Mersenne numbers: trees and loops thru LLT. Tony Reix ([email protected]) 2005, 6th of November In his book ”Solved and Unsolved Problems in Number Theory”, Chapter ”Supplementary Comments, Theorems, and Exercises”, page 215 (Edition 1962), Daniel Shanks provided some information about Mersenne numbers (Mq = 2q − 1 , q prime) and also some ”open problems”. One of these problems deals with the LLT (Lucas-Lehmer Test) and with using it for representating the relationships between all numbers 0 ≤ x < Mq as a graph. D. Shanks provided an example with q=5 and said: ”Develop a general theory for all prime Mp , proving the main theorems, if you can”. I’m studying this and I’ve been able to find properties and to study properties of Mersenne numbers with q=5, 7, 13, 17, 19, 31 . But I still do not have any proof for the property described by Shanks and for the ones I’ve found. Can you help ? The LLT says that a Mersenne number is prime iff it divides Sq−2 , where S0 = 4, Si+1 = Si2 − 2. Let call llt the function: llt : x 7→ x2 − 2. In the following, I will take the example q = 5 ; but everything is (expected to be) true for q and Mq prime. For simplifying, I will also take numbers x between 0 and (M q − 1)/2, since (Mq − x)2 − 2 ≡ x2 − 2 (mod Mq ). The problem: M −1 If you apply the llt function to numbers between 0 and q2 and draw the relationships, you get: - one binary tree (q=5: starting at: 4, 9, 10, 11, and finishing at 8-0) ; - and several loops (q=5: 2-2 , 1-1 , 12-13-12 , 3-7-16-6-3). This is summarized herafter. I -

know the following properties: The tree has 2q−2 roots. q+1 The tree ends with: 2 2 . The length of the loops is a divisor of q-1.

My question is: How to prove that the llt function generates 1 tree and these loops for all Mersenne primes ? And which theory can help studying the properties of these tree and loops ? As an example, how one can prove that the number of loops of lengh 30 for q=31 is 17894588 ?

Also, multiplying all numbers of a loop modulo Mq results as +/- 1 (q=5: 12*13=1 , 3*6*15*7=-1). Why ? Also, multiplying all numbers in a column of the tree results as ±2(q+1)/2 (q=5: 14*5=8, 4*9*10*11=-8). Why ? And, if you multiply all numbers of the tree, except 0, but including 2( q + 1)/2 and 2, the result is also +/- 1 . Why ? And also, multiplying 2 numbers of the same sub-branch in the tree gives a number in the associated sub-branch (q=5: 4*9=5, 10*11=14). Why ?

M5 = 31 4 14 9 8

0

2

10 5 11

3

7

15

13

12

6

1

q 3 5 7 13 17 19 31

1 2 2 2 2 2 2 2

2

3

1 1 1

4

5

6

8

9

10

12

15

16

18

30

1 2 2 2 2

4 9

1 3

165 30

6

4 4

2032 56

7252 48

2182

Table 1: Number of loops for Mersenne primes.

17894588

Total 2 4 8 180 2068 7316 17896832