A338646 Primes p such that 47^(p-1) == 1 + A*p (mod p^2) and |A/p| is a new record low.

2, 3, 5, 19, 37, 47, 38693, 44657, 148091, 178621, 692521, 4584379, 262148693, 347850691, 502176491, 1139746919, 1387837067, 5291181761, 92653098679, 202259581243

Offset: 1

Author

Felix Fröhlich, Apr 22 2021

Keywords

• nonn
• hard
• more

Comments

47 is the smallest b such that no base-b Wieferich prime, i.e., prime p such that b^(p-1) == 1 (mod p^2) is known (cf. Fischer).

The known terms of the sequence are base-47 near-Wieferich primes matching a definition of "nearness" introduced by Dorais and Klyve (cf. Dorais, Klyve, 2011).

If a base-47 Wieferich prime exists, then the sequence is finite and terminates at that prime.

Examples

			             p | abs(A/p) (frac) | abs(A/p) (dec)
----------------------------------------------------
             2 |  1/2            | 0.5
             3 |  1/3            | 0.333333333333333
             5 |  1/5            | 0.2
            19 |  2/19           | 0.105263157894736
            37 |  2/37           | 0.054054054054054
            47 |  1/2209         | 0.000452693526482
         38693 | 10/38693        | 0.000258444679916
         44657 |  4/44657        | 0.000089571623709
        148091 | 13/148091       | 0.000087783862625
        178621 |  1/178621       | 0.000005598445871
        692521 |  1/692521       | 0.000001443999532
       4584379 |  1/4584379      | 0.000000218132052
     262148693 | 39/262148693    | 0.000000148770530
     347850691 | 47/347850691    | 0.000000135115442
     502176491 | 51/502176491    | 0.000000101557920
    1139746919 | 75/1139746919   | 0.000000065804082
    1387837067 |  8/1387837067   | 0.000000005764365
    5291181761 |  3/5291181761   | 0.000000000566981
   92653098679 |  7/92653098679  | 0.000000000075550
  202259581243 |  5/202259581243 | 0.000000000024720
		

Links

• F. G. Dorais and D. Klyve, A Wieferich Prime Search up to 6.7 × 10^15, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.2.
• Richard Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Crossrefs

Cf. A339855.

Programs

PARI

my(a=0, ab=0, r=0); forprime(p=1, , a = (lift(Mod(47, p^2)^(p-1))-1)/p; ab=abs(a/p); if(r==0, r=ab; print1(p, ", "), if(ab < r, r=ab; print1(p, ", "))))

Extensions

a(19) from Felix Fröhlich, Jul 01 2021

a(20) from Felix Fröhlich, Jul 02 2021