

A238736


Balancing Wieferich primes: primes p that divide their Pell quotients, where the Pell quotient of p is A000129(p  (2/p))/p and (2/p) is a Jacobi symbol.


6




OFFSET

1,1


COMMENTS

Williams 1982 (p. 86), notes that p = 13, 31 and 1546463 are the only primes less than 10^8 for which the Pell quotient vanishes mod p. Elsenhans and Jahnel, "The Fibonacci sequence modulo p^2," p. 5, report in effect that there are no more such primes p < 10^9.
Williams 1991 (p. 440), and Sun 1995 pt. 3, Theorem 3.3, together prove a set of formulas connecting the Pell quotient with the Fermat quotient (base 2) (A007663) and harmonic numbers like H(floor(p/8)) (see example in the Formula section below). As is well known, the vanishing of the Fermat quotient (base 2) is a necessary condition for the failure of the first case of Fermat's Last Theorem (see discussion under A001220); and in light of a corresponding result of Dilcher and Skula concerning this type of harmonic number, the vanishing of the Pell quotient mod p is also a necessary condition for the failure of the first case of Fermat's Last Theorem.
There are no more terms up to 10^10.
Using the PARI script by Charles R Greathouse IV, I have extended the search from 10^10 to 10^12 without finding a further solution.  John Blythe Dobson, Mar 30 2015
Also primes p such that p^2 divides A001109((p  (2/p))/2).  Jianing Song, Oct 08 2018
From Felix Fröhlich, May 18 2019: (Start)
The term "balancing Wieferich prime" comes from Rout, 2016.
Primes p that satisfy the congruence B_{p(8/p)} == 0 (mod p^2), where B_i denotes the ith balancing number A001109(i) and (a/b) denotes the Jacobi symbol (cf. Rout, 2016, (1.6)).
Primes p such that the period of the balancing sequence (A001109) modulo p is equal to the period of the balancing sequence modulo p^2 (cf. Panda, Rout, 2014, p. 275).
Under the abc conjecture for the number field Q(sqrt(2)) there exist at least (log(x)/log(log(x)))*(log(log(log(x))))^m balancing nonWieferich primes <= x such that p == 1 (mod k) for any integers k > 2, m > 0 (cf. Dutta, Patel, Ray, 2019). This is an improvement of an earlier result stating there are at least log(x)/log(log(x)) balancing nonWieferich primes p == 1 (mod k) less than x (cf. Theorem 3.2 in Rout 2016). (End)


LINKS

Table of n, a(n) for n=1..3.
Karl Dilcher and Ladislav Skula, A new criterion for the first case of Fermat's Last Theorem, Mathematics of Computation, 64 (1995), 363392.
Utkal Keshari Dutta, Bijan Kumar Patel and Prasanta Kumar Ray, A brief remark on balancingWieferich primes, Mathematica, Vol. 60 (83), No. 1 (2018), 4853 [Subscription required].
Utkal Keshari Dutta, Bijan Kumar Patel and Prasanta Kumar Ray, Balancing nonWieferich primes in arithmetic progressions, Proceedings  Mathematical Sciences, Vol. 129, No. 2 (2019), Article 21, DOI:10.1007/s1204401804593.
AndreasStephan Elsenhans and Jörg Jahnel, The Fibonacci sequence modulo p^2  An investigation by computer for p < 10^14, arXiv 1006.0824 [math.NT], 2010.
Georges Gras, On the structure of the Galois group of the Abelian closure of a number field, arXiv 1212.3588 [math.NT], 2013.
Hao Pan, Lehmer's type congruences for lacunary harmonic sums, arXiv 0905.0941 [math.NT], 2009.
G. K. Panda and S. S. Rout, Periodicity of Balancing Numbers, Acta Mathematica Hungarica 143 (2014), 274286. Also on ResearchGate.
Sudhansu Sekhar Rout, Balancing nonWieferich primes in arithmetic progression and abc conjecture, Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 9 (2016), 112116.
ZhiHong Sun, Combinatorial sum ... and its applications in Number Theory, III (English version), originally published in Chinese in Journal of Nanjing University Mathematical Biquarterly, 12 (1995), 90102.
ZhiHong Sun, Five congruences for primes, Fibonacci Quarterly, 40 (2002), 345351.
H. C. Williams, The influence of computers in the development of number theory, Computers & Mathematics with Applications, 8 (1982), 7593.
H. C. Williams, Some formulas concerning the fundamental unit of a real quadratic field, Discrete Mathematics, 92 (1991), 431440.


FORMULA

The condition for p to be a member of this sequence is A000129(pe)/p == F(pe, 2)/p == 0 (mod p), where F(pe, 2) is the pe'th Fibonacci polynomial evaluated at the argument 2, and e = (2/p) is a Jacobi Symbol.
Let PellQuotient(p) = A000129(pe)/p, q_2 = (2^(p1)  1)/p = A007663(p) be the corresponding Fermat quotient of base 2, H(floor(p/8)) be a harmonic number, and e = (2/p) be a Jacobi Symbol. Then a result of Williams (1991), as refined by Sun (1995), shows that 2*PellQuotient(p) == 4*q_2  H(floor(p/8)) (mod p).


EXAMPLE

PellQuotient(13) = 6214 = 13*478; PellQuotient(31) = 3470274850 = 31*111944350.


MATHEMATICA

Select[Prime[Range[1000]], Mod[Fibonacci[#  JacobiSymbol[2, #], 2]/#, #] == 0 &]


PROG

(PARI) is(n)=isprime(n) && (Mod([2, 1; 1, 0], n^2)^(nkronecker(2, n)))[2, 1]==0 \\ Charles R Greathouse IV, Mar 04 2014


CROSSREFS

Cf. A000129, A001109.
Sequence in context: A214488 A247836 A159670 * A087511 A299449 A300087
Adjacent sequences: A238733 A238734 A238735 * A238737 A238738 A238739


KEYWORD

nonn,hard,more,bref


AUTHOR

John Blythe Dobson, Mar 04 2014


EXTENSIONS

Name amended by Felix Fröhlich, May 26 2019


STATUS

approved



