A338646 - OEIS

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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.

2, 3, 5, 19, 37, 47, 38693, 44657, 148091, 178621, 692521, 4584379, 262148693, 347850691, 502176491, 1139746919, 1387837067, 5291181761, 92653098679, 202259581243 (list; graph; refs; listen; history; text; internal format)

	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	`Table of n, a(n) for n=1..20.` `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)`
	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` `Adjacent sequences: A338643 A338644 A338645 * A338647 A338648 A338649`
	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`

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Last modified August 30 16:34 EDT 2024. Contains 375545 sequences. (Running on oeis4.)