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 its 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?

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

In the fist arithmetic sequence that you have written, we have 79 (a prime) on 27 (not a prime) th position.

Abhishek Sinha - 7 years 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.

Finn Hulse - 7 years ago

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

Sharky Kesa - 7 years ago

@Sharky Kesa – Yes they will... :D

@Finn Hulse – MWAHAHAHAHAHA!

@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 – Actually the cap is at 5-6 minutes. I'm just having a little fun. :D

@Finn Hulse – Good, for a second I was worried.

@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 – ??

@Finn Hulse – Like, you can try 1, 8, 15, 22, etc. It doesn't work as good as 3 though.

@Sharky Kesa – Mmm.

@Finn Hulse – How about you use 210? 1, 211, 421, etc.? It works to over 1000.

@Sharky Kesa – Holy crap it does. :O

@Finn Hulse – I just found it.

@Sharky Kesa – Wait no because 841, the 5th number, isn't prime.

Finn Hulse - 5 years, 8 months ago

@Finn Hulse – Where have you been? Also, yeah, it isn't. I think I was referring to some other pattern.

Sharky Kesa - 5 years, 8 months ago

@Sharky Kesa – Hehe, I've been just living life.

@Finn Hulse – Nothing special?

@Sharky Kesa – Started high school, started running, did swimming over the summer... Played way too much League of Legends. xD

@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 – 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 – 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 – 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 – I'm really bad at swimming. There was a massive 10m difference between 2nd and me.

@Sharky Kesa – Wait but you actually went 13:09 for 5K?

@Finn Hulse – Yes.

@Sharky Kesa – I came 7th.

@Sharky Kesa – Isn't the world record like 12:30?

@Sharky Kesa – There's a NUMB3RS episode about this. Have you seen that show?

Finn Hulse - 6 years, 12 months ago

@Finn Hulse – Nope.

Sharky Kesa - 6 years, 12 months ago

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 - 7 years ago

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

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 – Perhaps! That's why I put it out here.

@Finn Hulse Tell us how the "talk" went on , in the future! This sounds like a great speech

Elliott Macneil - 6 years, 11 months ago

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

Finn Hulse - 6 years, 11 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. :(

Sharky Kesa - 6 years, 10 months ago

@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 - 6 years, 10 months ago

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 - 5 years, 8 months ago

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.

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.

@Calvin Lin I'm interested in your response.

He responded XD

Yuxuan Seah - 7 years 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 XD

@Yuxuan Seah – Good idea! :D

@Finn Hulse – Yay :D

Yuxuan Seah - 6 years, 11 months ago

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 - 7 years ago

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

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?

Then what is the optimal case for $d=2$ ?

@Daniel Liu – Isn't that the question of the day? :D

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

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 - 6 years, 11 months ago

Thanks!

I wish to know this as well, you are 17 years old but you were 14 or 13 when you originally wrote this. How the heck did you manage to do this, I barely knew my times tables when I was 13.

César Castro - 3 years, 2 months ago

@César Castro – Hehe, I guess I was actually a hair over 13 when I wrote this (you can guess why my age says 17 on Brilliant.)

Finn Hulse - 3 years, 2 months ago

@Finn Hulse – Why were you even in school? Why don't you have like seven PhDs by now?

@César Castro – It doesn't take a genius to make an arithmetic sequence and observe an embedded pattern! I'm just 99% motivated and 1% gifted. :)

hey, but in d second AP, you havent considered 23!!!!

Aviroop Pal - 6 years, 10 months ago

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

Hey! How's it going, Finn?

I commented some days ago, but this time I come to ask for help.

I have a really crappy English class too. We're doing a unit on public speaking as well, 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.

I don't have a topic yet, but I wanted to see if you still have a written record of your speech on the Green-Tao Theorem. If we could talk somewhere else, I would really appreciate your help!

César Emanuel Castro - 3 years, 2 months ago

Email me finnhulse@gmail.com.

u have made a good pattern

Shubham Taipan - 6 years, 11 months ago

Thank you. :D

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$