At first, we [search] such cases
in the range of p<263,
[another computation log].
Pairs (L, e) in the next table was found.
| L | e | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| L=22-1*1 | e=1*2 | e=4*3 | e=5*4 | e=7*4 | e=11*4 | e=17*4 | e=29*4 | e=35*4 | e=40*4 | |
| L=23-1 | e=1*5 2 | h-p |
e=3*5 2 | h-p |
e=5*5 | e=13*5 | e=19*5 2 | h-p |
e=25*5 | e=49*6 | e=51*7 | ||
| L=25-1 | e=7*5 | |||||||||
| L=27-1 | e=1*5 | e=11*5 | e=17*5 | e=23*5 | e=53*7 | |||||
| L=213-1 | e=11*5 | e=19*6*5 | ||||||||
| L=217-1 | (No pair exists in the range.) | |||||||||
| L=219-1 | e=3*5 2 | h-p |
e=17*6 | e=29*6 2 | h-p |
e=33*6 | e=37*7 | |||||
| L=231-1 | e=9*7 | e=27*7 | ||||||||
| L=261-1 | (No pair exists in the range.) | |||||||||
*1: L = 22 - 1 = 3 is a unique Mersenne prime in which 2 is a primitive root modulo L.
*2: Since Pe = ∅ for e = 1, ¬(2|H-p) holds theoretically by Proposition 4.
*3: Since p(=97) ∈ Pe, computation for the case had been included in the computer-assisted proof of Theorem.
*4: Although the method of Lemma 1(II) could be used for the cases, the method of Lemma 1(I) was employed because the degree of the polynomials is small.
[computation]
*5: Computations for the cases are included in the [search].
*6: Computations for the cases are divided into many sub jobs and each sub job was executed in a thread of CPU in about 10 personal computers.
*7: Computations in gray cells needs very large amount of computation. They are not computed in this work.
pcf2:
Note on the class number of the p th cyclotomic field, II
Shoichi FUJIMA and Humio ICHIMURA