On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers

Diego Marques Departamento de Matemática Universidade de Brasília Brasília 70910-900 Brazil

                       diego@mat.unb.br

Abstract The m-th Cullen number C_{m} is a number of the form m2^{m} + 1 and the m-th Woodall number W_{m} has the form m2^{m} − 1. In 2003, Luca and Stănică proved that the largest Fibonacci number in the Cullen sequence is F_{4} = 3 and that F_{1} = F_{2} = 1 are the largest Fibonacci numbers in the Woodall sequence. A generalization of these sequences is defined by C_{m,s} = ms^{m} + 1 and W_{m,s} = ms^{m} − 1, for s > 1. In this paper, we search for Fibonacci numbers belonging to these generalized Cullen and Woodall sequences.

1 Introduction

A Cullen number is a number of the form m2^{m}+1 (denoted by C_{m}), where m is a nonnegative integer. The first few terms of this sequence are

1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, …

which is the OEIS [26] sequence \underline{A002064}. (This sequence was introduced in 1905 by Father Cullen [6] and it was mentioned in the well-known book of Guy [9, Section B20].) These numbers gained great interest in 1976, when Hooley [11] showed that almost all Cullen numbers are composite. However, despite their being very scarce, it is still conjectured that there are infinitely many Cullen primes. For instance, C_{6679881} is a prime number with more than 2 millions of digits (PrimeGrid, August 2009).

In a similar way, a Woodall number (also called Cullen number of the second kind) is a positive integer of the form m2^{m} − 1 (denoted by W_{m}). The first few terms of this sequence are

1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, …

which is the OEIS sequence \underline{A003261}. In a personal communication to Keller [13, p. 1739], Suyama asserted that Hooley’s method can be reformulated to show that it works for any sequence of numbers of the form m2^{m+a}+b where a and b are integers. In particular, Woodall numbers are almost all composites. However, it is conjectured that the set of Woodall primes is infinite. We remark that W_{3752948} is a prime number (PrimeGrid, December 2007).

These numbers can be generalized to the generalized Cullen and Woodall numbers which are numbers of the form

C_{m,s} = ms^{m} + 1 and W_{m,s} = ms^{m} − 1,

where m ≥ 1 and s ≥ 2. Clearly, one has that C_{m,2} = C_{m} and W_{m,2} = W_{m}, for all m ≥ 1. For simplicity, we call C_{m,s} and W_{m,s} an s-Cullen number and an s-Woodall number, respectively. Also, an s-Cullen or s-Woodal number is said to be trivial if it has the form s + 1 or s − 1, respectively, or equivalently when its first index is equal to 1. This family was introduced by Dubner [7] and is one of the main sources for prime number “hunters”. A prime of the form C_{m,s} is C_{139948,151} an integer with 304949 digits.

Many authors have searched for special properties of Cullen and Woodall numbers and their generalizations. In regards to these numbers, we refer to [8, 10, 13] for primality results and [17] for their greatest common divisor. The problem of finding Cullen and Woodall numbers belonging to other known sequences has attracted much attention in the last two decades. We cite [18] for pseudoprime Cullen and Woodall numbers, and [1] for Cullen numbers which are both Riesel and Sierpiński numbers.

In 2003, Luca and Stănică [16, Theorem 3] proved that the largest Fibonacci number in the Cullen sequence is F_{4} = 3 = 1 · 2^{1} + 1 and that F_{1} = F_{2} = 1 = 1 · 2^{1} − 1 are the largest Fibonacci numbers in the Woodall sequence. Note that these numbers are trivial Cullen and Woodall numbers (in the previous sense, i.e., m = 1).

In this paper, we search for Fibonacci numbers among s-Cullen numbers and s-Woodall numbers, for s > 1. Our main result is the following

Theorem 1. Let s be a positive integer. If (m, n, ℓ) is an integer solution of

F_{n} = ms^{m} + ℓ,  (1)

where ℓ ∈ {−1, 1} and m, n > 0, then

m < (6.2 + 1.9P (s)) log(3.1 + P (s)),  (2)

and

n < \frac{log((6.2 + 1.9P (s)) log(3.1 + P (s))s^{(6.2+1.9P (s)) log(3.1+P (s))} + 1)}{log α} + 2,

where P (s) denotes the largest prime factor of s and α = (1 + \sqrt{5})/2.

In particular, the above theorem ensures that for any given s ≥ 2, there are only finitely many Fibonacci numbers which are also s-Cullen numbers or s-Woodall numbers and they are effectively computable.

We should recall that ν_{p}(r) denotes the p-adic order (or valuation) of r which is the exponent of the highest power of a prime p which divides r. Also, the order (or rank) of appearance of n in the Fibonacci sequence, denoted by z(n), is defined to be the smallest positive integer k, such that n | F_{k} (some authors call it order of apparition, or Fibonacci entry point). We refer the reader to [19, 20, 21, 22, 23] for some results about this function. Let p be a prime number and set e(p) := ν_{p}(F_{z(p)}). By evaluating e(p), for primes p < 30, one can see that e(p) = 1. In fact, e(p) = 1 for all primes p < 2.8 · 10^{16} (PrimeGrid, March 2014). Moreover, the assertion e(p) = 1 for all prime p is equivalent to z(p) ≠ z(p^{2}), for all primes p (this is related to Wall’s question [28]). This question raised interest in 1992, when Sun and Sun [27] proved (in an equivalent form) that e(p) = 1 for all primes p, implies the first case of Fermat’s “last theorem”.

In view of the previous discussion, it seems reasonable to consider problems involving primes with e(p) = 1 (because of their abundance). Our next result deals with this kind of primes.

Theorem 2. There is no integer solution (m, n, s, ℓ) for Eq. (1) with n > 0, m > 1, ℓ ∈ {−1, 1} and s > 1 such that e(p) = 1 for all prime factor p of s.

In particular, the only solutions of Eq. (1), with the previous conditions, occur when m = 1 and have the form

(m, n, s, ℓ) = (1, n, F_{n} − ℓ, ℓ).

An immediate consequence of Theorem 2 and the fact that e(p) = 1 for all primes p < 2.8 · 10^{16} is the following

Corollary 3. There is no Fibonacci number that is also a nontrivial s-Cullen number or s-Woodall number when the set of prime divisors of s is contained in

{2, 3, 5, 7, 11, …, 27999999999999971, 27999999999999991}.

This is the set of the first 759997990476073 prime numbers.

Here is an outline of the paper. In Section 2, we recall some facts which will be useful in the proofs of our theorems, such as the result concerning the p-adic order of F_{n} and factorizations of the form F_{n} ± 1 = F_{a}L_{b}, with |a − b| ≤ 2. In the last section, we combine these mentioned tools, the fact that a common divisor of F_{a} and L_{b} is small and a lower bound for Fibonacci and Lucas numbers, to get an inequality of the form m < C log m which gives an upper bound for m in terms of C which in turn depends on the factorization of s. With this bound, we reduce the analysis of Eq. (1) for a finite number of cases which can be settled by using an approach used in a recent paper by Bugeaud, Luca, Mignotte and Siksek. By using these ingredients (with some more technicalities) we deal with the proofs of Theorems 1 and 2.

2 Auxiliary results

We cannot go very far into the lore of Fibonacci numbers without encountering its companion Lucas sequence (L_{n})_{n≥0} which follows the same recursive pattern as the Fibonacci numbers, but with initial values L_{0} = 2 and L_{1} = 1. First, we recall some classical and helpful facts which will be essential ingredients to prove Theorems 1 and 2.

Lemma 4. We have

(a) If d = gcd(m, n), then

gcd(F_{m}, L_{n}) = \begin{cases}L_{d}, & if m/d is even and n/d is odd; \\ 1 or 2, & otherwise. \end{cases}

(b) If d = gcd(m, n), then gcd(F_{m}, F_{n}) = F_{d}.

(c) (Binet’s formulae) If α = (1 + \sqrt{5})/2 and β = (1 − \sqrt{5})/2, then

F_{n} = \frac{α^{n} − β^{n}}{α − β} and L_{n} = α^{n} + β^{n}.

(d) α^{n−2} ≤ F_{n} ≤ α^{n−1} and α^{n−1} < L_{n} < α^{n+1} for all n ≥ 1.

(e) z(p) ≤ p + 1, for all prime numbers p.

Proofs of these assertions can be found in [14]. We refer the reader to [2, 12, 24] for more details and additional bibliography.

The equation F_{n} + 1 = y^{2} and more generally F_{n} ± 1 = y^{ℓ} with integer y and ℓ ≥ 2 have been solved in [25] and [5], respectively. The solution for the last equation makes appeal to Fibonacci and Lucas numbers with negative indices which are defined as follows: let F_{n} = F_{n+2} − F_{n+1} and L_{n} = L_{n+2} − L_{n+1}. Thus, for example, F_{−1} = 1, F_{−2} = −1, and so on. Bugeaud et al. [5, Section 5] used these numbers to give factorizations for F_{n} ± 1. Let us sketch their method for the convenience of the reader.

Since the Binet’s formulae remain valid for Fibonacci and Lucas numbers with negative indices, one can deduce the following result.

Lemma 5. For any integers a, b, we have

F_{a}L_{b} = F_{a+b} + (−1)^{b}F_{a−b}.

Proof. The identity α = (−β)^{−1} together with Lemma 4 (c) leads to

F_{a}L_{b} = \frac{α^{a} − β^{a}}{α − β} (α^{b} + β^{b}) = F_{a+b} + \frac{α^{a}β^{b} − β^{a}α^{b}}{α − β} = F_{a+b} + (−1)^{b}F_{a−b}.

Lemma 5 gives immediate factorizations for F_{n} ± 1, depending on the class of n modulo 4. For example, if n ≡ 0 (mod 4), then F_{n} + 1 = F_{(n/2)−1}L_{(n/2)+1}. In general, we have F_{n} ± 1 = F_{a}L_{b}, where 2a, 2b ∈ {n ± 2, n ± 1}.

We remark that the p-adic order of Fibonacci and Lucas numbers has been completely characterized. For instance, from the main results of Lengyel [15], we extract the following results.

Lemma 6. For n ≥ 1,

ν_{2}(F_{n}) = \begin{cases}0, & if n ≡ 1, 2 (mod 3); \\ 1, & if n ≡ 3 (mod 6); \\ 3, & if n ≡ 6 (mod 12); \\ ν_{2}(n) + 2, & if n ≡ 0 (mod 12), \end{cases} .

ν_{5}(F_{n}) = ν_{5}(n), and if p is prime ≠ 2 or 5, then

ν_{p}(F_{n}) = \begin{cases}ν_{p}(n) + e(p), & if n ≡ 0 (mod z(p)); \\ 0, & otherwise, \end{cases}

where e(p) := ν_{p}(F_{z(p)}).

Lemma 7. Let k(p) be the period modulo p of the Fibonacci sequence. For all primes p ≠ 5, we have

ν_{2}(L_{n}) = \begin{cases}0, & if n ≡ 1, 2 (mod 3); \\ 2, & if n ≡ 3 (mod 6); \\ 1, & if n ≡ 0 (mod 6) \end{cases}

and

ν_{p}(L_{n}) = \begin{cases}ν_{p}(n) + e(p), & if k(p) ≠ 4z(p) and n ≡ \frac{z(p)}{2} (mod z(p)); \\ 0, & otherwise. \end{cases}

Observe that the relation L_{n}^{2} = 5F_{n}^{2} + 4(−1)^{n} implies that ν_{5}(L_{n}) = 0, for all n ≥ 1. Now we are ready to deal with the proofs of our results.

3 The proofs

3.1 The proof of Theorem 1

In order to simplify our presentation, we use the familiar notation [a, b] = {a, a + 1, …, b}, for integers a < b.

The smallest value of (6.2 + 1.9P (s)) log(3.1 + P (s)) is 16.292 … (it happens when s is a power of 2). Thus, we may suppose m > 16 (however, for our purpose it suffices to consider m > 4).

We rewrite Eq. (1) as F_{n}−ℓ = ms^{m}. Since ℓ ∈ {−1, 1}, then Lemma 5 gives F_{a}L_{b} = ms^{m}, where 2a, 2b ∈ {n ± 2, n ± 1} and |a − b| ∈ {1, 2}. Observe that in this case gcd(F_{a}, L_{b}) = 1 or 3 (Lemma 4 (a)).

Case 1. If s ≢ 0 (mod 3). Since gcd(F_{a}, L_{b}) = 1 or 3 and 3 ∤ s, then, without loss of any generality, we can write s = p_{1}^{a_{1}} ⋯ p_{k}^{a_{k}}, with a_{i} ≥ 0, such that p_{1}^{a_{1}} ⋯ p_{t}^{a_{t}} divides F_{a} and p_{t+1}^{a_{t+1}} ⋯ p_{k}^{a_{k}} divides L_{b}, where p_{1}, …, p_{k} are distinct primes. Thus ν_{p_{i}}(F_{a}) ≥ a_{i}m, for i ∈ [1, t] and ν_{p_{j}}(L_{b}) ≥ a_{j}m, for j ∈ [t + 1, k], and on the other hand, Lemmas 6 and 7 imply

ν_{p_{i}}(F_{a}) ≤ ν_{p_{i}}(a) + (1 − δ_{p_{i},5} + δ_{p_{i},2})e(p_{i})

and

ν_{p_{j}}(L_{b}) ≤ max{ν_{p_{j}}(b) + e(p_{j}), 2},

where δ_{r,s} denotes the usual Kronecker delta. Since 1 − δ_{p_{i},5} + δ_{p_{i},2} ≤ 2 and m > 2, then

ν_{p_{i}}(F_{a}) ≤ ν_{p_{i}}(a) + 2e(p_{i}) and ν_{p_{j}}(L_{b}) ≤ ν_{p_{j}}(b) + e(p_{j}).

Thus one obtains that ν_{p_{i}}(a) ≥ a_{i}m − 2e(p_{i}), for all i ∈ [1, t] and ν_{p_{j}}(b) ≥ a_{j}m − e(p_{j}), for all j ∈ [t+1, k]. Since p_{1}, …, p_{k} are pairwise coprime, we have a ≥ p_{1}^{a_{1}m−2e(p_{1})} ⋯ p_{k}^{a_{k}m−2e(p_{t})} and b ≥ p_{t+1}^{a_{t+1}m−e(p_{t+1})} ⋯ p_{k}^{a_{k}m−e(p_{k})}. Hence F_{a}L_{b} = ms^{m} together with the estimates in Lemma 4 (d) yields

mp_{1}^{ma_{1}} ⋯ p_{k}^{ma_{k}} ≥ α^{a+b−3}
> α^{∏_{i=1}^{t} p_{i}^{ma_{i}−2e(p_{i})}+∏_{i=t+1}^{k} p_{i}^{ma_{i}−2e(p_{i})}−3},

where we used the inequality ma_{i} − e(p_{i}) > ma_{i} − 2e(p_{i}). Note that we may suppose that ma_{i} > 2e(p_{i}), for all i ∈ [1, k] (otherwise, we would have m ≤ 2e(p_{i}), for some i and Theorem 1 is proved). Also p_{i} ≥ 2, for all i ∈ [1, k] and then

∏_{i=1}^{t} p_{i}^{ma_{i}−2e(p_{i})} + ∏_{i=t+1}^{k} p_{i}^{ma_{i}−2e(p_{i})} ≥ ∑_{i=1}^{k} p_{i}^{ma_{i}−2e(p_{i})}.

Thus we have

4.3mp_{1}^{ma_{1}} ⋯ p_{k}^{ma_{k}} > α^{p_{1}^{ma_{1}−2e(p_{1})}} ⋯ α^{p_{k}^{ma_{k}−2e(p_{k})}},

where we have used that α^{3} < 4.3. But for m > 4, it holds that 4.3m < 2^{m} ≤ p_{1}^{m} and so

p_{1}^{m(a_{1}+1)} ⋯ p_{k}^{ma_{k}} > α^{p_{1}^{ma_{1}−2e(p_{1})}} ⋯ α^{p_{k}^{ma_{k}−2e(p_{k})}}.

If the inequality p_{i}^{ma_{i}−2e(p_{i})} > m(a_{i} + 1)(log p_{i})/ log α holds for all i ∈ [1, k], we arrive at the following absurdity

p_{1}^{m(a_{1}+1)} ⋯ p_{k}^{ma_{k}} > α^{p_{1}^{ma_{1}−2e(p_{1})}} ⋯ α^{p_{k}^{ma_{k}−2e(p_{k})}} > p_{1}^{m(a_{1}+1)} ⋯ p_{k}^{m(a_{k}+1)}.

Thus, there exists i ∈ [1, k], such that p_{i}^{ma_{i}−2e(p_{i})} ≤ m(a_{i}+1)(log p_{i})/ log α. Now, by applying the log function in the previous inequality together with a straightforward calculation, we obtain

\frac{m}{log m} ≤ \frac{1}{log p_{i}} + \frac{log(a_{i} + 1)}{a_{i} log m log p_{i}} + \frac{log (\frac{log p_{i}}{log α} )}{a_{i} log m log p_{i}} + \frac{2e(p_{i})}{a_{i} log m}.

Note that a_{i} ≥ 1, log p_{i} ≥ log 2 > 0.69 and log m ≥ log 3 > 1.09. Also, the functions x ↦ (log(x+1))/x and x ↦ (log(log x/ log α))/(log x) are nonincreasing for x ≥ 1 and x ≥ 4, respectively, then (log(a_{i} + 1))/a_{i} ≤ log 2 and (log(log p_{i}/ log α))/(log p_{i}) < 0.77. Therefore,

\frac{m}{log m} < 3.1 + 1.9e(p_{i}).

Since the function x ↦ x/ log x is increasing for x > e, it is a simple matter to prove that, for A > e,

\frac{x}{log x} < A implies that x < 2A log A.  (3)

A proof for that can be found in [3, p. 74].

By using (3) for x := m and A := 3.1 + 1.9e(p_{i}) > e, we have that

m < (6.2 + 3.8e(p_{i})) log(3.1 + 1.9e(p_{i})).  (4)

Since p_{i} ≤ P (s), then in order to get the desired inequality in (2), it suffices to prove that e(p) ≤ p/2 for all primes p. Clearly, the inequality holds for p = 2, so we may suppose p ≥ 3. Since p^{e(p)} | F_{z(p)}, we have p^{e(p)} ≤ F_{z(p)}. Suppose, towards a contradiction, that e(p) > p/2, then we use Lemma 4 (d) and (e) to obtain

p^{p/2} < p^{e(p)} ≤ F_{z(p)} ≤ α^{z(p)−1} ≤ α^{p}.

This yields that p < α^{2} < 2.619 which is impossible, since p ≥ 3. Thus e(p) ≤ p/2 ^{∗}. Thus

m < (6.2 + 1.9P (s)) log(3.1 + P (s)),

where we used that P (s) ≥ p_{i}, for all i ∈ [1, k].

Now, we use Lemma 4 (d) to get α^{n−2} ≤ F_{n} ≤ ms^{m} +1 and a straightforward calculation yields

n < \frac{log((6.2 + 1.9P (s)) log(3.1 + P (s))s^{(6.2+1.9P (s)) log(3.1+P (s))} + 1)}{log α} + 2,

where we used the upper bound for m in terms of s.

Case 2. If s ≡ 0 (mod 3). We can proceed as before unless gcd(F_{a}, L_{b}) = 3. In this case, for some suitable choice of ϵ_{1}, ϵ_{2} ∈ {0, 1}, with ϵ_{1} + ϵ_{2} = 1, we have

\frac{F_{a}}{3^{ϵ_{1}}} · \frac{L_{b}}{3^{ϵ_{2}}} = \frac{ms^{m}}{3}

and gcd(F_{a}/3^{ϵ_{1}}, L_{b}/3^{ϵ_{2}}) = 1. From this point on the proof proceeds along the same lines as the proof of previous case.

∗In fact, the same proof gives the sharper bound e(p) ≤ (p log α)/ log p. This bound together with the squeeze theorem gives lim_{p→∞} e(p)/p = 0.

3.2 The proof of Theorem 2

Let (m, n, s, ℓ) be a solution of Eq. (1) satisfying the conditions in the statement of Theorem 2 and suppose that m ≤ 16. Note that F_{n} = 4s^{4} ± 1 can be rewritten as F_{n} = (2s^{2})^{2} ± 1, but Bugeaud et al. [5, Theorem 2] listed all solutions of the Diophantine equation F_{n} ± 1 = y^{t}. A quick inspection in their list gives y = 1, 2 or 3, but none of these values have the form 2s^{2}, for s > 1. So, there is no solution for Eq. (1) when m = 4.

So, we wish to solve the equation F_{n} = ms^{m} ± 1, when m ∈ [2, 16]\{4}. As previously done, let us rewrite it as F_{a}L_{b} = ms^{m}. Note that m has at most 2 distinct prime factors and they belong to {2, 3, 5, 7, 11, 13}. Since gcd(F_{a}, L_{b}) = 1 or 3, we can deduce that

F_{a} = p_{1}^{α_{1}}p_{2}^{α_{2}}p_{3}^{α_{3}}(s_{1}^{r})^{p},  (5)

where p_{1}, p_{2}, p_{3}, p are primes less than 17, α_{1}, α_{2}, α_{3} ∈ {0, 1, 2, 3, 4}, s_{1} | s and m = pr.

However, in 2007, by combining some deep tools in number theory, Bugeaud, Luca, Mignotte and Siksek [4, Theorem 4 for m = 1], found (in particular) all solutions of the Diophantine equation

F_{t} = 2^{x_{1}} · 3^{x_{2}} ⋯ 541^{x_{100}}y^{p},

where x_{i} ≥ 0 and p is a prime number. More precisely, they proved that in this case, one has

t ∈ [1, 16] ∪ [19, 22] ∪ {24, 26, 27, 28, 30, 36, 42, 44}.  (6)

In particular, t ≤ 44.

Note that Eq. (5) is a particular case already treated by Theorem 4 of [4]. However, for convenience of the reader, we describe in a few words how these calculations can be performed. First, if m = 2, then we have the equation F_{a} = 2^{α_{1}} · 3^{α_{2}}s_{1}^{2} and after a quick inspection in (6), one can see that possible values for a do not give any solution for Eq. (1). In the case of m ≥ 3, we use that a ≤ 44 together with a ≥ (n − 2)/2 to get n ≤ 90 and so

3s^{3} − 1 ≤ ms^{m} + ℓ = F_{n} ≤ F_{90} = 2880067194370816120.

Therefore s ≤ 986492. Now by using Mathematica [29], one deduces that the Diophantine equation F_{n} = ms^{m} + ℓ, for 2 ≤ n ≤ 90, 3 ≤ m ≤ 16, ℓ ∈ {−1, 1} and 2 ≤ s ≤ 986492, has no any solution.

Let us suppose that m > 16. We remark that in order to get the inequality (4) in the proof of Theorem 1, we only assume that m > 4. Thus, we have

m < (6.2 + 3.8e(p_{i})) log(3.1 + 1.9e(p_{i})),

where p_{i} is a prime factor of s. However, by hypothesis, all prime factors p of s satisfy e(p) = 1. In particular, e(p_{i}) = 1 and so we have the following absurdity: 16 < m < 10 log 5 = 16.094 …. This completes the proof of the theorem.

4 Acknowledgement

The author would like to extend his gratitude to the referee for his/her useful comments that helped to improve the clarity of this paper. He also thanks to Josias Tschanz for correcting the English of the paper.

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2010 Mathematics Subject Classification: Primary 11B39, Secondary 11Dxx. Keywords: Fibonacci numbers, p-adic order, Cullen number, Woodall numbers.

(Concerned with sequences \underline{A000045}, \underline{A002064}, and \underline{A003261}.) Received April 9 2014; revised versions received August 11 2014; August 15 2014. Published in Journal of Integer Sequences, September 3 2014. Return to Journal of Integer Sequences home page.