meditationatae

Just another WordPress.com site

probable primes, degree 1024 polynomial

I’ve become intrigued by statistics of integer polynomial evaluations (large integers) with a fixed polynomial of a high degree, such as 1024, obtained by iterating 5 times over a degree 5 polynomial.  In the case at hand, there are about 8 times as many probable primes in the polynomial values for the indeterminate X ranging from 2 to 40000, compared to what “naive” heuristics would suggest.  I have no hope of understanding this, so I’m “cataloguing” it.  The Math software used is PARI/gp.

===========================

? count=0;for(Y=2,40000, if(ispseudoprime(p1024(Y)), count=count+1;print(Y,” “,count )))

150 1
2674 2
2821 3
3576 4
4726 5
4892 6
5373 7
6083 8
6927 9
8819 10
8967 11
9479 12
11275 13
11785 14
11896 15
12649 16
13283 17
13918 18
16977 19
19087 20
19773 21
21993 22
23449 23
26893 24
28859 25
30046 26
32260 27
34791 28
36292 29
36509 30
36522 31
37953 32
38622 33
39304 34
time = 14h, 52min, 39,892 ms.

==============================

 

? sum(X=2, 40000, 1/log(p1024(X)))
time = 1,514 ms.
%8 = 4.1337644215965725897596893455519230085

 

==================================
? 34/%8
time = 0 ms.
%9 = 8.2249486260923094575862169468665882626

 

==============================
? \u
j =
  (X)->X^4-X^2+1

p1024 =
  (Z)->j(j(j(j(j(Z)))))

Advertisements

Written by meditationatae

June 8, 2013 at 1:02 pm

Posted in History

One Response

Subscribe to comments with RSS.

  1. The polynomial being iterated, j(X) = X^4 – X^2 +1, has degree four. Then p1024(X) := j(j(j(j(j(X))))), has degree 4^5 = 1024 in X.

    meditationatae

    June 8, 2013 at 1:05 pm


Comments are closed.

%d bloggers like this: