ABC implies there are infinitely many non-Fibonacci-Wieferich primes

Journal Pre-proof ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes

Wayne Peng

PII:

S0022-314X(19)30410-X

DOI:

https://doi.org/10.1016/j.jnt.2019.11.010

Reference:

YJNTH 6447

To appear in:

Journal of Number Theory

Received date:

24 March 2017

Revised date:

22 October 2019

Accepted date:

23 November 2019

Please cite this article as: W. Peng, ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes, J. Number Theory (2019), doi: https://doi.org/10.1016/j.jnt.2019.11.010.

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ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes Wayne Peng UR Mathematics, 915 Hylan Building, University of Rochester, RC Box 270138 Rochester, NY 14627

Abstract We define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. We show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterl´e-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture.

1. Introduction In this paper, we study the periodic properties of a rational recurrence sequence modulo a positive integer. We use the Fibonacci sequence {Fn }n≥0 to illustrate the question we are going to study here. The nth Fibonacci number Fn is defined by the recurrence relation Fn = Fn−1 + Fn−2

for all n ≥ 2

with initial values F0 = 0 and F1 = 1. Consider the sequence {Fn mod m}n≥0 for some integer m. Since Z/mZ is a finite set and Fn is defined by the recurrence relation, the sequence {Fn mod m} will be periodic or preperiodic, i.e. there is an n0 such that Fn+π ≡ Fn mod m for some positive integer π and for all n ≥ n0 . If the integer n0 is 0, then we say the sequence is periodic. For example, say p = 7, the sequence {Fn mod 7}n≥0 is 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, . . . which is a periodic sequence of period 16. The Chinese remainder theorem suggests that it is sufficient to study the case where m is a power of prime p. D.D. Wall [Wal60] studied this sequence and proved that {Fn mod pe }n≥0 is periodic for all primes p. He observed that the period of {Fn mod p}n≥0 and the period of {Fn mod p2 }n≥0 are different for all primes p up to 1999, and formulated the following conjecture. Conjecture 1 (Wall’s conjecture). Let π(m) be the period of the sequence {Fn mod m}n≥0 . Then, we have π(p) = π(p2 )

for all primes p.

(1)

However, our heuristic suggests that Wall’s conjecture may be wrong (see Conjecture 5). Next, we want to show conditionally that Equation (1) is true for infinitely many primes in much more general settings. We need some definitions to formally state our results. A degree m linear recurrence sequence is a sequence X = {xn }n≥0 defined by a recurrence relation xn+m = c0 xn + c1 xn+1 + c2 xn+2 + · · · + cn−1 xn+m−1

∀n ≥ 0

¯ where ci ∈ Q. ¯ In our previous paper [GP15], we have already dealt with initial values x0 , x1 , . . . , xm−1 ∈ Q, with the case m = 2. We would like to call a given 2m-tuple (a1 , . . . , am , b1 , . . . , bm ) a recurrence tuple if ¯ \ {0} a1 , . . . , am , b1 , . . . , bm ∈ Q Preprint submitted to Elsevier

December 16, 2019

with ai = aj for all i = j. A sequence X = {xn }n≥0 is a linear recurrence sequence of degree m generated by the recurrence tuple if the closed form of the terms of the sequence is xn = b1 an1 + b2 an2 + · · · + bm anm

∀n ≥ 0.

See [Bru10] for more details. We call K = Q(a1 , . . . , am , b1 , . . . , bm ) the field of the tuple (a1 , . . . , am , b1 , . . . , bm ), and let OK be the ring of integers of K. We will call an ideal I of OK non-degenerate relative to the recurrence tuple (a1 , . . . , am , b1 , . . . , bm ) if vp (ai ) = 0 and vp (bi ) = 0 for all i for all prime factor p of I. We define πX (I) := the period of the sequence {xn mod I}. A prime ideal p ⊂ OK is called an X -base Fibonacci-Wieferich prime if πS (p) = πS (p2 ). We will give an equivalent definition later in the introduction for πX (I) since πX (I) is closely related to the order of elements ai in the modulo ring OK /I for all i. Our first main theorem is that there are infinitely many primes which are not X -base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterl´e-Szpiro [Szp90, Mas02, Oes88] for number fields. We call a number field K an abc-field if the abc-conjecture is true for K. Theorem 2. Let K be the field of the tuple (a1 , . . . , am , b1 , . . . , bm ), and let X = {b1 an1 + · · · + bm anm }n≥0 . Then, there are infinitely many non-X -base Fibonacci-Wieferich primes, assuming K is an abc-field. When we look at the problem closely, we find out that πX (I) is actually determined by the order of ai for all i in the modulo ring OK /I, bi are redundant information in our questions. More precisely, we have πX (I) = lcm{ordI (ai ) | i = 1, . . . , m} if I is non-degenerate relative to (a1 , . . . , am , b1 , . . . , bm ). Details will be given in Proposition 9. Therefore, when we consider the equation πX (p) = πX (p2 ), we actually deal with the equation ordp (ai ) = ordp2 (ai ) for all i, and this generalizes Wieferich primes. For α ∈ Q we say p is an α-Wieferich prime if p satisfies the equation αp−1 ≡ 1 mod p2 . Definition 3. Let X = {a1 , a2 , . . . , am }, and let K be a number field containing all ai . An X-base FibonacciWieferich prime p is a prime ideal of OK satisfying NK/Q (p)−1

ai

≡1

mod p2

for all i

where NK/Q is the usual norm. From the discussion above Definition 3, we find that πX (I) = πX (I) if X is a recurrence sequence generated by the recurrence tuple X. Hence, when we think πX (I), we can deal with it like the period of a sequence X modulo I instead of the order of elements {a1 , . . . , am }, but Definition 3 is much more easy to handle. For obtaining a logarithmic lower bound on the number of X-base Fibonacci-Wieferich prime, we define Wα (B) := {prime ideal p ⊂ OK | N (P) ≤ log B, and p is not an α-Wieferich prime} and similarly WX (B) := {prime ideal p ⊂ OK | N (P) ≤ log B, and p is not an X-base Fibonacci-Wieferich prime}. We note that WX (B) = ∪m i=1 Wai (B). Therefore, a lower bound for Wai (B) is naturally a lower bound for WX (B). Then, we follow Silverman’s [Sil88] closely to give a logarithmic lower bound. 2

Theorem 4. For any non zero algebraic number α ∈ X \ ∂D1 (0) where ∂D1 (0) is the boundary of the complex unit circle with center at origin, there exists a constant Cα > 0 such that we have |WX (B)| ≥ |Wα (B)| ≥ Cα log B for all sufficiently large B assuming K is an abc-field. Note that if we let α be the golden ratio and give X = {α, −α−1 }, this situation corresponds to the Fibonacci sequence. Wall conjectured that every prime is a non-X-base Fibonacci-Wieferich prime, or a non-Fibonacci-Wieferich prime for short, and he gave his computational results to support his conjecture. However, heuristic results([GP15], [CDP97], [MR07] or [EJ10]) suggest that we should instead have the following conjecture, which contradicts Wall’s conjecture. ¯ Conjecture 5. Let X = {a1 , . . . , am } ⊂ Q. 1. If the rank of the multiplicative group a1 , . . . , am  is 1, then there are infinitely many X-base FibonacciWieferich primes. 2. If the rank of the multiplicative group a1 , . . . , am  is greater than 1, then there are finitely many X-base Fibonacci-Wieferich primes. Note that the rank of the multiplicative group is not depended on the number fields in where ai lies. Moreover, according to the fundamental theorem of abelian groups, an abelian group can be expressed as the product of free part and torsion part, we define the rank of an abelian group to be the rank of free part. Since the rank of the multiplicative group α, −α is 1, we would have infinitely many Fibonacci-Wieferich primes by assuming Conjecture 5 to be true, which contradicts Wall’s conjecture. The geometric point of view on this problem can be explained in terms of dynamical systems. Let V (K) be a variety over a number field K, and let φ : V (K) → V (K) be a morphism of V into itself. For an integer n ∈ N, denote by φn the n-th compositional iterate φ ◦ · · · ◦ φ; for any point P ∈ V (K), we let Oφ (P ) := {φn (P )}n be a (forward) φ-orbit of P . We are curious about the relation between the sequence Oφ (P ) mod p and Oφ (P ) mod p2 . We will discuss this in the last section. We briefly list what we will do in each section. In Section 2, we will give an introduction of the abcconjecture for number fields and the history of Wall’s conjecture. Following, Section 3, we prove Theorem 2. Section 4 uses the method of [Sil88] to get a logarithmic lower bound, Theorem 4. Finally, Section 5 talks about the heuristic of why Conjecture 5 is true from the algebraic aspect and the geometric aspect. 2. Some history about Wall’s conjecture and the abc-conjecture for number fields Wieferich primes were first introduced in the study of Fermat’s Last Theorem by Wieferich [Wie09] and Mirimanoff [Mir11]. Later, Zhi-Hong Sun and Zhi-Wei Sun [SS92] that if Wall’s conjecture is true, then Fermat’s Last Theorem holds. Fibonacci-Wieferich primes are sometimes called Wall-Sun-Sun primes. The search for Wall-Sun-Sun primes is still ongoing, but nothing has been found up to 28 × 1015 , as reported by PrimeGrid [Pri] in 2014. Although PrimeGrid has extended the search to a very large number, it is still not known unconditionally whether or not there are infinitely many primes such that π(p) = π(p2 ) (where π(·) is as in (1)). Silverman introduced a method to show that there are infinitely many non-α-Wieferich primes, assuming the traditional abc-conjecture, and our idea is to mimic Silverman’s proof for our purpose. The abc-conjecture is very useful for dealing with these kinds of questions. P. Ribenboim and G. Walsh [RW99] used the abcconjecture to show that there are only finitely many powerful terms in the Lucas sequence and the Fibonacci sequence, and lately Minoru Yabuta [Yab07] further generalized this result. Instead of just studying the second digits of the p-adic expansion, [DW16] proves a conjecture of Erdos relative to Wieferich primes. I would like to briefly introduce Silverman’s trick. We start with the traditional abc-conjecture.

3

Conjecture 6 (The abc-conjecture on rational numbers). If we have three integers a, b, c ∈ Z \ {0} which satisfy a + b + c = 0, then for every ε > 0 there is a constant Cε > 0 such that the following inequality holds max{|a|, |b|, |c|} ≤ Cε rad(a, b, c)1+ε where rad(a, b, c) :=

 p|abc

p.

This version is often called the abc-conjecture of Masser-Oesterl´e-Szpiro. On the other hand, the abcconjecture for function fields is a theorem of Stothers [Sto81], Mason [Mas84], and Silverman [Sil84]. In fact, the discovery of the abc-theorem for function fields is earlier than the abc-conjecture for rational numbers and then for number fields (see [GT02] for more historical introduction). The abc-conjecture is often useful in the study of arithmetic randomness questions. It is especially useful for controlling the squarefree part of a given sequence. Definition 7. Let N be an integer. We define κ(N ) as the following  κ(N ) = p. pN

Note that κ(N ) is generally not equal to rad(N ). We use a simple example to illustrate the idea we are going to carry out in the proof. For a sequence xn = an + bn with a, b ∈ N \ {1, 0}, we have, for ε > 0, max{an , bn , xn } ≤ C rad(an bn xn )1+ε . The left hand side of the inequality increases as n increases, so the right hand side should also increase. Note that an and bn do not provide new prime factors, so the source of new primes comes from xn . Further assuming that κ(xn ) is bounded for n > 0, we should have xn ≥ rad(xn )2 for sufficiently large n since most prime factors p of xn divide xn twice. However, xn ≥ rad(xn )2 contradicts the inequality given by the abc-conjecture when we choose ε < 1. In 1998, Paul Vojta [Voj98] formulated a new conjecture, and, as a consequence of this conjecture, the abc-conjecture and the n-tuple abc-conjecture for an arbitrary number field K follow as special cases. Readers can also refer to [GNT13] which is an application of Vojta’s abc-conjecture of the universal form. For the abc conjecture for number fields, we set the following: K is a number field, and let OK be the ring of integers of K over Z. p is a prime ideal of OK , and let I be an ideal of OK . NK/Q (I) := |OK /I|. GK := {σ : K → C | σ is an embedding}. V (f ) is the zero set defined by the function f .  For an element x ∈ K, NK/Q (x) := σ∈GK σ(x). We define N (·) := log |NK/Q (·)|/[K : Q] where [K : Q] is the degree of K. Note N (·) is independent of the choice of K. 8. MK is the set of all finite and infinite places on K. For a finite place v ∈ MK and for a prime ideal p which is correspondent to the place v, we denote v by vp sometimes for clarifying the correspondent relation, and we define for x ∈ K. xvp := NK/Q (p)−vp (x)

1. 2. 3. 4. 5. 6. 7.

Let | · | be the usual absolute value. If v ∈ MK is an infinite place, then we define xv := |σ(x)|e for every non-conjugate embedding σ ∈ GK with e = 1 if σ is real, and e = 2 if σ is complex.

4

9. We define the absolute logarithmic height of any 3-tuple (x1 , x2 , x3 ) in K 3 \ {0} to be h(x1 , x2 , x3 ) :=

 1 log max{x1 v , x2 v , x3 v }. [K : Q] v∈MK

We also define a set of ideals with respect to the point (x1 , x2 , x3 ) to be IK (x1 , x2 , x3 ) := {p ⊂ OK | if vp (xi ) = vp (xj ) for some 1 ≤ i, j ≤ 3}, and let rad(x1 , x2 , x3 ) :=

1 [K : Q]





log NK/Q (p) =

p∈IK (x1 ,x2 ,x3 )

log N (p).

p∈IK (x1 ,x2 ,x3 )

With all of these notations, the abc-conjecture says the following. Conjecture 8. For any ε > 0, there is a constant Cε > 0, depending only on ε, such that, for any tuple ¯ | X + Y + Z = 0} and V (f ) is the zero (a, b, c) ∈ H \ V (X) ∪ V (Y ) ∪ V (Z) where H := {[X : Y : Z] ∈ P2 (Q) set of a polynomial f , we have h(a, b, c) < (1 + ε) rad(a, b, c) + Cε . A height function can be thought of as a function to measure the arithmetic complexity of a point on an affine space, a projective space or an algebraic variety. With the language of height, the abc-conjecture says that the complexity of a given point on the hyperplane X + Y + Z = 0 is bounded by those primes relative to the point, up to a constant and a scalar 1 + ε. Vojta’s conjecture could be thought of as an abc-conjecture in the context of algebraic geometry. A height function for a projective geometry depends on a divisor, and if we choose our divisor to be [0], [1] and [∞], Masser-Oesterl´e-Szpiro conjecture follows as a special case. Vojta’s conjecture also implies the n-tuple abc-conjecture for any number field. There is another version of n-tuple abc-conjecture for integers, which was proposed by J. Browkin and J. Brzezi´ nski [BB94]. 3. There are infinitely many non-X-base Fibonacci-Wieferich primes We set the following notations for this section and for the following sections: 1. Let (a1 , . . . , am , b1 , . . . , bm ) be a recurrence tuple. 2. Let X = {xn }n≥0 be a sequence generated by the recurrence tuple (a1 , . . . , am , b1 , . . . , bm ), i.e. xn = b1 an1 + b2 an2 + · · · bm anm

for all n ≥ 0.

3. The reduction map ¯· : OK → OK /I is troublesome since ai are not necessarily algebraic integers. Fortunately, if ai are not degenerate relative to I, then we can define the reduction map on the projective space ¯· : P1 (K) → P1 (OK /I). More precisely, we have [α : 1] = [α : α ] where α and α are in OK and α OK , α OK and I ¯ by the are coprime, so α mod I and α mod I are well-defined in OK /I. In the end, we define α embedding map and the projection map proj

¯·

K → P1 (K) → P1 (OK /I) → OK /I. ¯ ) to denote the order of α ¯ ∈ (OK /I e )× for a proper ideal I. We use ordI e (α e 4. Let πX (I ) be the length of the period of the sequence {xn mod I e }n≥0 . Proposition 9 shows that the πX is well-defined.

5

The followings are some well-known facts about the notations we define above. First, the sequence X is ¯ for all n ≥ 0 a recurrence sequence of degree m, i.e. xn+m = c0 xn + c1 xn+1 + · · · + cm−1 xn+m−1 with ci ∈ Q with proper initial values. Second, an ideal I of the integral ring OK where K is the field of the recurrence tuple (a1 , . . . , am , b1 , . . . , bm ) can be expressed uniquely as a product of prime ideals pe11 · · · perr . Note that for all but finitely many prime ideals are non-degenerate relative to the recurrence tuple, ¯m , ¯b1 . . . , ¯bm are units in the and for those ideals I non-degenerate relative to the recurrence tuple, a ¯1 , . . . , a e ring OK /I OK for any integer e ≥ 1. Thus, we can consider the order of ai in the multiplicative group (OK /I e OK )× . With all of these arguments, we are able to give the following proposition. Proposition 9. For any ideal I = pe11 · · · perr of K non-degenerate relative to the recurrence tuple (a1 , . . . , am , b1 , . . . , bm ), the sequence {xn mod I e }n≥0 is periodic. Then, we have πX (I) = lcm{πX (pe11 ), πX (p2 e2 ), . . . , πX (perr )} = lcm{ordpei (aj ) | for all i, j}. i

= lcm{πX (pe11 ), πX (p2 e2 ), . . . , πX (perr )}. Thus, we xn = b1 an1 + · · · bm anm is of degree m, the sequence

Proof. The Chinese Remainder Theorem implies πX (I) assume I is a power of a prime ideal. Since the sequence {xn mod I} is periodic if and only if the following system of equations holds. ⎧ ≡ x0 mod I xπ ⎪ ⎪ ⎪ ⎪ ⎨xπ+1 ≡ x1 mod I . .. ⎪ ⎪ . ⎪ ⎪ ⎩ xπ+m−1 ≡ xm−1 mod I It implies that

⎡ ⎢ ⎢ ⎢ ⎣

b1 b1 a 1 .. .

b2 b2 a 2 .. .

··· ··· .. .

b1 am−1 1

b2 am−1 2

···

⎤ ⎡ ⎤ aπ1 x0 ⎥ ⎢ aπ2 ⎥ ⎢ x1 ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ .. ⎥ ≡ ⎢ .. ⎥ ⎦⎣ . ⎦ ⎣ . ⎦ aπm xm−1 bm am−1 m bm bm am .. .

⎤⎡

mod I

where π := πX (I). Since I is non-degenerate relative to (a1 , . . . , am , b1 , . . . , bm ), ai are all distinct and bi are nonzero, the matrix is invertible. Therefore, we have ⎡ π ⎤ ⎡1⎤ a1 1⎥ ⎢ aπ2 ⎥ ⎢ ⎥ ⎢ ⎥≡⎢ mod I. ⎢ ⎣ . . . ⎦ ⎣ .. ⎥ .⎦ aπm 1 Hence, it is clear that π ≥ lcmi {ordI (ai )}. The inequality for the other direction is trivial. A quick conclusion of this theorem is that the period should divide the order of the multiplicative group (OK /p)× where p is a prime ideal, and this conclusion generalizes [Wal60]. Corollary 10. Let K be the field of a recurrence tuple (a1 , . . . , am , b1 , . . . , bm ), and let p be a prime ideal of K which is non-degenerate relative to the tuple. Then, we have πX (p) divides NK/Q (p) − 1. Let p be a prime integer which is non-degenerate relative to the tuple, and let X be a rational sequence, i.e. every term xn is rational. Then, πX (p) divides pe−1 (pf − 1) where f is the inertia degree of p over K and e is the ramified degree of p over K.

6

Proof. Obviously, the length of period is independent of the choice of fields, i.e. πX (pOK ) = πX (pOL ) for K ⊆ L. By the proposition 9, it trivially implies the first part of the corollary. For the second part, since K is Galois over Q, the prime decomposition of pOK = pe1 · · · per where the inertia degrees of pi are same for all i. It implies (OK /pOK )× = (OK /pe1 )× × · · · × (OK /per )× , and it is clear that the multiplicative order of elements in (OK /pOK )× divides the order of a component (OK /pe1 )× which is pe−1 (pf − 1) where f is the inertia degree. Since all but finitely many prime integers p are non-degenerate relative to the recurrence tuple, Corollary 10 holds for all but finitely many primes p. Proposition 9 also indicates that X -base Fibonacci-Wieferich primes are generalized Wieferich primes. This proposition generalizes in the sense that, instead of considering the order of a single element, we consider the order of a set of elements. Therefore, a prime which is not an α-Wieferich prime is also not an X -base Fibonacci-Wieferich prime where α is in X = {a1 , a2 , . . . , am }. The following lemma proves this idea. Lemma 11. Let p be a prime which is non-degenerate relative to the recurrence tuple. If we have ani − 1 ∈ p

and

ani − 1 ∈ p2

for some integer n and i, then we have πX (p) = πX (p2 ). Proof. It is equivalent to show that if π := πX (p) = πX (p2 ), then we will have ani − 1 ∈ p2 for all n and i satisfying ani − 1 ∈ p which trivially follows from Proposition 9. We define the function κ in Definition 7 for a number field. Let IK (x) := {p ∈ Spec(OK ) | vp (x) = 1}  p. κK (x) = p∈IK (x)

Theorem 2. Let K be the field of the tuple (a1 , . . . , am , b1 , . . . , bm ), and let X = {b1 ai1 + · · · + bm aim }i≥0 . Then, there are infinitely many non-X -base Fibonacci-Wieferich primes, assuming K is an abc-field. Proof. By lemma 11, it is sufficient to show that, for any i, there are infinitely many non-ai -Wieferich primes, or equivalently Un := κK (ani − 1) is unbounded. For the sake of contradiction, we assume N (Un ) < B for some constant B, and also write (αn − 1)OK = Un Vn for some fractional ideal Vn . By the definition of the height function, we have a natural inequality N (Un Vn ) ≤ h(αn , α, (α)n − 1). On the other hand, using Conjecture 8, for any ε > 0, there exists a constant Cε such that h(αn , 1, αn − 1) ≤ (1 + ε) rad(αn , 1, αn − 1) + Cε . Note that rad(αn , 1, αn − 1) =

 α(αn −1)∈p

1 N (p) ≤ N (α) + N (Un ) + N (Vn ), 2

so there exists a constant Cε,α such that (1 − ε)N (Vn ) ≤ εN (Un ) + Cε,α . 2

(2)

However, for ε < 1, Inequality (2) could not hold once the N (Un ) is bounded. Hence, we conclude that Un is unbounded. 7

4. A logarithmic lower bound We define the followings for this section: 1. Let α ∈ K with |α| = 1 for any number field K if we do not specify. 2. The height function h : K → R≥0 of any element α ∈ K is defined as h(α) :=

 1 log max{αv , 1}. [K : Q] v∈MK

3. The approximation function λv : K → R≥0 with respect to a place v is defined as, for every α, λv (α) = 4. 5. 6. 7. 8.

1 log max{αv , 1}. [K : Q]

∞ Let MK be the set of all infinity places of K. ϕ is the Euler totient function. Φn is the n-th cyclotomic polynomial. Let Wα (B) = {prime ideal p | N (p) ≤ log B, and p is not an α − Wieferich prime}. mp := ordp (α) if vp (α) = 0.

To simplify notations, we let un = κK ((αn − 1)) and Un = κK ((αn − 1)) such that there are ideals vn , wn , Vn and Wn in Spec(OK ) such that (αn − 1)OK = un vn w−1 n ; Φn (α)OK = Un Vn Wn−1 . By Proposition 9 and Lemma 11, we have {prime ideal p|p is not an X − base Fibonacci-Wieferich prime} =

m 

Wai (∞)

i=1

with X = {a1 , . . . , am }. Thus, a logarithmic lower bound for |Wai | for some i is also a lower bound for non-X-base Fibonacci-Wieferich primes. Write αOK = IJ −1 where ideas I and J are coprime and where K is a field that contains X. We should note that the radical of the ideal wn and the ideal Wn are both contained in the radical of J since the poles of (αn − 1) and (Φn (α)) are the poles of (α). Beside, since ∞ without loss of generality. Wα (B) = Wα−1 (B), we can further assume λv (α) > 0 for some v ∈ MK Lemma 12. If we assume (n)IJ + p = OK and p ⊇ Un , then we have mp = n

αNK/Q (p)−1 ≡ 1

and

mod p2 .

Proof. Since p ⊇ Un , we have Φn (α) ≡ 0 mod p, which also implies αn ≡ 1 mod p. Moreover, since (n)IJ + p = OK , the polynomial X n − 1 is separable modulo p. Therefore, for all divisors d of n, Φd (α) ≡ 0 mod p, i.e. mp = n. Since p2 ⊇ Un , αn ≡ 1 + up mod p2 for some u ∈ (OK /p)× . It follows that αNK/Q (p)−1 ≡ (1 + up)

NK/Q (p)−1 n

≡1+

NK/Q (p) − 1 up n

mod p2 ,

which completes our proof. Lemma 13. We have |Wα (B)| ≥ |{n ≤ (log B − log 2)/h(α) | N (Un ) ≥ log(n) + N (IJ)}|.

8

Proof. Given an n ≤ (log B − log 2)/h(α) with N (Un ) ≥ log(n) + N (IJ), there exists a prime ideal pn ⊇ Un with (n)IJ + pn = OK , so we have m pn = n

and

αNK/Q (pn )−1 ≡ 1

mod p2n

by Lemma 12. Since n ≤ (log B − log 2)/h(α), we have |N (pn )| ≤ h(αn − 1) ≤ nh(α) + log 2 ≤ log B. Moreover, if we find pn = pn , then

n = mpn = mpn = n

which shows that, for every integer n with N (Un ) ≥ log(n) + N (IJ), we can construct a unique prime ideal in Wα . We also need to generalize Lemma 5 of [Sil88], which shows that we can find an absolutely constant c > 0 such that Φn (a, b) ≥ ecϕ(n) for all n ≥ 2. The details of Silverman’s proof can be found in the lecture note [Sil89]. In our case, n ≥ 2 will be weakened to n large enough, but a weaker lemma would not affect our final result because we are only care about n sufficiently large. This lemma is an easy application of equidistribution on the primitive roots of unity. In general, if we let f : C → C be a continuous function, then   i (i,n)=1 f (ζn ) → f (z)dz φ(n) ∂D1 (0) where ∂D1 (0) is the boundary of the complex unit circle. Readers can find more details in [BIR08]. Lemma 14. We can find a constant c > 0 such that N (Φn (α)) ≥ cϕ(n). for all sufficiently large n. Proof. Since the primitive n-th roots of unity ζni with (i, n) = 1 are equidistribution on the unit circle, we have   log(ζ i − σ(α))   1 1 n → log(z − σ(α))dz N (Φn (α)) = [K : Q] φ(n) [K : Q] ∂D1 (0) σ∈GK (i,n)=1

σ∈GK

when n goes to infinity. By Jensen’s formula, it implies  N (Φn (α)) → λv (α) φ(n) ∞

as

n → ∞,

v∈MK

so N (Φn (α))/φ(n) is greater than zero for some large enough n which completes our proof. Lemma 15. Fix δ > 0. Then |{n ≤ Y | φ(n) ≥ δn}| ≥ (

6 − δ)Y + O(log Y ). π2

(The big-O constant is absolute.) Proof. See [Sil88]. Lemma 16. If we assume K is an abc-field, then, for all ε > 0, there exists a constant Cε,α satisfying N (Vn ) ≤ nεh(α) + Cε,α . 9

Proof. Since vn ⊆ Vn , we only need to prove a similar estimation for vn . By Conjecture 8, given ε > 0, there exists a constant Cε satisfying h(αn , 1, αn − 1) ≤ (1 + ε) rad(αn , 1, αn − 1) + Cε .

(3)

On the left hand side of Inequality (3), we have h(αn − 1) ≤ h(αn , 1, αn − 1).

(4)

On the other side of Inequality (3), we have  (1 + ε) rad(α , 1, α − 1) ≤ n

n

Because of N (un ) + N (vn ) − N (wn ) = Inequality (4) and (5), we have

 σ∈GK

N (vn ) (1 + ε)(N (IJ) + N (un ) + 2

 .

(5)

log |σ(α)n − 1|/[K : Q] ≤ h(αn − 1). Now, following from



N (vn ) h(α − 1) ≤ (1 + ε) N (IJ) + h(α − 1) + N (wn ) − 2 n

n

 + Cε .

Therefore, we have 2ε 2 h(αn − 1) + 2(N (IJ) + N (wn )) + Cε 1+ε (1 + ε) 2ε log 2 2εn 2 h(α) + + 2(N (IJ) + N (wn )) + Cε . ≤ 1+ε 1+ε (1 + ε)

N (vn ) ≤

Replacing Cε by Cε,α := 2(ε log 2 + Cε )/(1 + ε) + 2(N (IJwn )), and replacing ε by ε/(2 − ε), we have the desired result. Theorem 4. For any non zero algebraic number α ∈ X \ ∂D1 (0) where ∂D1 (0) is the boundary of the complex unit circle with center at origin, there exists a constant Cα > 0 such that we have |WX (B)| ≥ |Wα (B)| ≥ Cα log B for all sufficiently large B assuming K is an abc-field. Proof. Let c be an absolute constant obtained from Lemma 14, and let c1 , c2 , . . . be constant not depending on n. By Lemma 13, for each integer n ≤ (log B − log 2)/h(α) with N (Un ) > log(n) + N (IJ), we can find a unique prime pn in Wα (B) such that pn ⊇ Un with (n)IJ + pn = OK . Thus, the lower bound for |{n ≤ (log B − log 2)/h(α) : N (Un ) ≥ log(n) + N (IJ)}| is also a lower bound for |Wα (B)|. Let Cε,α be the constant given in Lemma 16. Note that nεh(α) + N (Un ) + Cε,α ≥ N (Vn ) + N (Un ) = |N (Φn (α))|, so we have |N (Un )| ≥ cϕ(n) − nεh(α) − Cε,α for large enough n by Lemma 14. Hence, if cϕ(n) − nεh(α) − c2 ≥ log(n) + N (IJ) holds, then N (Un ) is greater than log(n) + N (IJ). Thus, we have |Wu (B)| ≥ |{n ≤ (log B − log 2)/h(α) | (6) holds.}|. 10

(6)

Now, Inequality (6) is equivalent to cϕ(n) − nεh(α) − c2 − log(n) − N (IJ) ≥ 0. Finally, we give some arbitrary fixed δ > 0, and let ε = cδ/2h(α), and further suppose that n satisfies ϕ(n) ≥ δn. Then, we have cϕ(n) − nεh(α) − c2 − log(n) − N (IJ) ≥

1 cδn − c2 − log(n) − N (IJ), 2

(7)

and there exists some integer n0 := n0 (δ, c) only depended on δ and c such that the right hand side of Inequality (7) is positive whenever n ≥ n0 . Consequently, for those n satisfying n ≥ n0 and φ(n) ≥ δn simultaneously, it will also satisfy Inequality (6). Therefore, we have |Wu (B)| ≥ |{n0 ≤ n ≤ (log B − log 2)/h(α) | ϕ(n) ≥ δn}| 6 ≥ ( 2 − δ)(log B − log 2)/h(α) + O(log(log B − log 2)/h(α)) − n0 . π by Lemma 15. Since we are free to choose δ > 0, the proof is completed. Corollary 17. Assuming abc, we have |{p ≤ B | p is not a Fibonacci-Wieferich prime}| ≥ O(log B). Although the set of non-Fibonacci-Wieferich primes is infinite assuming abc, we not know if abc implies that the Dirichlet density of the set of non-Fibonacci-Wieferich primes is greater than 0. 5. Heuristic result and the conjecture 1. Let Gm (K) := K × = K \ {0}. ¯ i.e. an element of the group could be 2. Let a1 , . . . , am  be a multiplicative group generated by ai ∈ Q, m written as a finite product i=1 aei i where ei are some integers. 3. By the fundamental theorem of abelian groups, an abelian group G is isomorphic to Z/mZ × Zr for some m and r, where Z/m is the torsion part of G and Zr = Z × Z × · · · × Z is the free part of G. We define the rank of G, rank G, to be r. 4. Let V be a smooth variety. Let dim V be the dimension of the variety V . 5. Let φ : V → V be a smooth endomorphism, and let q be a point in V . The orbit Oφ (q) of q under the map φ is a sequence {φn (q)}n≥0 where we use φn to represent the nth compositional iterate of φ. I would like to separate this section into two subsections. Both subsections are for explaining Conjecture 5. One of the subsections is from the arithmetic point of view to make the heuristic argument of Conjecture 5, and the other is from the geometric point of view. 5.1. Arithmetic point of view Fermat’s little theorem says that 2p−1 ≡ 1 mod p for every prime p, so it must be that 2p−1 ≡ 1 + kp p mod p2 for some 0 ≤ kp ≤ p − 1. If we are to assume that kp is distributed randomly between 0 and p − 1, the possibility of kp ≡ 0 mod p would be 1/p. Therefore, the expected number of Wieferich primes below Y is given by  1/p ≈ log log Y. p≤Y

This is the well-known heuristic argument that show why people conjectured that the number of Wieferich primes is infinite even though we only find few of them. 11

Given a number field K and an element α ∈ K, since αN (p)−1 ≡ 1 mod p, we must have αN (p)−1 ≡ 1 + kp p

mod p2

where kp ∈ OK /p. Supposing kp distributed randomly, we would have the possibility 1/NK/Q (p) for getting kp ≡ 0 mod p. Therefore, the expected number of α-Wieferich primes with the norm smaller than Y is given by  1 , NK/Q (p) p;NK/Q (p)≤Y

which also tends to infinity as Y goes to infinity. Given a recurrence tuple (a1 , . . . , am , b1 . . . , bm ), πX (p) = πX (p2 ) happens if kp ≡ 0 mod p simultaneously for each ai . Moreover, the multiplicative relation among ai affects the expected number. For example, Let α be a quadratic unit, and let α ¯ be its conjugates so αα ¯ = 1 or −1. Then, we see that αp−1 ≡ 1 2 p−1 2 ¯ ≡ 1 mod p . Therefore, the expected number of {α, α ¯ }-base mod p for some prime p if and only if α Fibonacci-Wieferich primes is equal to the expected number of {α} − base Fibonacci primes. For the similar reason, the expected number of X -base Fibonacci-Wieferich primes with the norm under Y should be 

1

p;NK/Q (p)≤Y

NK/Q (p)r

where r = rank a1 , . . . , am−1 . 5.2. Geometric point of view A recurrence sequence can be considered as a dynamical system. Let qi := (xi , xi+1 , . . . , xi+m ) be a point ¯ m . Then, the matrix on Q ⎡ ⎤ 0 1 0 ··· 0 ⎢0 0 1 ··· 0⎥ ⎢ ⎥ ⎢ ⎥ M := ⎢. . . . . . . . . . . . . . .⎥ ⎢ ⎥ ⎣0 0 0 ··· 1⎦ c 1 c 2 c 3 · · · cm is a linear transformation for which M (qi ) = qi+1 for all i ≥ 0. We should note that the length of the period of the sequence X mod pe is equal to the length of the period of the orbit OM (q0 ) modulo pe . Then, we want to ask the following question: if the kp is randomly distributed, what is the possibility of π = πX (p) = πX (p2 )? Equivalently, we want to know the possibility for that the congruence M π (q0 ) ≡ q0 holds. We definitely have

⎛

mod p2

x0 + k0,p p x1 + k1,p p .. .

⎞

⎜ ⎟ ⎜ ⎟ M π (q0 ) ≡ ⎜ ⎟ ⎝ ⎠ xm−1 + km−1,p p

mod p2

by the definition of π. Let V be the Zariski closure of OM (q0 ). Suppose V is of dimension d, so we can freely choose d many coordinates, after given those coordinates, there are bounded choices of the remaining coordinates. Beside, there are NK/Q (p) many choices of ki,p for each freely chosen coordinate, so the possibility of k0,p ≡ k1,p ≡ · · · ≡ km−1,p ≡ 0 mod p is O(1/NK/Q (p)d ). Therefore, as long as the dim V = 1, we expect that there are infinitely many X -base Fibonacci-Wieferich primes.

12

The rank of the multiplicative group a1 , . . . , am  should be equal to the dimension of V , and we are going to prove this at the end of this paper. Our main idea is the following. The eigenvalues of our matrix M are ai which are distinct by assumption, so the matrix M is diagonalizable with a diagonal matrix ⎤ ⎡ a1 0 · · · 0 ⎢ 0 a2 · · · 0⎥ ⎥ ⎢ A=⎢ ⎥. ⎣. . . . . . . . . . . . ⎦ 0

0

···

am

I.e. We can find an invertible matrix B such that M = B −1 AB. Let ri = B(qi ). This implies A(ri ) = ri+1 , so we have a commutative diagram B qi −−−−→ ri ⏐ ⏐ ⏐ ⏐ . M A B

qi+1 −−−−→ ri+1 It is clear that the length of the period of the orbit OM (q0 ) modulo pe is equal to the periodof the orbit e OA (r0 ) modulo pe for any prime p and any integer e. Moreover, if we can find ej ∈ N such that j∈I aj j = 1 for some index set I, then we have  (j)  e i  (j) (ri )ej = aj j (r0 )ej j∈I

j∈I

j∈I

where we use r(j) to represent the j-th coordinate of a point r, i.e. ri belongs to the variety defined by the   e (j) equation j∈I Yj j = ± j∈I (r0 )ej where Yi are variables. We need a lemma in [Lau84]. Lemma 18 (Laurent). Let V ⊆ Gnm be a subvariety. Let G be a multiplicative subgroup of Gnm . If V ∩ G is Zariski dense in V , then V is a subgroup of Gnm . Another lemma is a well-known result in representation theory (see [TY05]), and we only present the part we need. Lemma 19. If a subvariety V is a proper subgroup of Gnm , then V contains in the zero set of the polynomial X1e1 · · · Xnen − 1 for some ei ∈ Z. Proposition 20. We have rank a1 , a2 , . . . , am  = dim {qi }. Proof. First, we assume that the {a1 , a2 , . . . , am } are multiplicative independent, which means rank a1 , a2 , . . . , am  = m. Let A be the diagonal matrix for the matrix M . We then consider A as a vector (a1 , a2 , . . . , am ) ∈ Gm m since all ai are not zero. Thus, {ri } = {Ai r0 } can be interpreted as the orbit of r0 under the action of {Ai } in the group Gm m. m−1 T 2 ] is an eigenvector of the matrix We need to show that r0 ∈ Gm m . First, we note that [1, ai , ai , . . . , ai M with the eigenvalue ai since ai is a root of the characteristic polynomial p(x) = xm − cm xm−1 − cm−1 xm−2 − · · · − c1 . It follows that

⎡ ⎢ ⎢ B −1 = ⎢ ⎣

1 a1 .. .

1 a2 .. .

··· ··· .. .

1 am .. .

am−1 1

am−1 2

···

am−1 m

13

⎤ ⎥ ⎥ ⎥. ⎦

Finally, r0 = [b1 , b2 , . . . , bm ]T since q0 = [x0 , x1 , . . . , xm−1 ] = B −1 r0 . Therefore, r0 ∈ Gm m by the assumption that bi are not zero. k Let W be a nontrivial subvariety of Gm m such that A r0 ∈ W for infinitely many k, and which is equivalent −1 k to saying that A ∈ r0 W for infinitely many k where the r0−1 W is still a variety. Since ai are not root of unity, {Ak } contains infinitely many distinct points; therefore, r0−1 W is a subgroup of Gm m by Lemma 18. However, since we assume ai are multiplicatively independent, the above conclusion contradicts Lemma 19. Hence, W is Gm m since W is the Zariski closure of a not empty set. If a1 , . . . , am  are multiplicatively dependent, we let W be the subvariety defined by all multiplicative relations, and follow the same argument which also implies W = {qi }. The whole paper is about some concrete cases of the following question. Let K be a number field, V be a variety over K, and φ : V → V be an endomorphism. Given an initial point q and a prime ideal p of K, we want to know whether the length of the periodic cycle of the sequence {φn (q) mod p}n≥0 is equal to the one of the sequence {φn (q) mod p2 }n≥0 or not. It is natural to ask this question for which φ is a polynomial over K = Q. However, it is also very easy to give an example which the length of modulo p and modulo p2 are equal for a high proportion of primes. For example, let φ1 (x) = x3 + 1 be a polynomial over Q, and let the initial point q = 1. We can split the φ1 : Fp → Fp into two different maps x3

+1

Fp −−−−→ Fp −−−−→ Fp . If 3 does not divide p − 1, then the first part of φ1 is a permutation which implies φ1 is permutation on Fp . Therefore, every elements of Fp is periodic under φ1 . Then, for any prime p congruent to 2 modulo 3, 1 is on the forward orbit of 0 modulo p where 0 is periodic modulo p(see [GTZ07] for more details). You can immediately conclude that there are half of the primes p for which #{φn1 (1) mod p}n≥0 = #{φn1 (1) mod p2 }n≥0 = #{φn1 (1) mod p3 }n≥0 . A similar idea can be applied to argue that the φ2 (x) = x3 + 2 has #{φn2 (2) mod p}n≥0 = #{φn2 (2) mod p2 }n≥0 = #{φn2 (2) mod p3 }n≥0 for p congruent to 2 modulo 3. However, we note that if we suppose the closure of S = {(φn1 (1), φn2 (2))}n≥0 is a nontrivial subvariety fixed by (φ1 , φ2 ), it follows that there are infinitely many preperiodic points on the closure of S by [GNY15]. Furthermore, by [Mim13], we can conclude that the Julia sets of φ1 and φ2 need to be physically same, but [SS95] shows that φ1 and φ2 cannot have the same Julia set which means that the Zariski closure of {(φn1 (1), φn2 (2))}n≥0 is A2 which is of dimension 2. Thus, if we naively apply Conjecture 5, we would have finite many primes satisfying #{(φn1 (1) mod p, φn2 (2) mod p)}n≥0 = #{(φn1 (1) mod p2 , φn2 (2) mod p2 )}n≥0 which is definitely not true for p congruent to 2 mod 3. N We propose the following question to fit our conjecture into a more general case. Let Φ : PN Q → PQ , N n denoted by Φ = [φ0 : . . . : φN ], and let α ∈ PQ . Express the coordinate of Φ (α) p-adically. Asking whether #{Φn (α) mod pe }n≥0 = #{Φn (α) mod pe+1 }n≥0 is the same as asking if Φl (α) ≡ Φk (α) mod pe implies that Φl (α) ≡ Φk (α) mod pe+1 . This is equivalent to asking the following: if the first e digits of Φl (α) and Φk (α) are equal, do we know that the (e + 1)st digit must be equal as well? The above example shows that in some cases there may be some arithmetical method allowing us to control the next digit or perhaps even the next few digits. However, we should have some randomness beyond some digit. In our example above, we should not be able to control the fourth digit in terms of the first three digits. More precisely, we cannot say much about when #{(φn1 (1) mod p3 , φn2 (2) mod p3 )}n≥0 = #{(φn1 (1) mod p4 , φn2 (2) mod p4 )}n≥0 . This analysis shows indicates if we really want to define more general Wieferich primes fitting in a dynamical context, we should first ask which digits have the property of randomness relative to the previous digits. References [BB94] J. Browkin and J. Brzezi´ nski, Some remarks on the abc-conjecture, Math. Comp. 62 (1994), no. 206, 931–939. 14

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16