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

%I A338646 #19 Jul 15 2021 02:50:01
%S A338646 2,3,5,19,37,47,38693,44657,148091,178621,692521,4584379,262148693,
%T A338646 347850691,502176491,1139746919,1387837067,5291181761,92653098679,
%U A338646 202259581243
%N A338646 Primes p such that 47^(p-1) == 1 + A*p (mod p^2) and |A/p| is a new record low.
%C A338646 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).
%C A338646 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).
%C A338646 If a base-47 Wieferich prime exists, then the sequence is finite and terminates at that prime.
%H A338646 F. G. Dorais and D. Klyve, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html">A Wieferich Prime Search up to 6.7 × 10^15</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.2.
%H A338646 Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt">Thema: Fermatquotient B^(P-1) == 1 (mod P^2)</a>
%e A338646              p | abs(A/p) (frac) | abs(A/p) (dec)
%e A338646 ----------------------------------------------------
%e A338646              2 |  1/2            | 0.5
%e A338646              3 |  1/3            | 0.333333333333333
%e A338646              5 |  1/5            | 0.2
%e A338646             19 |  2/19           | 0.105263157894736
%e A338646             37 |  2/37           | 0.054054054054054
%e A338646             47 |  1/2209         | 0.000452693526482
%e A338646          38693 | 10/38693        | 0.000258444679916
%e A338646          44657 |  4/44657        | 0.000089571623709
%e A338646         148091 | 13/148091       | 0.000087783862625
%e A338646         178621 |  1/178621       | 0.000005598445871
%e A338646         692521 |  1/692521       | 0.000001443999532
%e A338646        4584379 |  1/4584379      | 0.000000218132052
%e A338646      262148693 | 39/262148693    | 0.000000148770530
%e A338646      347850691 | 47/347850691    | 0.000000135115442
%e A338646      502176491 | 51/502176491    | 0.000000101557920
%e A338646     1139746919 | 75/1139746919   | 0.000000065804082
%e A338646     1387837067 |  8/1387837067   | 0.000000005764365
%e A338646     5291181761 |  3/5291181761   | 0.000000000566981
%e A338646    92653098679 |  7/92653098679  | 0.000000000075550
%e A338646   202259581243 |  5/202259581243 | 0.000000000024720
%o A338646 (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, ", "))))
%Y A338646 Cf. A339855.
%K A338646 nonn,hard,more
%O A338646 1,1
%A A338646 _Felix Fröhlich_, Apr 22 2021
%E A338646 a(19) from _Felix Fröhlich_, Jul 01 2021
%E A338646 a(20) from _Felix Fröhlich_, Jul 02 2021