Sunday, May 4, 2014
Spontaneus symetry breaking of prime numbers.
Let's imagine for a moment that we start adding "+1" for each natural number. Whenever we reach a prime number we switch the sum from "+1" to "1". If prime numbers are randomly distributed we expect the total sum to oscillate around 0.
Take for example the first 20 numbers (in bold prime numbers, where we start switching back):
 Total
3 3 
2 2 2 2 
1 1 1 1 
000
1 1 1 
2 2 2 
3 

0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 
* * * * * * * 
Let's see what we get in practice.
Next python code simulate the previous process/algorithm/"thing":
1 def isprime(n):
2 n*=1.0
3 if n%2==0 and n!=2 or n%3==0 and n!=3:
4 return False
5 for b in range(1,int((n**0.5+1)/6.0+1)):
6 if n%(6*b1)==0:
7 return False
8 if n %(6*b+1)==0:
9 return False
10 return True
11
12 START=1
13 total=0
14 sum = 1
15 for END in range(5000000,30000001,5000000):
16 totalNeg=0
17 totalPos=0
18
19 for i in range(START, END):
20 total = total + sum
21 if total > 0:
22 totalPos += 1
23 if total < 0:
24 totalNeg += 1
25 if isprime(i):
26 sum = 1 * sum
27 START = END+1
28
29 rat = totalPos*1./totalNeg
30 if rat > 1:
31 rat = 1./rat
32
33 print total, total*1./END, totalPos, totalNeg, rat
Every 5000000 numbers we print some statistics (line 33).
In the program output, the first column prints the "total" variable, which shows the total sum at a given time. We expect such column to be evenly distributed between negative and positive numbers. The second column shows the ratio between the total sum and the current natural number, that we expect to be close to zero. The third column will show how many times the total sum was possitive, with the fourth column showing how many times it was negative, finally the fith column shows the ratio between the times the total sum was possitive and the time it was negative. We expect column 3 to be quite similar to column 4, and so fith column close to 1.
Let's see what our program shows in practice:
SUM Current / SUM
1757 0.00292833333333 90292 509554 0.177198098729
3393 0.00308454545455 90292 1009554 0.0894375139913
6197 0.003873125 90292 1509554 0.059813693316
9685 0.0046119047619 90292 2009554 0.0449313628795
10401 0.00400038461538 90292 2509554 0.0359793015014
Not exactly the result I was waiting for. Intuition failed (as is always the case with prime numbers). The sum never oscillates and cross back to zero creating what scientifics will call an spontaneous break of symmetry.
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1 comment:
Very interesting. I love prime numbers and all misteries around them. Take a look to this post about Ulam's Spiral and my own implementation of a similar approach.
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