A339855 -id:A339855

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A241014

Let p be the n-th prime, then a(n) = A/p where A is the smallest number (in absolute value) such that F_{p-(p/5)} == A (mod p^2) with F_n = A000045(n) and (p/5) the Legendre symbol.

+10
11

1, 1, 1, 3, 5, 3, -1, 3, -8, -3, -6, 13, -2, -4, 16, -25, 10, -13, 7, -16, -15, -30, 21, 5, 37, -4, 22, 24, 26, -53, 13, 64, 58, -22, -29, 60, 44, -3, 44, -43, -5, -50, 94, 31, -56, 5, -99, 3, -73, 18, 29, 5, -59, -1, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

a(n) is the smallest A such that p is a near-Wall-Sun-Sun prime. A gives the value of F_p-(p/5) modulo p^2 and a value of 0 would indicate a Wall-Sun-Sun prime. A244801 is similar but always gives the positive A, while this sequence gives A with the smallest absolute value.

a(1), with p=2, is technically ambiguous between 1 and -1, so a(1)=1 is by convention. Clearly this cannot happen for n>1 (where p^2 is odd). - Jeppe Stig Nielsen, Sep 09 2021

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp., 76 (2007), 2087-2094.

MATHEMATICA

Array[(#3 - #2 Boole[#3 > #2/2])/#1 & @@ {#, #^2, Mod[Fibonacci[# - KroneckerSymbol[#, 5]], #^2]} &@ Prime[#] &, 55] (* Michael De Vlieger, Sep 08 2021 *)

PROG

(PARI) forprime(p=2, 1e2, a=fibonacci(p-kronecker(p, 5))%p^2; if(a>p^2/2, a-=p^2); a=a/p; print1(a, ", "))

(PARI) a(n)=my(p=prime(n)); centerlift(((Mod([1, 1; 1, 0], p^2))^(p-kronecker(p, 5))))[1, 2]/p \\ Charles R Greathouse IV, Aug 21 2014

CROSSREFS

Cf. A113650, A195988, A244801, A339855, A347565.

KEYWORD

sign

AUTHOR

Felix Fröhlich, Aug 13 2014

STATUS

approved

A347565

Primes p such that A241014(A000720(p)) is +1 or -1.

+10
2

2, 3, 5, 17, 251, 733, 1063, 123863, 1677209, 6336823451747417, 104868559750360787, 7665762181374748069

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Very near misses for Wall-Sun-Sun primes.

LINKS

Table of n, a(n) for n=1..12.

CROSSREFS

Cf. A241014, A113650, A296240, A339855.

KEYWORD

nonn,hard,more

AUTHOR

Jeppe Stig Nielsen, Sep 06 2021

STATUS

approved

A338646

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

+10
1

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.

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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