Introduction

Let \((F_n)_{n \ge 1}\) be the sequence of Fibonacci numbers, which is defined by the initial conditions \(F_1=F_2=1\) and by the linear recurrence \(F_n=F_{n-1} + F_{n-2}\) for \(n \ge 3\). It is well known [22] that every positive integer n can be written as a sum of distinct non-consecutive Fibonacci numbers, that is, \(n = \sum _{i=1}^m d_i F_i\), where \(m \in \mathbb {N}\), \(d_i \in \{0, 1\}\), and \(d_i d_{i+1} = 0\) for all \(i \in \{1, \ldots , m-1\}\). This is called the Zeckendorf representation of n and, apart from the equivalent use of \(F_1\) instead of \(F_2\) or vice versa, is unique.

The Zeckendorf representation of integer sequences has been studied in several works. For instance, Filipponi and Freitag [6, 7] studied the Zeckendorf representation of numbers of the form \(F_{kn}/F_n\), \(F_n^2/d\) and \(L_n^2/d\), where \(L_n\) are the Lucas numbers and d is a Lucas or Fibonacci number. Filipponi, Hart, and Sanchis [8, 13, 14] analyzed the Zeckendorf representation of numbers of the form \(mF_n\). Filipponi [8] determined the Zeckendorf representation of \(m F_n F_{n + k}\) and \(mL_n L_{n + k}\) for \(m \in \{1,2,3,4\}\). Bugeaud [3] studied the Zeckendorf representation of smooth numbers. The study of Zeckendorf representations has been also approached from a combinatorial point of view [1, 9, 12, 21]. Moreover, generalizations of the Zeckendorf representation to linear recurrences other than the sequence of Fibonacci numbers have been considered [4, 5, 10, 11, 16].

For all integers a and \(m \ge 1\) with \(\gcd (a, m) = 1\), let \((a^{-1} \bmod m)\) denote the least positive multiplicative inverse of a modulo m, that is, the unique \(b \in \{1, \dots , m\}\) such that \({ab \equiv 1} \pmod m\). Prempreesuk, Noppakaew, and Pongsriiam [17] determined the Zeckendorf representation of \((2^{-1} \bmod F_n)\), for every positive integer n that is not divisible by 3. (The condition \(3 \not \mid n\) is necessary and sufficient to have \(\gcd (2, F_n) = 1\).) In particular, they showed [17, Theorem 3.2] that

$$\begin{aligned} (2^{-1}\!\!\! \mod F_n) = {\left\{ \begin{array}{ll} \sum \nolimits _{k \,=\, 0}^{(n - 7) / 2} F_{n - 3k - 2} + F_3 &{} \text { if } n \equiv 1 \mod 3 ;\\ \sum \nolimits _{k \,=\, 0}^{(n - 8) / 2} F_{n - 3k - 2} + F_4 &{} \text { if } n \equiv 2 \mod 3 ; \end{array}\right. } \end{aligned}$$

for every integer \(n \ge 8\). We extend their result by determining the Zeckendorf representation of the multiplicative inverse of a modulo \(F_n\), for every fixed integer \(a \ge 3\) and every positive integer n with \(\gcd (a, F_n) = 1\). Precisely, we prove the following result.

Theorem 1.1

Let \(a \ge 3\) be an integer. Then there exist integers \(M, n_0, i_0 \ge 1\) and periodic sequences \(\varvec{z}^{(0)}, \dots , \varvec{z}^{(M - 1)}\) and \(\varvec{w}^{(1)}, \dots , \varvec{w}^{(i_0)}\) with values in \(\{0, 1\}\) such that, for all integers \(n \ge n_0\) with \(\gcd (a, F_n) = 1\), the Zeckendorf representation of \((a^{-1} \bmod F_n)\) is given by

$$\begin{aligned} (a^{-1} \bmod F_n) = \sum _{i \,=\, i_0}^{n - 1} z_{n - i}^{(n \bmod M)} F_i + \sum _{i \,=\, 1}^{i_0 - 1} w_n^{(i)} F_i . \end{aligned}$$

From the proof of Theorem 1.1 it follows that \(M, n_0, i_0\), \(\varvec{z}^{(0)}, \dots , \varvec{z}^{(M - 1)}\), and \(\varvec{w}^{(1)}, \dots , \varvec{w}^{(i_0)}\) can be computed from a (see also Remark 4.1 at the end of the paper).

Preliminaries on Fibonacci numbers

Let us recall that for every integer \(n \ge 1\) it holds the Binet formula

$$\begin{aligned} F_n = \frac{\varphi ^n - \overline{\varphi }^n}{\sqrt{5}} , \end{aligned}$$

where \(\varphi := (1 + \sqrt{5}) / 2\) is the Golden ratio and \(\overline{\varphi } := (1 - \sqrt{5}) / 2\) is its algebraic conjugate. Furthermore, it is well known that for every integer \(m \ge 1\) the Fibonacci sequence \((F_n)_{n \ge 1}\) is (purely) periodic modulo m. Let \(\pi (m)\) denote its period length, or the so-called Pisano period.

The next lemma gives a formula for the inverse of a modulo \(F_n\).

Lemma 2.1

For all integers \(a \ge 1\) and \(n \ge 3\) with \(\gcd (a, F_n) = 1\), we have that

$$\begin{aligned} (a^{-1} \bmod F_n) = \frac{b F_n + 1}{a} , \end{aligned}$$

where \(b := (-F_r^{-1} \bmod a)\) and \(r := (n \bmod \pi (a))\).

Proof

Since \(r \equiv n \pmod {\pi (a)}\), we have that \(F_r \equiv F_n \pmod a\). In particular, it follows that \(\gcd (a, F_r) = \gcd (a, F_n) = 1\). Hence, \(F_r\) is invertible modulo a, and consequently b is well defined. Moreover, we have that

$$\begin{aligned} bF_n + 1 \equiv -F_r^{-1} F_r + 1 \equiv 0 \pmod a , \end{aligned}$$

and thus \(c := (bF_n + 1) / a\) is an integer. On the one hand, we have that

$$\begin{aligned} ac \equiv b F_n + 1 \equiv 1 \pmod {F_n} . \end{aligned}$$

On the other hand, since \(b \le a - 1\) and \(n \ge 3\), we have that

$$\begin{aligned} 0 \le c \le \frac{(a - 1) F_n + 1}{a} = F_n - \frac{F_n - 1}{a} < F_n . \end{aligned}$$

Therefore, we get that \(c = (a^{-1} \bmod F_n)\), as desired.

Preliminaries on base-\(\varphi \) expansion

We need some basic results regarding the so-called base-\(\varphi \) expansion of real numbers, which was introduced by Bergman [2] in 1957 (see also [19]), and which is a particular case of non-integer base expansion (see, e.g., [15, 18]). Let \(\mathfrak {D}\) be the set of sequences in \(\{0, 1\}\) that have no two consecutive terms equal to 1, and that are not ultimately equal to the periodic sequence \(0, 1, 0, 1, \dots \). Then for every \(x \in {[0, 1)}\) there exists a unique sequence \(\varvec{\delta }(x) = (\delta _i(x))_{i \in \mathbb {N}}\) in \(\mathfrak {D}\) such that \(x = \sum _{i = 1}^\infty \delta _i(x) \varphi ^{-i}\). Precisely, \(\delta _i(x) = \lfloor T^{(i)}(x) \rfloor \) for every \(i \in \mathbb {N}\), where \(T^{(i)}\) denotes the ith iterate of the map \(T : {[0, 1)} \rightarrow {[0, 1)}\) defined by \(T(\hat{x}) := (\varphi \hat{x} \bmod 1)\) for every \(\hat{x} \in {[0, 1)}\). Furthermore, letting \(\mathcal {F} := \mathbb {Q}(\varphi ) \cap {[0,1)}\), if \(x \in \mathcal {F}\) then \(\varvec{\delta }(x)\) is ultimately periodic. In particular, if \(x \in \mathcal {F}\) is given as \(x = x_1 + x_2 \varphi \), where \(x_1, x_2 \in \mathbb {Q}\), then the preperiod and the period of \(\varvec{\delta }(x)\) can be effectively computed by finding the smallest \(i \in \mathbb {N}\) such that \(T^{(i)}(x) = T^{(j)}(x)\) for some \(j \in \mathbb {N}\) with \(j < i\). Conversely, for every ultimately periodic sequence \(\varvec{d} = (d_i)_{i \in \mathbb {N}}\) in \(\mathfrak {D}\) we have that the number \(x = \sum _{i=1}^\infty d_i \varphi ^{-i}\) belongs to \(\mathcal {F}\), and \(x_1, x_2 \in \mathbb {Q}\) such that \(x = x_1 + x_2 \varphi \) can be effectively computed in terms of the preperiod and period of \(\varvec{d}\) by using the formula for the sum of the geometric series. Moreover, in the case that x is a rational number in [0, 1) then \(\varvec{\delta }(x)\) is purely periodic [20].

The next lemma collects two easy inequalities for sums involving sequences in \(\mathfrak {D}\).

Lemma 3.1

For every sequence \((d_i)_{i \in \mathbb {N}}\) in \(\mathfrak {D}\) and for every \(m \in \mathbb {N} \cup \{\infty \}\), we have:

  1. (1)

    \(\sum _{i = 1}^m d_i \varphi ^{-i} \in {[0, 1)}\) and

  2. (2)

    \(\sum _{i = 1}^m d_i (-\varphi )^{-i} \in {(-1, \varphi ^{-1})}\).

Proof

Since \((d_i)_{i \in \mathbb {N}}\) belongs to \(\mathfrak {D}\), there exists \(k \in \mathbb {N}\) such that \(d_k = d_{k + 1} = 0\). Let k be the minimum integer with such property. Then

$$\begin{aligned} \sum _{i = 1}^\infty d_i \varphi ^{-i}&= \sum _{i = 1}^{k - 1} d_i \varphi ^{-i} + \sum _{i = k + 2}^{\infty } d_i \varphi ^{-i} < \sum _{j = 1}^{\lfloor k / 2\rfloor } \varphi ^{-(2j - 1)} + \sum _{i = k + 2}^{\infty } \varphi ^{-i} \\&= \left( 1 - \varphi ^{-2 \lfloor k / 2 \rfloor }\right) + \varphi ^{-k} \le 1 , \end{aligned}$$

and ()(1) is proved. Let us prove ()(2). On the one hand, we have

$$\begin{aligned} \sum _{i = 1}^m d_i (-\varphi )^{-i} \le \sum _{j = 1}^m d_{2j} \varphi ^{-2j} < \sum _{j = 1}^\infty \varphi ^{-2j} = \varphi ^{-1} , \end{aligned}$$

where the second inequality is strict because \(\mathfrak {D}\) does not contain sequences that are ultimately equal to \((0, 1, 0, 1, \dots )\). On the other hand, similarly, we have

$$\begin{aligned} \sum _{i = 1}^m d_i (-\varphi )^{-i} \ge -\sum _{j = 1}^m d_{2j - 1} \varphi ^{-(2j - 1)} > -\sum _{j = 1}^\infty \varphi ^{-(2j - 1)} = -1 . \end{aligned}$$

Thus ()(2) is proved.

The following lemma relates base-\(\varphi \) expansion and Zeckendorf representation.

Lemma 3.2

Let N be a positive integer and write \(N = x \varphi ^m / \sqrt{5}\) for some \(x \in \mathcal {F}\) and some integer \(m \ge 2\). Then the Zeckendorf representation of N is given by

$$\begin{aligned} N = \sum _{i = 1}^{m - 1} \delta _{m - i}(x) F_i . \end{aligned}$$

Moreover, we have \(\delta _m(x) = 0\).

Proof

Let \(R := N - \sum _{i = 1}^{m - 1} \delta _{m - i} (x) F_i\). We have to prove that \(R = 0\). Since R is an integer, it suffices to show that \(|R| < 1\). We have

$$\begin{aligned} \sqrt{5} N&= x \varphi ^m = \sum _{i = 1}^\infty \delta _i(x) \varphi ^{m - i} = \sum _{i = 1}^m \delta _i(x) \varphi ^{m - i} + \sum _{i = m + 1}^\infty \delta _i(x) \varphi ^{m - i} \\&= \sum _{i = 0}^{m - 1} \delta _{m - i}(x) \varphi ^i + \sum _{i = 1}^\infty \delta _{i + m}(x) \varphi ^{-i} \\&= \sum _{i = 0}^{m - 1} \delta _{m - i}(x) (\varphi ^i - \overline{\varphi }^i) + \sum _{i = 0}^{m - 1} \delta _{m - i}(x) \overline{\varphi }^i + \sum _{i = 1}^\infty \delta _{i + m}(x) \varphi ^{-i} \\&= \sqrt{5} \sum _{i = 1}^{m - 1} \delta _{m - i}(x) F_i + \sum _{i = 0}^{m - 1} \delta _{m - i}(x) (-\varphi )^{-i} + \sum _{i = 1}^\infty \delta _{i + m}(x) \varphi ^{-i} . \end{aligned}$$

Hence, we get that

$$\begin{aligned} \sqrt{5} R = \sum _{i = 0}^{m - 1} \delta _{m - i}(x) (-\varphi )^{-i} + \sum _{i = 1}^\infty \delta _{i + m}(x) \varphi ^{-i} . \end{aligned}$$

For the sake of contradiction, suppose that \(\delta _m(x) = 1\). Then \(\delta _{m+1}(x) = 0\) and, by Lemma 3.1, it follows that

$$\begin{aligned} \sqrt{5} R = 1 + \sum _{i = 1}^{m - 1} \delta _{m - i}(x) (-\varphi )^{-i} + \sum _{i = 2}^\infty \delta _{i + m}(x) \varphi ^{-i} \in (1 - 1 + 0, 1 + \varphi ^{-1} + \varphi ^{-1}) = (0, \sqrt{5}) , \end{aligned}$$

which is a contradiction, since R is an integer.

Therefore, \(\delta _m(x) = 0\) and, again by Lemma 3.1, we have

$$\begin{aligned} \sqrt{5} R = \sum _{i = 1}^{m - 1} \delta _{m - i}(x) (-\varphi )^{-i} + \sum _{i = 1}^\infty \delta _{i + m}(x) \varphi ^{-i} \in (-1 + 0, \varphi ^{-1} + 1) \subseteq (-\sqrt{5}, \sqrt{5}) , \end{aligned}$$

so that \(|R| < 1\), as desired.

The next lemma regards the base-\(\varphi \) expansions of the sum of two numbers.

Lemma 3.3

Let \(x, y \in {[0, 1)}\), \(m \in \mathbb {N}\), and put \(v := x + y \varphi ^{-m}\). Suppose that there exists \(\lambda \in \mathbb {N}\) such that \(\lambda + 2 \le m\) and \(\delta _\lambda (x) = \delta _{\lambda + 1}(x) = 0\). Then, putting

$$\begin{aligned} w := \sum _{i = \lambda + 2}^\infty \delta _i(x) \varphi ^{-i} + \sum _{i = m + 1}^\infty \delta _{i - m}(y) \varphi ^{-i} , \end{aligned}$$

we have that \(v, w \in {[0, 1)}\) and

$$\begin{aligned} \delta _i(v) = {\left\{ \begin{array}{ll} \delta _i(x) &{}\text { if } i \le \lambda , \\ \delta _i(w) &{}\text { if } i > \lambda , \end{array}\right. } \end{aligned}$$
(1)

for every \(i \in \mathbb {N}\).

Proof

From Lemma 3.1()(1), we have that

$$\begin{aligned} 0 \le w< \varphi ^{-(\lambda + 1)} + \varphi ^{-m} < \varphi ^{-(\lambda + 1)} + \varphi ^{-(\lambda + 2)} = \varphi ^{-\lambda } . \end{aligned}$$

Hence, \(w \in {[0,\varphi ^{-\lambda })} \subseteq {[0, 1)}\) and so \(w = \sum _{i = \lambda + 1}^\infty \delta _i(w) \varphi ^{-i}\). Therefore, recalling that \(\delta _{\lambda + 1}(x) = 0\), we get that

$$\begin{aligned} v&= x + y\varphi ^{-m} = \sum _{i = 1}^\infty \delta _i(x) \varphi ^{-i} + \sum _{i = 1}^\infty \delta _i(y) \varphi ^{-i-m} = \sum _{i = 1}^\infty \delta _i(x) \varphi ^{-i} + \sum _{i = m + 1}^\infty \delta _{i-m}(y) \varphi ^{-i} \\&= \sum _{i = 1}^{\lambda } \delta _i(x) \varphi ^{-i} + w = \sum _{i = 1}^{\lambda } \delta _i(x)\varphi ^{-i} + \sum _{i = \lambda + 1}^{\infty } \delta _i(w)\varphi ^{-i} , \end{aligned}$$

which is the base-\(\varphi \) expansion of v. (Note that \(\delta _\lambda (x) = 0\).) In particular, by Lemma 3.1()(1), we have that \(v \in {[0, 1)}\). Thus (1) follows.

Proof of Theorem 1.1

Fix an integer \(a \ge 3\). Let us begin by defining \(M, n_0, i_0\), and \(\varvec{z}^{(0)}, \dots , \varvec{z}^{(M - 1)}\). Put \(M := \pi (a)\). For each \(r \in \{0, \dots , M - 1\}\) with \(\gcd (a, F_r) = 1\), let \(b_r := (-F_r^{-1} \bmod a)\), \(x_r := b_r / a\), and \(\varvec{z}^{(r)} := \varvec{\delta }(x_r)\). Note that \(x_r \in (0, 1)\). Since \(x_r\) is a positive rational number, we have that \(\varvec{z}^{(r)}\) is a (purely) periodic sequence belonging to \(\mathfrak {D}\). Let \(\ell \) be the least common multiple of the period lengths of \(\varvec{z}^{(0)}, \dots , \varvec{z}^{(M - 1)}\), and put \(i_0 := \ell + 3\). Finally, let \(n_0 := \max \{i_0 + 1, \lceil \log (2a) / \!\log \varphi \rceil \}\).

Pick an integer \(n \ge n_0\) with \(\gcd (a, F_n) = 1\) and, for the sake of brevity, put \(r := (n \bmod M)\). From Lemma 2.1 and Binet’s formula (2), we get that

$$\begin{aligned} (a^{-1} \bmod F_n) = \frac{b_r F_n + 1}{a} = \frac{b_r(\varphi ^n - \overline{\varphi }^n)}{\sqrt{5} a} + \frac{1}{a} = (x_r + y_n \varphi ^{-n}) \frac{\varphi ^n}{\sqrt{5}} , \end{aligned}$$
(2)

where

$$\begin{aligned} y_n := \frac{\sqrt{5}}{a} - x_r (-\varphi )^{-n} . \end{aligned}$$

Since \(n \ge n_0\), it follows that \(y_n \in (0, 1)\) and \(x_r + y_n \varphi ^{-n} \in (0, 1)\). Therefore, from (2) and Lemma 3.2, we get that

$$\begin{aligned} (a^{-1} \bmod F_n) = \sum _{i \,=\, 1}^{n - 1} \delta _{n - i}(x_r + y_n \varphi ^{-n}) F_i . \end{aligned}$$

Since \(\varvec{\delta }(x_r)\) is (purely) periodic and belongs to \(\mathfrak {D}\), we have that \(\varvec{\delta }(x_r)\) contains infinitely many pairs of consecutive zeros. Furthermore, since the period length of \(\varvec{\delta }(x_r)\) is at most \(\ell \), we have that among every \(\ell +1\) consecutive terms of \(\varvec{\delta }(x_r)\) there are two consecutive zero. In particular, there exists \(\lambda = \lambda (r)\) such that \(n - \ell - 3 \le \lambda \le n - 2\) and \(\delta _{\lambda }(x_r) = \delta _{\lambda + 1}(x_r) = 0\). Consequently, by Lemma 3.3, we get that \(\delta _i(x_r + y_n \varphi ^{-n}) = \delta _i(x_r)\) for each positive integer \(i \le \lambda \) and, a fortiori, for each positive integer \(i \le n - i_0\). Therefore, we have that

$$\begin{aligned} (a^{-1} \bmod F_n)&= \sum _{i \,=\, i_0}^{n - 1} \delta _{n - i}(x_r) F_i + \sum _{i \,=\, 1}^{i_0 - 1} \delta _{n - i}(x_r + y_n \varphi ^{-n}) F_i \\&= \sum _{i \,=\, i_0}^{n - 1} z_{n - i}^{(r)} F_i + \sum _{i \,=\, 1}^{i_0 - 1} w_n^{(i)} F_i , \nonumber \end{aligned}$$
(3)

where \(\varvec{w}^{(1)}, \cdots , \varvec{w}^{(i_0)}\) are the sequences defined by \(w_n^{(i)} := \delta _{n - i}(x_r + y_n \varphi ^{-n})\). Note that, by construction,

$$\begin{aligned} z_1^{(r)}, z_2^{(r)}, \dots , z_{n - i_0}^{(r)}, w_n^{(i_0-1)}, w_n^{(i_0-2)}, \dots , w_n^{(1)} \end{aligned}$$

is a string in \(\{0,1\}\) with no consecutive zeros. Hence, (3) is the Zeckendorf representation of \((a^{-1} \bmod F_n)\).

It remains only to prove that \(\varvec{w}^{(1)}, \cdots , \varvec{w}^{(i_0)}\) are periodic. By (3) and the uniqueness of the Zeckendorf representation, it suffices to prove that

$$\begin{aligned} R(n) := (a^{-1} \bmod F_n) - \sum _{i \,=\, i_0}^{n - 1} z_{n - i}^{(r)} F_i = \sum _{i \,=\, 1}^{i_0 - 1} w_n^{(i)} F_i \end{aligned}$$
(4)

is a periodic function of n. From the last equality in (4), we have that \(0 \le R(n) < \sum _{i \,=\, 1}^{i_0 - 1} F_i\). (Actually, one can prove that \(0 \le R(n) < F_{i_0}\), but this is not necessary for our proof.) Fix a prime number \(p > \max \{a, \sum _{i \,=\, 1}^{i_0 - 1} F_i\}\). It suffices to prove that R(n) is periodic modulo p. Recalling that \((a^{-1} \bmod F_n) = (b_r F_n + 1) / a\) and that the sequence of Fibonacci numbers is periodic modulo p, it follows that \((a^{-1} \bmod F_n)\) is periodic modulo p. Hence, it suffices to prove that \(R^\prime (n) := \sum _{i = i_0}^{n - 1} z_{n - i}^{(r)} F_i\) is periodic modulo p. Using that \(\varvec{z}^{(r)}\) has period length dividing \(\ell \), we get that

$$\begin{aligned} R^\prime (n + \ell M) - R^\prime (n)&= \sum _{i \,=\, i_0}^{n + \ell M - 1} z_{n + \ell M - i}^{((n + \ell M) \bmod M)} F_i - \sum _{i = i_0}^{n - 1} z_{n - i}^{(r)} F_i \\&= \sum _{i \,=\, i_0}^{n + \ell M - 1} z_{n + \ell M - i}^{(r)} F_i - \sum _{i \,=\, i_0}^{n - 1} z_{n - i}^{(r)} F_i \\&= \sum _{i \,=\, n}^{n + \ell M - 1} z_{n + \ell M - i}^{(r)} F_i + \sum _{i \,=\, i_0}^{n - 1} (z_{n + \ell M - i}^{(r)} - z_{n - i}^{(r)}) F_i \\&= \sum _{j \,=\, 1}^{\ell M} z_{j}^{(r)} F_{n + \ell M - j} , \end{aligned}$$

which is a linear combination of sequences that are periodic modulo p. Hence \(R^\prime (n)\) is periodic modulo p. The proof is complete.

Remark 4.1

The proof of Theorem 1.1 provides a way to compute the positive integers \(M, i_0, n_0\) and the periods of the periodic sequences \(\varvec{z}^{(0)}, \dots , \varvec{z}^{(M - 1)}\) and \(\varvec{w}^{(1)}, \dots , \varvec{w}^{(i_0)}\). Indeed, going through the proof, we have that: \(M = \pi (a)\) is the Pisano period of a, which can be computed in an obvious way; \(\varvec{z}^{(r)} = \varvec{\delta }\big ((-F_r^{-1}\!\!\! \mod a) / a\big )\) and so the period of \(\varvec{z}^{(r)}\) can be computed as explained at the beginning of Section 3; \(i_0\) and \(n_0\) have simple formulas in terms of \(\ell \), which is the least common multiple of the period lengths of \(\varvec{z}^{(0)}, \dots , \varvec{z}^{(M - 1)}\). Finally, the periods of \(\varvec{w}^{(1)}, \dots , \varvec{w}^{(i_0)}\) can be computed from (4) and the fact that R(n) is periodic with period length at most \(\pi (p)^2 \ell M\), which follows from the arguments after (4). However, note that proceeding in this way might be impractical, since \(\ell \) might be exponential in M, and thus p might be double exponential in M; making the search for the periods of \(\varvec{w}^{(1)}, \dots , \varvec{w}^{(i_0)}\) extremely long.