March 30th, 2018, 03:35 PM  #1 
Newbie Joined: Mar 2018 From: United States Posts: 28 Thanks: 2  Primes and Factoring
One way to create an infinite sequence of prime numbers: Start with an odd prime. Find the first number whose 2cycle has a length equal to that prime. That will be your next prime. Repeat the above. Example: 11, 23, 47, . . . . 
March 30th, 2018, 04:35 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 598 Thanks: 366 Math Focus: Dynamical systems, analytic function theory, numerics 
Another way to generate a list of infinite primes is to start with any prime and find the first number larger than it which is also prime. That will be your new prime. Repeat the above. Example: 7,11,13,... Also, what does the 2cycle of an integer mean? 
March 30th, 2018, 05:38 PM  #3  
Newbie Joined: Mar 2018 From: United States Posts: 28 Thanks: 2  Quote:
In the method I presented there is no necessity to know at the start that there is an infinite number of primes. That will come out of the process. Nor will you have to verify that any given number is a prime. That also comes out of the process. What you do have to do is find the length of 2cycles. Which brings me to: Quote:
For example, the 2cycle for 7 is 2, 4, 1. Thus its length is 3. Finally, I omitted details, so what I said was not intended to be a proof, just a statement.  
April 1st, 2018, 03:19 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
What you said originally was "Find the first number whose 2cycle has a length equal to that prime. That will be your next prime.". The first number, after 3, that has 2cycle equal to 3 is 7. But the next prime is 5, not 7.

April 1st, 2018, 05:23 AM  #5  
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  Quote:
I suspect that Penrose is not trying to construct a list of all primes. He is just trying to generate an infinite list of primes. Look at his first example. 11, 23, 47. The list above ignores 13, 17, 19, 29, 31, 37, 41, and 43. Penrose does not bother to prove his assertion. Nor does he explain what the mathematical utility of his list is. Indeed, he does not even explain how to apply his method. How is 23 related to the 2cycle of 11?  
April 1st, 2018, 06:59 AM  #6  
Newbie Joined: Mar 2018 From: United States Posts: 28 Thanks: 2  Quote:
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This code is VBA and runs on a spreadsheet, but anyone can easily translate to their favorite language. Option Explicit Dim i As Long Dim j As Long Dim k As Long Dim n As Long Dim CyclicValue As Long Dim CyclicOrder As Long Dim row As Long Dim col As Long Sub CycleOne() row = 5 col = 1 n = 23 Cells(1, col) = n CyclicValue = 2 Cells(4, col) = CyclicValue CyclicOrder = 1 Do CyclicValue = (2 * CyclicValue) Mod n Cells(row, col) = CyclicValue CyclicOrder = CyclicOrder + 1 If CyclicValue = 1 Then Cells(2, col) = CyclicOrder Exit Do End If row = row + 1 Loop End Sub Output: 23 11 2 4 8 16 9 18 13 3 6 12 1  
April 1st, 2018, 02:05 PM  #7  
Senior Member Joined: Aug 2012 Posts: 2,259 Thanks: 682  Quote:
These numbers get big quickly. After 47 I found 2351 and then 4703, is that what you get? After that I could not find the next one in a search up to $100,000$. How far do I have to go, and how do I know I can find some $n$ such that $ord_2(n) = 4703$, let alone that it must be prime? It would be interesting to have a proof or a counterexample about your conjecture. First, is it true that we can always find the next number in this sequence? And if so, must it be prime? (Edit) My slow Python program is chugging along and hasn't found the next number after 4703 under 162,000. Now I have to either spend the rest of the day optimizing my program, which still won't help because I have no way of knowing if there even is a next number after 4703. Or perhaps someone has some clever theoretical ideas here ... or perhaps OP can say if he got to 4703 and what happened next. Last edited by Maschke; April 1st, 2018 at 03:04 PM.  
April 1st, 2018, 03:23 PM  #8 
Banned Camp Joined: Apr 2016 From: Australia Posts: 244 Thanks: 29 Math Focus: horses,cash me outside how bow dah, trash doves 
I too was once a huge fan of making quirky observations and limiting the detail of my explanation as a means of avoiding saying something silly which has a negative impact on the perception I want to give the reader, that I understand all details of my initial statement and the reader must bow to me in all my glory. But I changed my mind and decided I prefer explicit rigorous explanation and proof of every result I present, if I cannot do so or cannot be bothered in that instance, it becomes the question of a competition for which participants win nothing, but were originally promised a prize that only the lower class or homeless may appreciate. Anyway repost a rigorous explanation in 24 hours, or I am going to every park in my locality and throwing sand in children's eyes. 
April 1st, 2018, 07:03 PM  #9  
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  Quote:
In any case, you need to prove that the indicated number is necessarily a prime. Please do so.  
April 2nd, 2018, 02:37 AM  #10  
Newbie Joined: Mar 2018 From: United States Posts: 28 Thanks: 2  Quote:
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