## 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.

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? 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)))))

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.

meditationataeJune 8, 2013 at 1:05 pm