

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.


7




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.
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



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



KEYWORD

nonn,hard,more,bref


AUTHOR



EXTENSIONS



STATUS

approved



