A287147 Primes p that set a new record for the size of the smallest b > 1 such that b^(p-1) == 1 (mod p^2).

2, 3, 7, 13, 17, 31, 53, 179, 271, 311, 503, 569, 587, 1231, 1307, 1543, 1931, 2647, 2711, 3089, 3917, 4919, 5879, 6491, 8933, 9137, 11437, 13411, 14431, 16657, 21599, 26053, 29129, 57367, 58481, 62071, 62971, 68351, 70639, 109721, 156967, 193811, 216211

Offset: 1

Author

Felix Fröhlich, May 20 2017

Keywords

• nonn

Comments

Primes p such that A039678(i) reaches record values, where i is the index of p in A000040.

Records of (A185103 restricted to primes). - Joerg Arndt, May 29 2017

Crossrefs

Cf. A000040, A039678, A185103, A248865, A278611.

Programs

Mathematica

Function[s, Prime@ Position[s, #][[1, 1]] & /@ Union@ FoldList[Max, s]]@ Table[Function[p, b = 2; While[PowerMod[b, p - 1, p^2] != 1, b++]; b]@ Prime@ n, {n, 10^3}] (* Michael De Vlieger, May 21 2017 *)

PARI

minb(n) = my(b=2); while(Mod(b, n^2)^(n-1)!=1, b++); b
my(r=0); forprime(p=1, , if(minb(p) > r, print1(p, ", "); r=minb(p)))

Python

from itertools import islice
from sympy import nextprime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A287147_gen(): # generator of terms
    c, p = 5, 3
    yield 2
    while True:
        d = nthroot_mod(1,p-1,p**2,True)[1]
        if d > c:
            c = d
            yield p
        p = nextprime(p)
A287147_list = list(islice(A287147_gen(),15)) # Chai Wah Wu, May 18 2022