# On the closed form of Z([p.2.sup.k]), p = [2.sup.q] - 1.

[section] 1. Introduction

The pseudo Smarandache function, Z(n), introduced by Kashihara [1], is as follows:

Definition 1.1. For any integer n > 1, the pseudo Smarandache function Z(n) is the smallest positive integer m such that 1 + 2 + ... + m = m(m+1)/2 is divisible by n. Thus,

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where [Z.sup.+] is the set of all positive integers.

Some of the properties satisfied by Z(n) are given in Majumdar [2], which also gives the explicit forms of Z(n) in some particular cases. It seems that there is no single closed form expression of Z(n).

Of particular interest is the values of Z([p.2.sup.k]), where p is a prime and k [member of] [Z.sup.+]. Majumdar [2] gives the explicit forms of Z([p.2.sup.k]) for p = 3, 5, 7, 11, 13, 17, 19, 31. In this paper, we derive the explicit form of Z([p.2.sup.k]) when p is a prime of the form p = [2.sup.q] - 1. This is given in the next section.

[section] 2. Closed form expression of Z([p.2.sup.k]), p = [2.sup.q] - 1

First note that, for any integer [alpha] [greater than or equal to] 1,

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Therefore, it follows by induction on a that p divides [2.sup.qa] - 1 for any integer a [greater than or equal to] 1.

The closed form expression of Z([p.2.sup.k]), when p = [2.sup.q] - 1, is given in the theorem below. Theorem 2.1. Let p be a prime of the form p = [2.sup.q] - 1, q [greater than or equal to] 1. Then

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Proof. First note that, if p = [2.sup.q] - 1 is prime, then by the Cataldi-Fermat Theorem, q must be a prime (see, for example, Theorem 4 in Daniel Shanks [3]). Now, by definition,

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Here, [2.sup.k+1] must divide one of m and m + 1, and p must divide the other. We now consider all the possible cases below:

Case (1): When k is of the form k = qa for some integer a [greater than or equal to] 1. Let p = 2P + 1. Now, since

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it follows that p divides [P.2.sup.k+1] + 1, so that [p.2.sup.k+1] divides [P.2.sup.k+1](P.2k+1 + 1). Therefore, the minimum m in (1) can be taken as [P.2.sup.k+1], and hence,

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Case (2): When k is of the form k = qa + 1 for some integer a [greater than or equal to] 0. Here,

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so that, p divides [2.sup.k+q-1] - 1 and hence, [p.2.sup.k+1] divides [2.sup.k+q-1] ([2.sup.k+q-1] - 1). Thus, in this case, the minimum m in (1) may be taken as [2.sup.k+q-1] - 1, so that Z([p.2.sup.k]) = [2.sup.k+q-1] - 1.

Case (3): When k is of the form k = qa + 2 for some integer a [greater than or equal to] 0. In this case, since

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it follows that, [p.2.sup.k+1] divides [2.sup.k+q-2]([2.sup.k+q-2] - 1), and hence, Z([p.2.sup.k]) = [2.sup.k+q-2] - 1.

Case (q): When k is of the form k = qa + q - 1 for some integer a [greater than or equal to] 0. Here,

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so that [p.2.sup.k+1] divides [2.sup.k+1]([2.sup.k+1] - 1), and consequently, Z([p.2.sup.k]) = [2.sup.k+1] - 1. All these complete the proof of the theorem.

[section] 3. Some special cases

Some special cases of Theorem 2.1 are Z([3.2.sup.k]) (corresponding to q = 2), Z([7.2.sup.k]) (corresponding to q = 3), and Z([31.2.sup.k]) (corresponding to q = 4). The explicit forms of Z([3.2.sup.k]), Z([7.2.sup.k]) and Z([31.2.sup.k]) are given below.

Corollary 3.1. For any integer k [greater than or equal to] 1,

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Proof. Since this case corresponds to q = 2, q divides k if and only if k is even. The result then follows from Theorem 2.1 immediately.

Corollary 3.2. For any integer k [greater than or equal to] 1,

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Proof. This case corresponds to q = 3, and so, there are three possibilities, namely, k is one of the three forms k = 3a, 3a + 1, 3a + 2. Then, appealing to Theorem 2.1, we get the desired expression for Z (7.2k).

Corollary 3.3. For any integer k [greater than or equal to] 1,

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Proof. Here, k can be one of the five forms k = 5a, 5a +1, 5a + 2, 5a + 3, 5a + 4. When k = 5a, by Theorem 2.1, Z([31.2.sup.k]) = [30.2.sup.k] = [15.2.sup.k+1]. Similarly, the other four cases follow from Theorem 2.1.

References

[1] Kashihara Kenichiro, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, USA, 1996.

[2] A. A. K. Majumdar, Wandering in the World of Smarandache Numbers, ProQuest Information and Learning, USA, 2010.

[3] Daniel Shanks, Solved and Unsolved Problems in Number Theory, Spartan Books, Washington D.C., USA, 1(1964).

A. A. K. Majumdar

Ritsumeikan Asia-Pacific University, 1-1 Jumonjibaru, Beppu-shi, 8748577, Japan