INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A23

A GENERALIZATION OF THE SMARANDACHE FUNCTION TO SEVERAL VARIABLES

Norbert Hungerbühler

Department of Mathematics, University of Fribourg, Pérolles, 1700 Fribourg, Switzerland

                       norbert.hungerbuehler@unifr.ch

Ernst Specker

Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland

                           specker@math.ethz.ch

Received: 1/16/06, Accepted: 6/30/06, Published: 10/06/06

Abstract

We investigate polyfunctions in several variables over ℤ_{n}. We show in particular how the problem of determining the cardinality of the ring of these functions leads to a natural generalization of the classical Smarandache function.

1. Introduction

Let us consider the ring ℤ_{n} := ℤ/nℤ, n > 1, and a function

f : ℤ_{n}^{d} → ℤ_{n}

of d variables in ℤ_{n} with values in ℤ_{n}. Such a function is called a polyfunction if there exists a polynomial

p ∈ ℤ_{n}[x_{1}, …, x_{d}]

such that

f(x) ≡ p(x) mod n ∀x = ⟨x_{1}, …, x_{d}⟩ ∈ ℤ_{n}^{d}.

The set of polyfunctions of d variables in ℤ_{n} with values in ℤ_{n}, equipped with pointwise addition and multiplication, is a ring with unit element. We denote this ring by G_{d}(ℤ_{n}), or, for simplicity, by G(ℤ_{n}) in the case of only one variable.

In the present article, we investigate polyfunctions in several variables over ℤ_{n}. We show in particular how the problem of determining the cardinality of the ring of these functions leads to a natural generalization of the classical Smarandache function (named after [17])

\begin{matrix}s : ℕ & → ℕ \\ n & ↦ s(n) := min\{k ∈ ℕ : n | k!\}, \end{matrix}  (1)

which was studied by Lucas in [10] for powers of primes, and by Kempner in [8] and Neuberg in [12] for general n. Indeed, s(n) is the minimal degree of a normed polynomial which vanishes (as a function) identically in ℤ_{n} (see [5]). The key is then to reformulate the above definition by setting

s(n) = |{k ∈ ℕ_{0} : n ∤ k!}|.

This definition then generalizes in a natural way to d > 1 dimensions (see (10) and (11)), where the number can be interpreted as the number of irreducible monomials x^{k} modulo n (see Section 5).

The number of polyfunctions in G_{d}(ℤ_{n}) is multiplicative in n (see Section 5). It therefore suffices to compute the values for n = p^{m}, p prime. By analysing the structure of the additive group of G_{d}(ℤ_{p^{m}}), which is completely described in Proposition 7, we find

|G_{d}(ℤ_{p^{m}})| = p^{∑_{i=1}^{m} s_{d}(p^{i})}

(see Theorem 6). However, the factors p^{s_{d}(p^{i})} do not correspond to additive subgroups of G_{d}(ℤ_{p^{m}}).

In Section 3 we present a characterization which allows us to test whether a given function f : ℤ_{n}^{d} → ℤ_{n} is a polyfunction, and if so, to determine a polynomial representative of f. In Section 4 we characterize the units in the ring G_{d}(ℤ_{n}).

We conclude this introduction with a short overview on the history of polyfunctions. The study of polyfunctions in one variable goes back to Kempner who discussed polyfunctions over ℤ_{n} in connection with Kronecker modular systems [9]. He also gave a formula for the number of polyfunctions over ℤ_{n}. Later, Carlitz investigated properties of polyfunctions over ℤ_{p^{n}} for p prime [2]. Keller and Olson gave a simplified proof of Kempner’s formula [7] and also determined the number of polyfunctions which represent a permutation of ℤ_{p^{n}}. Null-polynomials over ℤ_{n} (i.e., polynomials which represent the zero-function) have been investigated by Singmaster [15]. Certain aspects of polyfunctions in several variables over ℤ_{n} were addressed in [11]. Recently, polyfunctions from ℤ_{n} to ℤ_{m} have attracted increasing attention (see [3], [4] and [1]). The focus there is to find conditions on the pair ⟨m, n⟩ such that all functions (or certain subclasses) from ℤ_{n} to ℤ_{m} are polyfunctions. In [13] and [14] polyfunctions over a general ring were discussed: the question asked being “for which rings R one can find a ring S, such that all functions on R can be represented by polynomials over S?”

2. Notation, Definitions and Basic Facts

In order to keep the formulas short, we use the following multi-index notation. For k = ⟨k_{1}, k_{2}, …, k_{d}⟩ ∈ ℕ_{0}^{d} and x := ⟨x_{1}, x_{2}, …, x_{d}⟩, let

x^{k} := ∏_{i=1}^{d} x_{i}^{k_{i}}

and

k! := ∏_{i=1}^{d} k_{i}!.

Furthermore, we write

|k| := ∑_{i=1}^{d} k_{i}

and

(\begin{matrix}x \\ k \end{matrix}) := ∏_{i=1}^{d} \binom{x_{i}}{k_{i}}.

Let e_{i} := ⟨0, …, 0, 1, 0, …, 0⟩ ∈ ℤ_{n}^{d}, with the 1 at place i. Then, we define the (forward) partial difference operator Δ by

Δ_{i}g(x) := g(x + e_{i}) − g(x)
Δ_{i}^{0} := identity
Δ_{i}^{k} := Δ_{i} ∘ Δ_{i}^{k−1}.

For a multi-index k, let

Δ^{k} := Δ_{1}^{k_{1}} ∘ … ∘ Δ_{d}^{k_{d}}.

Notice that the Δ operators commute and that Δ^{k^{1}} ∘ Δ^{k^{2}} = Δ^{k_{1}+k_{2}}. We recall that

Δ^{r}g(x) = ∑_{k⩽r}g(x + r − k)(−1)^{|k|}(\begin{matrix}r \\ k \end{matrix}),  (2)

where k ⩽ r means 0 ⩽ k_{i} ⩽ r_{i} (see e.g. [16]). A polynomial p equals its “Taylor expansion”

p(x) = ∑_{|k|⩽deg(p)}Δ^{k}p(0)(\begin{matrix}x \\ k \end{matrix})  (3)

(see e.g. [6]). Observe, that the monomial x^{l} defines by ((x + n)^{l})_{n∈ℤ} for any fixed x an arithmetic sequence of order l. Therefor, one easily checks by induction, that

Δ^{r}x^{l} = \{ \begin{matrix}0 & if r > l, \\ r! & if r = l. \end{matrix}  (4)

Hence, the summation in (3) can be restricted to the shadow of p, i.e., the multi-indices k with the property that 0 ⩽ k ⩽ r for a monomial x^{r} in p. Indeed, if k does not belong to the shadow of p, then Δ^{k}p(0) = 0 by (4).

It is well known (see e.g. [6]) that a polynomial p has integer coefficients if and only if the condition

k! | Δ^{k}p(0)  (5)

holds for all k in the shadow of p (for other values of k, the condition (5) is trivially satisfied by the previous remark).

3. Characterization of Polyfunctions

Let f : ℤ_{n}^{d} → ℤ_{n} be a polyfunction, i.e., there exists a polynomial p ∈ ℤ_{n}[x_{1}, …, x_{d}] such that

f(x) ≡ p(x) mod n for all x ∈ ℤ_{n}^{d}.  (6)

Since for all x ∈ ℤ_{n}

∏_{i=0}^{n−1}(x − i) = 0 in ℤ_{n},

we may assume, without loss of generality, that the degree of p is, in each variable separately, strictly less than n. Thus, in ℤ_{n} we have for arbitrary x ∈ ℤ_{n}^{d},

f(x) =^{by (6)} p(x)
=^{by (3)} ∑_{k_{i}<n} Δ^{k}p(0)(\begin{matrix}x \\ k \end{matrix})

=^{by (6)} \underbrace{∑_{k_{i}<n} Δ^{k}f(0)(\begin{matrix}x \\ k \end{matrix})}_{=:h(x)} .

Hence, the polynomial h represents f, but it does not necessarily have integer coefficients. However, observing (5) and exploiting the fact that in ℤ_{n},

Δ^{k}p(0) = Δ^{k}f(0)

holds for all k, we obtain:

Lemma 1 If f : ℤ_{n}^{d} → ℤ_{n} is a polyfunction, then (i) for all multi-indices k with components k_{i} < n, there exist α_{k} ∈ ℤ such that for the

numbers β_{k} := Δ^{k}f(0) + α_{k}n,

k! | β_{k},  (7)

and

(ii) the polynomial ∑_{k_{i}<n} β_{k}(\begin{matrix}x \\ k \end{matrix}) has integer coefficients and represents f.

From (7) it follows, that

(n, k!) | Δ^{k}f(0) ^{1}  (8)

for all k with k_{i} < n. We will show now that this condition characterizes polyfunctions. To this end, we consider an arbitrary function f : ℤ_{n}^{d} → ℤ_{n}. Since there exists an interpolation polynomial for f, with degree in each variable strictly less than n, which agrees with f on the set {0, 1, …, n − 1}^{d}, we infer from (3) that, in ℤ_{n},

f(x) = ∑_{k_{i}<n} Δ^{k}f(0)(\begin{matrix}x \\ k \end{matrix})

for all x ∈ ℤ_{n}^{d}. If condition (8) is satisfied for f, we find coefficients β_{k} = Δ^{k}f(0) + α_{k}n, as above in Lemma 1(i), such that k! | β_{k}. Hence, in ℤ_{n}

f(x) = ∑_{k_{i}<n} β_{k}(\begin{matrix}x \\ k \end{matrix}) mod ( ∑_{k=0}^{n−1} β_{k}\binom{x}{k}, n),

for all x ∈ ℤ_{n}^{d}. In other words, condition (8) implies that f is a polyfunction and we have the following characterization:

Theorem 2 f : ℤ_{n}^{d} → ℤ_{n} is a polyfunction over ℤ_{n} if and only if (n, k!) | Δ^{k}f(0) for all multi-indices k with k_{i} < n.

4. The Inverse of a Polyfunction

Let f : ℤ_{n}^{d} → ℤ_{n}. Then f is invertible (i.e., there exists a function g : ℤ_{n}^{d} → ℤ_{n}, such that for all x ∈ ℤ_{n}^{d} there holds f(x)g(x) = 1) if and only if Image(f) ⊂ U(ℤ_{n}). Here, U(ℤ_{n}) denotes the multiplicative group of units in ℤ_{n}. We want to show that the same characterization holds for invertible polyfunctions over ℤ_{n}.

Proposition 3 A polyfunction f : ℤ_{n}^{d} → ℤ_{n} is invertible in the ring of polyfunctions (and hence a unit in G_{d}(ℤ_{n})) if and only if

Image(f) ⊂ U(ℤ_{n}).

Proof. The necessity of the condition is trivial. In order to prove that it is also sufficient, let k := lcm{ord(x) | x ∈ U(ℤ_{n})}^{2}. Then, if p denotes a polynomial representing f, we have

p^{k}(x) = 1 in ℤ_{n}

for all x ∈ ℤ_{n}^{d}. Hence, the polynomial p^{k−1} represents the inverse of f. □

5. The Number of Polyfunctions

Let a be an element of ℤ_{n}. We say, the monomial ax^{k} ∈ ℤ_{n}[x] is reducible (modulo n) if a polynomial p(x) ∈ ℤ_{n}[x] exists with deg(p) < |k| such that ax^{k} ≡ p(x) mod n for all x ∈ ℤ_{n}^{d}. Moreover, we say that ax^{k} is weakly reducible (modulo n) if ax^{k} ≡ p(x) mod n for all x ∈ ℤ_{n}^{d}, for a polynomial p ∈ ℤ_{n}[x] with deg(p) ⩽ |k| (instead of deg(p) < |k|), and such that x^{k} (or a multiple of it) does not appear as a monomial in p.

The following lemma characterizes the tuples k for which ax^{k} is (weakly) reducible.

Lemma 4 (i) If ax^{k} ∈ ℤ_{n}[x] is weakly reducible modulo n, then n | ak!. (ii) If n | ak!, then ax^{k} is reducible modulo n.

In particular, a monomial is reducible if and only if it is weakly reducible.

Proof. (i) We assume, that p(x) reduces ax^{k} weakly. Hence, q(x) := ax^{k} − p(x) is a nullpolynomial (i.e., a polynomial which represents the zero-function) in d variables over ℤ_{n}. Then, we write q in the form

q(x) = ∑_{\begin{matrix}l∈ℕ_{0}^{d} \\ |l|⩽|k| \end{matrix}}q_{l}x^{l}  (9)

for suitable coefficients q_{l} ∈ ℤ_{n}, with q_{k} = a. Using the linearity of the Δ operator, we obtain that, modulo n,

0 = Δ^{k}q(x) =^{(9)} ∑_{\begin{matrix}l∈ℕ_{0}^{d} \\ |l|⩽|k| \end{matrix}}q_{l}Δ^{k}x^{l} =^{(4)} ak!.

In fact, all terms in the above sum with l ≠ k vanish by (4), since |l| ⩽ |k| and l ≠ k implies that k is not in the shadow of x^{l}. And the only remaining term, Δ^{k}x^{k}, equals k!, again by (4).

2 lcm(M) is the least common multiple of all integer numbers in a finite set M. ord(x) denotes the order of an element x in a finite multiplicative group G, i.e., ord(x) is the smallest number k ∈ ℕ such that x^{k} = 1.

(ii) We assume, that n | ak!. Then, the polynomial

q(x) := a ∏_{i=1}^{d} ∏_{l=1}^{k_{i}}(x_{i} + l) = ak!\binom{x + k}{k}

is a null-polynomial over ℤ_{n} and the term of maximal degree is ax^{k}. Hence, q(x) − ax^{k} reduces to ax^{k}. □

Lemma 4 allows us to count the number of monomials x^{k}, k ∈ ℕ_{0}^{d}, which are not reducible. Let

S_{d}(n) := {k ∈ ℕ_{0}^{d} : n ∤ k!}  (10)

denote the set of multi-indices k such that x^{k} is not reducible modulo n. Its cardinality is the natural generalization of the Smarandache function to the case of several variables:

s_{d}(n) := |S_{d}(n)|.  (11)

Of course, for d = 1 the function s_{1} agrees with the usual number theoretic Smarandache function (see introduction)—except for n = 1, since s(1) = 1, but s_{1}(1) = 0. Actually, by defining s(n) := min{k ∈ ℕ_{0} : n | k!} (i.e., the minimum is taken over k ∈ ℕ_{0} rather than over k ∈ ℕ), this discrepancy could be removed. Incidentally, Kempner originally defined s(1) = 1 in [8], but changed to s(1) = 0 in [9]. The following table displays s_{d}(n) for the first few values of d and n.

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ s_{1} & 0 & 2 & 3 & 4 & 5 & 3 & 7 & 4 & 6 & 5 & 11 & 4 & 13 \\ s_{2} & 0 & 4 & 9 & 12 & 25 & 9 & 49 & 16 & 27 & 25 & 121 & 13 & 169 \\ s_{3} & 0 & 8 & 27 & 32 & 125 & 27 & 343 & 56 & 108 & 125 & 1331 & 39 & 2197 \\ s_{4} & 0 & 16 & 81 & 80 & 625 & 81 & 2401 & 176 & 405 & 625 & 14641 & 113 & 28561 \end{matrix}

Table 1: Values of s_{d}(n)

Before we now start to compute the number of Ψ_{d}(p^{m}) poyfunctions in G_{d}(ℤ_{p^{m}}), it is useful to include a general remark. The notion of the ring of polyfunctions G(ℤ_{n}) generalizes in a natural way to the ring G(R) of polyfunctions over an arbitrary ring R. If R and S are commutative rings with unit element, then G(R ⊕ S) and G(R) ⊕ G(S) are isomorphic as rings in the obvious way. In particular, since ℤ_{n} ⊕ℤ_{m} ≅ ℤ_{nm} if m and n are relatively prime, we have that G(ℤ_{nm}) ≅ G(ℤ_{n}) ⊕ G(ℤ_{m}) if (m, n) = 1.

Analogously in several variables, we have the decomposition G_{d}(ℤ_{mn}) ≅ G_{d}(ℤ_{m})⊕G_{d}(ℤ_{n}) if (m, n) = 1. This means, e.g., that the number Ψ_{d}(n) of polyfunctions in G_{d}(ℤ_{n}) is multiplicative in n. Therefore, we may restrict ourselves to the case n = p^{m} for p prime.

Now, the strategy to count the number of polyfunctions is to seek a unique standard representation of such functions by a polynomial. Such a representation is given in Proposition 5 below. Then, we will just have to count these representing polynomials. Let us first consider the case of one variable. Obviously,

∏_{i=1}^{s_{1}(n)}(x − i) = \binom{x + s_{1}(n)}{s_{1}(n)}s_{1}(n)!

is a normed^{3} null-polynomial in G(ℤ_{n}), and from Lemma 4 it follows in particular that there is no polynomial of smaller degree with this property. Therefore, every polyfunction in one variable over ℤ_{n} has a (not necessarily unique) representing polynomial of degree strictly less than s_{1}(n) (and here s_{1}(n) cannot be replaced by a smaller number). Basically by the same argument, Lemma 4 allows us to construct a unique representation of every polyfunction in d variables over ℤ_{p^{m}}.

Proposition 5 Every polyfunction f ∈ G_{d}(ℤ_{p^{m}}) has a unique representation of the form

f(x) ≡ ∑_{i=1}^{m} p^{m−i} ∑_{k∈S_{d}(p^{i})} α_{ki}x^{k}  (12)

where α_{ki} ∈ ℤ_{p}.

Proof. It is common to write n = ∏ p^{ν_{p}(n)} for the prime decomposition of a positive integer n. We adopt this notation and write

ν_{p}(k!) = max{x ∈ ℕ_{0} : p^{x} | k!}

for the number of factors p in k!. Notice that ν_{p}(k!) < i if and only if k ∈ S_{d}(p^{i}). Then, as an immediate consequence of Lemma 4, we obtain, that every polyfunction f ∈ G_{d}(ℤ_{p^{m}}) has a unique representation of the form

f(x) ≡ ∑_{\begin{matrix}k∈ℕ_{0}^{d} \\ ν_{p}(k!)<m \end{matrix}} α_{k}x^{k},  (13)

where α_{k} ∈ {0, 1, …, p^{m−ν_{p}(k!)} − 1}. Since, on the other hand, every number α_{k} ∈ {0, 1, …, p^{m−ν_{p}(k!)} − 1} has a unique representation of the form

α_{k} = ∑_{{i⩽m: k∈S_{d}(p^{i})}}p^{m−i}α_{ki}

for certain coefficients α_{ki} ∈ ℤ_{p}, we can rewrite (13) such that we obtain (12). □

As an immediate consequence of Proposition 5, we now get the formula for the number of poyfunctions in the following theorem. Observe that we use the notation exp_{p} a := p^{a} for better readability.

3i.e., its leading coefficient is 1

Theorem 6 The number of polyfunctions in G_{d}(ℤ_{p^{m}}), p prime, is given by

Ψ_{d}(p^{m}) = exp_{p}(∑_{i=1}^{m} s_{d}(p^{i})).

Example. To compute the number of polyfunctions Ψ_{2}(8) in two variables over ℤ_{8}, we need:

S_{2}(2) = {⟨k_{1}, k_{2}⟩ : 0 ⩽ k_{1} ⩽ 1, 0 ⩽ k_{2} ⩽ 1}
s_{2}(2) = 4
S_{2}(4) = {⟨k_{1}, k_{2}⟩ : 0 ⩽ k_{1} ⩽ 3, 0 ⩽ k_{2} ⩽ 3, k_{1}k_{2} < 4}
s_{2}(4) = 12
S_{2}(8) = {⟨k_{1}, k_{2}⟩ : 0 ⩽ k_{1} ⩽ 3, 0 ⩽ k_{2} ⩽ 3}
s_{2}(8) = 16.

This gives Ψ_{2}(8) = 2^{4+12+16} = 2^{32}. ◯

Notice that the formulas (13) and (12) reflect the structure of the additive group of G_{d}(ℤ_{p^{m}}). In fact

A_{dk}(ℤ_{p^{m}}) := {f ∈ G_{d}(ℤ_{p^{m}}) : f(x) ≡ αx^{k}, α ∈ ℤ_{p^{m−νp(k!)}}} ≅ ℤ_{p^{m−νp(k!)}}

are additive subgroups in G_{d}(ℤ_{p^{m}}) and hence, by (13):

Proposition 7 (G_{d}(ℤ_{p^{m}}), +) ≅ ⨁_{\begin{matrix}k∈ℕ_{0}^{d} \\ ν_{p}(k!)<m \end{matrix}} ℤ_{p^{m−νp(k!)}}.

As an immediate consequence of Theorem 6 and Proposition 7, we note the following identity:

Corollary 8 ∑_{i=1}^{m} s_{d}(p^{i}) = ∑_{k∈S_{d}(p^{m})}(m − ν_{p}(k!)) = m s_{d}(p^{m}) − ∑_{k∈S_{d}(p^{m})} ν_{p}(k!).

For completeness, we add an explicit formula for Ψ_{d}(n) = |G_{d}(ℤ_{n})| for general n. We start from the identity

Ψ_{d}(n) = Ψ_{d}(∏_{i=1}^{k} p_{i}^{ν_{p_{i}}(n)}) = ∏_{i=1}^{k} Ψ_{d}(p_{i}^{ν_{p_{i}}(n)}).

By taking the logarithm on both sides and using Theorem 6 we obtain

ln Ψ_{d}(n) = ∑_{i=1}^{k} ln Ψ_{d}(p_{i}^{ν_{p_{i}}(n)})

= ∑_{i=1}^{k} ln p_{i} ∑_{j=1}^{ν_{p_{i}}(n)} s_{d}(p_{i}^{j}).  (14)

Observe that the Mangoldt function

Λ : ℕ → ℕ, x ↦ \begin{cases}ln p & if x = p^{k}, p prime, k ⩾ 1 \\ 0 & else \end{cases}

allows us to simplify (14) further and to obtain

ln Ψ_{d}(n) = ∑_{i=1}^{k} ∑_{j=1}^{ν_{p_{i}}(n)} s_{d}(p_{i}^{j})Λ(p_{i}^{j}).

Since the Mangoldt function is zero on all numbers which are not powers of primes, this last expression can be interpreted as a sum over all divisors of n. Moreover, since Λ(1) = 0, the value of s_{d}(1) is irrelevant. Hence, using the Dirichlet convolution

(f ∗ g)(n) = ∑_{d|n} f(\frac{n}{d})g(d)

with f ≡ 1 and g = s_{d} Λ, we arrive at

ln Ψ_{d}(n) = (1 ∗ (s_{d} Λ))(n).

Hence, we have the following Theorem:

Theorem 9 The number Ψ_{d}(n) of polyfunctions in G_{d}(ℤ_{n}), n > 1, is given by

Ψ_{d}(n) = e^{1∗(s_{d} Λ)(n)}.

6. The Towers of Hanoï

The Smarandache function can be used to solve the Towers of Hanoï problem. In Theorem 6, for p = 2 and one variable, we need the numbers

s(2^{k}).

Let us consider the first difference sequence

a_{k} := s(2^{k}) − s(2^{k−1}), k = 1, 2, 3, …

The sequence starts with

(a_{k})_{k∈ℕ} = (2, 2, \underbrace{0}_{ε_{1}}, 2, 2, \underbrace{0, 0}_{ε_{2}}, 2, 2, \underbrace{0}_{ε_{3}}, 2, 2, \underbrace{0, 0, 0}_{ε_{4}}, 2, 2, \underbrace{0}_{ε_{5}}, 2, 2, …).

Two 2s alternate with groups of ε_{k} 0s. The sequence

(ε_{k})_{k∈ℕ} = (1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5,
1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, …),

with the property that 2^{ε_{k}} divides exactly 2k, is now indeed the solution of the Towers of Hanoï. It provides the number of the disk, which is to be relocated in the k-th move.

Alternatively, knowing the solution of the Towers of Hanoï one has an efficient way to compute s(2^{k}).

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