilliant and reading stephen hawking 's book. I am even aiming and have promised to myself that i would reach level 4 and level 5 in all topics Sriram Venkatesan · 2 years, 11 months ago

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@Sriram Venkatesan Thanks! Finn Hulse · 2 years, 11 months ago

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@Finn Hulse You're lucky. They don't have anything like that in Australia. Sharky Kesa · 2 years, 11 months ago

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@Finn Hulse We don't have open lectures. Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa HOLY CRAP. Where have you been living? Dude that is terrible! That is obscene! That is absurd! That is despicable! :O Finn Hulse · 2 years, 11 months ago

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@Sharky Kesa Plus, most people aren't interested in math. Hmph. Sharky Kesa · 2 years, 11 months ago

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My sense is that this pattern works because you "chose" the starting value \(a_1\), and also that values are small enough for you to "see a pattern".

For example, if we used \( a_1 = 2, d = 3 \) as opposed to \( a_1 = -1, d=3 \), we will have primes at the positions \(1, 2, 4, 6, 10, \ldots \), which doesn't highlight the pattern you are looking for. Calvin Lin Staff · 2 years, 11 months ago

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@Calvin Lin I've adressed this in the last paragraph. Daniel had a similar response. Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Just because the initial cases look like there is a pattern, doesn't necessarily mean that there is such a pattern. Perhaps if you compile 100 - 1000 terms, that will give you more insight as to whether or not this is true. Calvin Lin Staff · 2 years, 11 months ago

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@Calvin Lin Perhaps! That's why I put it out here. Finn Hulse · 2 years, 11 months ago

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Well, consider \(a_1=2\) and \(d=3\). This gives primes at positions \(1,2,4,6,10\) so...

In addition, \(a_1=1\) and \(d=2\) gives primes at positions \(2,3,4,6,7,9,10\). I don't really see any pattern (or primes) here. Daniel Liu · 2 years, 11 months ago

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@Daniel Liu Look at the the first sequence given. If \(a_1=-1\), then all positions will be shifted back, so that instead the primes will lie on

\[2, 3, 5, 7, 11\]

as shown in the note as well. So obviously \(a_1\) can vary, and doing such will "shift" the results. So \(a_1\) is really depends on what \(d\) is to create the optimal prime sequence. Am I being unclear? Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Also it makes sense that there should exist an optimal \(a_1\) because both distributions of primes follow the same general logarithmic scale (at least my intuition).

However, you have yet to define "optimal". Every sequence has an optimal case; however, how optimal does this optimal case need to be? Daniel Liu · 2 years, 11 months ago

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@Daniel Liu There is no set definition. There is only approximations and rough estimates. This is Number Theory, not Algebra. Primes are not cute little patterns. Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Thus, your conjecture cannot be proven false. Good Game, sir.

But joking aside, I still think this noticing is a bit trivial. It's kind of like using the distribution of primes to approximate the distribution of primes. When defining a word, you can't use the word itself.

In addition:

Primes are not cute little patterns.

I think that contradicts your original post. Daniel Liu · 2 years, 11 months ago

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@Daniel Liu Yes... This is true. Actually it's very true. But the distribution of primes is constant and calculate-able. Think about the set theory behind it. By the way have you ever seen NUMB3RS? Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Can you explain what "set theory" is behind the distribution of primes? I'm a little confused. Daniel Liu · 2 years, 11 months ago

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@Finn Hulse Then what is the optimal case for \(d=2\)? Daniel Liu · 2 years, 11 months ago

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@Daniel Liu Isn't that the question of the day? :D Finn Hulse · 2 years, 11 months ago

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@Calvin Lin I'm interested in your response. Finn Hulse · 2 years, 11 months ago

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@Finn Hulse He responded XD Yuxuan Seah · 2 years, 11 months ago

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@Yuxuan Seah And you can actually test this out by using programming to try out different values, and show the teacher all the cases up to 1 trillion. That would turn it into a 45-hour presentation though XD Yuxuan Seah · 2 years, 11 months ago

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@Yuxuan Seah Good idea! :D Finn Hulse · 2 years, 11 months ago

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@Finn Hulse Yay :D Yuxuan Seah · 2 years, 11 months ago

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u have made a good pattern Shubham Taipan · 2 years, 10 months ago

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@Shubham Taipan Thank you. :D Finn Hulse · 2 years, 10 months ago

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