This email showed up on the "Khan Academy" (aka home school academy!) email list. I have no idea how old this person is, but he seems to be interested in learning more.
From: Travis Rogers <cyanember@msn.com> To: <khan-academy-comments@googlegroups.com> Subject: infinite prime pattern Date: Wed, 1 Feb 2012 22:02:43 -0800
Well, here is two way of many to look at the pattern I found. first (not much faster, if at all, than conventional methods) square root of N =low Z(rounded up), N/3 = high Z(rounded up). N/ each Z from low to high =X and the remainders = Y N/each Y if none = whole number then N=prime. e.g say N=37 sq root 6.something so 7. 37/3= 12.3 so 13. 37/7 = x5y2 37/8= x4y5 37/9= x4y1 37/10= x3y7 37/11= x3y4 37/12= x3y1 37/13= x2y9
now with this method you can ignore any y's that are not prime's greater than 5 or are repeats from x(5,4,3,2) or z(7,8,9,10,11,12,13) so in this case 37 can be confirmed without further calculation.
the second way I looked at the pattern seems to work much faster and the pattern is found in what is left behind - 4 6,8,9 6,8,10,12,14,15,16 18,20,22,24,25 20,21,24,26,27,28,30,32,33,35,36 30,32,35,40,42,44,45,48,49 34,36,40,42,44,45,48,50,51,52,54,56,58,60,63,64 38,39,40,42,44,45,48,50,54,56,57,60,63,66,70,72,74,75,76,78,80,81 42,44,45,48,50,52,54,55,60,62,64,65,66,70,72,75,77,80,82,84,85,88,90,92,93,95,96,99,100
note 46 hasn't been found yet so accurate only to less than 44 but as you can see the first line is the numbers left out of the eliminated in the first 10 squares, and the second is the actual primes in the first 10 squares.
1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,63,67,68,69,71,73,79,83,86,87,89,91,94,97,98, 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
Okay now to try and explain how to see the pattern. One of my biggest problems is thinking in three dimensions, and that is exactly where the pattern lies. Each number squared will eliminate a finite number of non primes (with the exception of 1, which by the standards of this pattern would be prime) so lets look at the first few squares. 1+
1+2+ 3*4-
1+2+3+ 4*5*6- 7*8-9-
1+ 2+ 3+ 4- 5* 6- 7* 8- 9* 10- 11* 12- 13* 14- 15- 16- +=confirmed prime - =eliminated in this square *=prime candidate according to this square
Z always equals the top right corner. x1 is the row below the top row ( or what I usually call the prime row) y's are the numbers in the prime row. If a number makes it to the prime row without being eliminate then it is a prime. to tell if a number is eliminated you must divide it by x and y if either comes out whole then it is not prime and is eliminated(for all numbers below the prime row). it is important to figure out each square separately and not automatically eliminate the numbers you know aren't primes. the pattern really comes to life when you build a 3d model even to just 10^2 (I bet Legos would work awesome) but once you build the model you then look at each xy xz and yz planes the patterns are impossible to miss. there are some obvious things like (if n = top right corner) x1 to n-1 diagonal on the xy plane never has any prime candidates while x=any y1 is always a prime candidate and x=any y=n never is a candidate. the way I found the above elimination patte rn was taking the lowest even number that has not been eliminated yet and starting at y1 of whatever x that number fell on. then just listed each number eliminated in the square from that starting point up. it creates some repeats but never eliminates a prime and as far as I can tell doesn't let anything slip through the cracks. I didn't test this real far but looks real promising.
but look at cross section of say x2 and z it will look like
o x x o x o x o x o x x o x x x o x o x o x o x x o x o x o x o x o x x x o x o x x o x o x x x o x o x
o is candidate x is non-candidate the first two rows are obvious all o's and all x's. third row x 2 o's fourth row all x's fifth x 4 o's 6th all x's 7th x 6 o's etc..
I wish I had more time to work on stuff like this but last week I put in 70+ hours at my regular job and I just can't dedicate much time to it. matter of fact this e-mail is taking up about 90% of my awake time today not spent working. I just wanted to share this with people who appreciate knowledge, and if anyone in the world can poke holes in it it will be you guys :) I hope somebody there will find the time to play with this a little.
Travis Rogers cyanember@msn.com