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?

## Comments

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TopNewestIn the fist arithmetic sequence that you have written, we have 79 (a prime) on 27 (not a prime) th position. – Abhishek Sinha · 2 years, 7 months ago

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– Finn Hulse · 2 years, 7 months ago

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.Log in to reply

– Sharky Kesa · 2 years, 7 months ago

If anyone could do that, the world would literally have to bow down to their knees.Log in to reply

– Finn Hulse · 2 years, 7 months ago

There's a NUMB3RS episode about this. Have you seen that show?Log in to reply

– Sharky Kesa · 2 years, 7 months ago

Nope.Log in to reply

– Finn Hulse · 2 years, 7 months ago

Yes they will... :DLog in to reply

– Finn Hulse · 2 years, 7 months ago

MWAHAHAHAHAHA!Log in to reply

– Sharky Kesa · 2 years, 7 months ago

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.Log in to reply

– Finn Hulse · 2 years, 7 months ago

Actually the cap is at 5-6 minutes. I'm just having a little fun. :DLog in to reply

– Sharky Kesa · 2 years, 7 months ago

Good, for a second I was worried.Log in to reply

– Sharky Kesa · 2 years, 7 months ago

How about you use the common difference as a prime which isn't a factor of \(10^n\)? 7, 13 17, etc.Log in to reply

– Finn Hulse · 2 years, 7 months ago

??Log in to reply

– Sharky Kesa · 2 years, 7 months ago

Like, you can try 1, 8, 15, 22, etc. It doesn't work as good as 3 though.Log in to reply

– Finn Hulse · 2 years, 7 months ago

Mmm.Log in to reply

– Sharky Kesa · 2 years, 7 months ago

How about you use 210? 1, 211, 421, etc.? It works to over 1000.Log in to reply

– Finn Hulse · 1 year, 4 months ago

Wait no because 841, the 5th number, isn't prime.Log in to reply

– Sharky Kesa · 1 year, 4 months ago

Where have you been? Also, yeah, it isn't. I think I was referring to some other pattern.Log in to reply

– Finn Hulse · 1 year, 4 months ago

Hehe, I've been just living life.Log in to reply

– Sharky Kesa · 1 year, 4 months ago

Nothing special?Log in to reply

– Finn Hulse · 1 year, 4 months ago

Started high school, started running, did swimming over the summer... Played way too much League of Legends. xDLog in to reply

– Sharky Kesa · 1 year, 4 months ago

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.Log in to reply

– Finn Hulse · 1 year, 4 months ago

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?Log in to reply

– Sharky Kesa · 1 year, 4 months ago

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.Log in to reply

13 minute 5k however is extremely impressive. If you moved to USA you could be #1! – Finn Hulse · 1 year, 4 months ago

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– Sharky Kesa · 1 year, 4 months ago

I'm really bad at swimming. There was a massive 10m difference between 2nd and me.Log in to reply

– Finn Hulse · 1 year, 4 months ago

Wait but you actually went 13:09 for 5K?Log in to reply

– Sharky Kesa · 1 year, 4 months ago

Yes.Log in to reply

– Sharky Kesa · 1 year, 4 months ago

I came 7th.Log in to reply

– Finn Hulse · 1 year, 4 months ago

Isn't the world record like 12:30?Log in to reply

– Finn Hulse · 2 years, 7 months ago

Holy crap it does. :OLog in to reply

– Sharky Kesa · 2 years, 7 months ago

I just found it.Log in to reply

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, 4 months ago

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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, 4 months ago

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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, 4 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, 6 months ago

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– Finn Hulse · 2 years, 6 months ago

I only got to talk for 10 minutes and then my teacher made me sit down. D;Log in to reply

– Sharky Kesa · 2 years, 6 months ago

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. :(Log in to reply

– Finn Hulse · 2 years, 6 months ago

The thing is, it was the last day of school and all 30 of my classmates also needed to present... :OLog in to reply

hey, but in d second AP, you havent considered 23!!!! – Aviroop Pal · 2 years, 6 months ago

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– Finn Hulse · 2 years, 6 months ago

Yes, I've overlooked it only because it doesn't help prove the point I'm making.Log in to reply

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, 7 months ago

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– Finn Hulse · 2 years, 7 months ago

Thanks!Log in to reply

– Finn Hulse · 2 years, 7 months ago

I went to a lecture on the Green-Tao theorem at the College of W+M when I was in 6th grade.Log in to reply

– Sharky Kesa · 2 years, 7 months ago

You're lucky. They don't have anything like that in Australia.Log in to reply

– Finn Hulse · 2 years, 7 months ago

You don't have any colleges? I didn't have to like sign up or anything I just walked in it was an open lecture. :PLog in to reply

– Sharky Kesa · 2 years, 7 months ago

We don't have open lectures.Log in to reply

– Finn Hulse · 2 years, 7 months ago

HOLY CRAP. Where have you been living? Dude that is terrible! That is obscene! That is absurd! That is despicable! :OLog in to reply

– Sharky Kesa · 2 years, 7 months ago

Plus, most people aren't interested in math. Hmph.Log in to reply

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, 7 months ago

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– Finn Hulse · 2 years, 7 months ago

I've adressed this in the last paragraph. Daniel had a similar response.Log in to reply

– Calvin Lin Staff · 2 years, 7 months ago

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.Log in to reply

– Finn Hulse · 2 years, 7 months ago

Perhaps! That's why I put it out here.Log in to reply

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, 7 months ago

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

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, 7 months ago

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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, 7 months ago

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– Finn Hulse · 2 years, 7 months ago

There is no set definition. There is only approximations and rough estimates. This is Number Theory, not Algebra. Primes are not cute little patterns.Log in to reply

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:

I think that contradicts your original post. – Daniel Liu · 2 years, 7 months ago

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– Finn Hulse · 2 years, 7 months ago

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?Log in to reply

– Daniel Liu · 2 years, 7 months ago

Can you explain what "set theory" is behind the distribution of primes? I'm a little confused.Log in to reply

– Daniel Liu · 2 years, 7 months ago

Then what is the optimal case for \(d=2\)?Log in to reply

Isn't that the question of the day? :D – Finn Hulse · 2 years, 7 months ago

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@Calvin Lin I'm interested in your response. – Finn Hulse · 2 years, 7 months ago

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– Yuxuan Seah · 2 years, 7 months ago

He responded XDLog in to reply

– Yuxuan Seah · 2 years, 7 months ago

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 XDLog in to reply

– Finn Hulse · 2 years, 7 months ago

Good idea! :DLog in to reply

– Yuxuan Seah · 2 years, 7 months ago

Yay :DLog in to reply

u have made a good pattern – Shubham Taipan · 2 years, 6 months ago

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– Finn Hulse · 2 years, 6 months ago

Thank you. :DLog in to reply