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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.
1
2, 3, 5, 19, 37, 47, 38693, 44657, 148091, 178621, 692521, 4584379, 262148693, 347850691, 502176491, 1139746919, 1387837067, 5291181761, 92653098679, 202259581243
OFFSET
1,1
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.
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.
EXAMPLE
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
PROG
(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, ", "))))
CROSSREFS
Cf. A339855.
Sequence in context: A191044 A344948 A079376 * A215312 A215316 A215308
KEYWORD
nonn,hard,more
AUTHOR
Felix Fröhlich, Apr 22 2021
EXTENSIONS
a(19) from Felix Fröhlich, Jul 01 2021
a(20) from Felix Fröhlich, Jul 02 2021
STATUS
approved