*Article* **Iterating the Sum of Möbius Divisor Function and Euler Totient Function**

#### **Daeyeoul Kim 1, Umit Sarp 2,\* and Sebahattin Ikikardes <sup>2</sup>**


Received: 15 October 2019; Accepted: 6 November 2019; Published: 10 November 2019

**Abstract:** In this paper, according to some numerical computational evidence, we investigate and prove certain identities and properties on the absolute Möbius divisor functions and Euler totient function when they are iterated. Subsequently, the relationship between the absolute Möbius divisor function with Fermat primes has been researched and some results have been obtained.

**Keywords:** Möbius function; divisor functions; Euler totient function

**MSC:** 11M36; 11F11; 11F30

#### **1. Introduction and Motivation**

Divisor functions, Euler *ϕ*-function, and Möbius *μ*-function are widely studied in the field of elementary number theory. The absolute Möbius divisor function is defined by

$$\mathcal{U}(n) := |\sum\_{d|n} d\mu(d)|.$$

Here, *n* is a positive integer and *μ* is the Möbius function. It is well known ([1], p. 23) that

$$\wp(n) = \sum\_{d|n} \mu(d) \frac{n}{d'},$$

where *ϕ* denotes the Euler *ϕ*-function (totient function). If *n* is a square-free integer, then *U*(*n*) = *ϕ*(*n*). The first twenty values of *U*(*n*) and *ϕ*(*n*) are given in Table 1.


**Table 1.** Values of *U*(*n*) and *ϕ*(*n*) (1 ≤ *n* ≤ 20).

Let *U*<sup>0</sup> (*n*) := *n*, *U* (*n*) := *U*<sup>1</sup> (*n*) and *Um* (*n*) := *Um*−<sup>1</sup> (*U* (*n*)), where *m* ≥ 1.

Next, to study the iteration properties of *Um* (*n*) ( resp., *ϕ<sup>m</sup>* (*n*)), we say the order (resp., class) of *n*, *m*-gonal (resp., *m* -gonal) absolute Möbius ( resp., totient) shape numbers, and shape polygons derived from the sum of absolute Möbius divisor (resp., Euler totient) function are as follows.

**Definition 1.** *(Order Notion) To study when the positive integer Um* (*n*) *is terminated at one, we consider a notation as follows. The order of a positive integer n* > 1 *denoted Ord*2(*n*) = *m, which is the least positive integer m when Um* (*n*) = 1 *and Um*−<sup>1</sup> (*n*) = 1*. The positive integers of order* 2 *are usually called involutions. Naturally, we define Ord*2(1) = 0*. The first* 20 *values of Ord*2(*n*) *and C*(*n*) + 1 *are given by Table 2. See [2].*

*n* **1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20** *Ord*2(*n*) 012122312 2 3 2 3 3 2 1 2 2 3 2 *C*(*n*) + 1012232333 3 4 3 4 3 4 4 5 3 4 4

**Table 2.** Values of *Ord*2(*n*) and *C*(*n*) + 1 (1 ≤ *n* ≤ 20).

**Remark 1.** *Define <sup>ϕ</sup>*0(*n*) = *n, <sup>ϕ</sup>*1(*n*) = *<sup>ϕ</sup>*(*n*) *and <sup>ϕ</sup>k*(*n*) = *<sup>ϕ</sup>*(*ϕk*−1(*n*)) *for all <sup>k</sup>* ≥ <sup>2</sup>*. Shapiro [2] defines the class number C*(*n*) *of n by that integer C such that ϕC*(*n*) = 2*. Some values of Ord*2(*n*) *are equal to them of C*(*n*) + 1*. Shapiro [2] defined C*(1) + 1 = *C*(2) + 1 = 1*. Here, we define C*(1) + 1 = 0 *and C*(2) + 1 = 1*. A similar notation of Ord*(*n*) *is in [3].*

**Definition 2.** *(Absolute Möbius m-gonal shape number and totient m -gonal shape number) If Ord*2(*n*) = *m* − 2 *(resp., C*(*n*) + 1 = *m , we consider the set* {(*i*, *Ui*(*n*))|*i* = 0, ..., *m* − 2} *(resp.,* {(*i*, *ϕi*(*n*))|*i* = 0, ..., *m* − 2} *and add* (0, 1)*. We then put Vn* = {(*i*, *Ui*(*n*))|*i* = 0, ..., *m* − 2 } ∪ {(0, 1)} *(resp., Rn* = {(*i*, *ϕi*(*n*))|*i* = 0, ..., *m* − 2 } ∪ {(0, 1)})*. Then we find a m-gon (resp., m -gon) derived from Vn (resp., Rn). Here, we call n an absolute Möbius m-gonal shape number (resp., totient m -gonal shape number derived from U and Vn (resp., ϕ and Rn) except n* = 1*.*

**Definition 3.** *(Convexity and Area) We use same notations, convex, non-convex, and area in [3]. We say that n is an absolute Möbius m-gonal convex (resp., non-convex) shape number with respect to the absolute Möbius divisor function U if* {(*i*, *Ui*(*n*))|*i* = 0, ..., *m* − 2 } ∪ {(0, 1)} *is convex (resp., non-convex). Let A*(*n*) *denote the area of the absolute Möbius m-gon derived from the absolute Möbius m-gonal shape number. Similarly, we define the totient m -gonal convex (resp., non-convex) shape number and B*(*n*) *denote the area of the totient m -gon.*

**Example 1.** *If n* = 2 *then we obtain the set of points V*<sup>2</sup> = *R*<sup>2</sup> = {(0, 2),(1, 1),(0, 1)}*. Thus,* 2 *is an absolute Möbius* 3*-gonal convex number with A* (2) = <sup>1</sup> <sup>2</sup> *. See Figure 1. See Figures 2–4 for absolute Möbius n-gonal shape numbers and totient n-gonal shape numbers with n* = 2, 3, 4, 5*. The first 19 values of A*(*n*) *and B*(*n*) *are given by Table 3.*

**Figure 4.** *U*(5) and *ϕ*(5).

**Table 3.** Values of *A*(*n*) and *B*(*n*) (2 ≤ *n* ≤ 20).


Kim and Bayad [3] considered the iteration of the odd divisor function *S*, polygon shape, convex, order, etc.

In this article, we considered the iteration of the absolute Möbius divisor function and Euler totient function and polygon types.

Now we state the main result of this article. To do this, let us examine the following theorem. For the proof of this theorem, the definitions and lemmas in the other chapters of this study have been utilized.

**Theorem 1.** *(Main Theorem) Let p*1,..., *pu be Fermat primes with p*<sup>1</sup> < *p*<sup>2</sup> ... < *pu,*

$$\begin{aligned} F\_0 &:= \{ p\_1, \ldots, p\_u \}, \\ F\_1 &:= \left\{ \prod\_{i=1}^t p\_i \mid p\_i \in F\_0, \ 1 \le t \le 5 \right\}, \\ F\_2 &:= \left\{ \prod\_{j=1}^r p\_{i\_j} \mid p\_{i\_j} \in F\_0, \ p\_1 \le p\_{i\_1} < p\_{i\_2} \ldots < p\_{i\_r} \le p\_{u\_r}, r \le u \right\} - (F\_0 \cup F\_1). \\\\ &\text{If } \text{Ond } (w) = 1 \text{ or } 2 \text{ then a partition function } w > 1 \text{ is} \end{aligned}$$

*If Ord*2(*m*) = 1 *or* 2 *then a positive integer m* > 1 *is*

⎧ ⎪⎨ ⎪⎩ *an absolute Möbius* <sup>3</sup>*-gonal (triangular) shape number*, *if m* = <sup>2</sup>*<sup>k</sup> or m* ∈ *<sup>F</sup>*<sup>1</sup> *an absolute Möbius* 4*-gonal convex shape number*, *if m* ∈ *F*<sup>0</sup> − {3} *or m* ∈ *F*<sup>2</sup> *an absolute Möbius* 4*-gonal non-convex shape number*, *otherwise*.

**Remark 2.** *Shapiro [2] computed positive integer m when C*(*m*) + 1 = 2. *That is, m* = 3, 4, 6. *Let C*(*m*) + 1 = 1 *or* 2. *Then*


**2. Some Properties of** *U*(*n*) **and** *ϕ*(*n*)

It is well known [1,4–14] that Euler *ϕ*-function have several interesting formula. For example, if (*x*, *y*) = 1 with two positive integers *x* and *y*, then *ϕ*(*xy*) = *ϕ*(*x*)*ϕ*(*y*). On the other hand, if *x* is a multiple of *y*, then *ϕ*(*xy*) = *yϕ*(*x*) [2]. In this section, we will consider the arithmetic functions *U*(*n*) and *ϕ*(*n*).

**Lemma 1.** *Let n* = *pe*<sup>1</sup> <sup>1</sup> *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> ··· *<sup>p</sup>er <sup>r</sup> be a factorization of n, where pr be distinct prime integers and er be positive integers. Then,*

$$U(n) = \prod\_{i=1}^{r} (p\_i - 1).$$

**Proof.** If *n* = *pe*<sup>1</sup> <sup>1</sup> *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> ··· *<sup>p</sup>er <sup>r</sup>* is an arbitrary integer, then we easily check

$$\begin{aligned} \mathcal{U}(n) &= \left| \sum\_{d|n} \mu(d)d \right| \\ &= \left| 1 - p\_1 - p\_2 - \dots - p\_r + p\_1 p\_2 + \dots + (-1)^n p\_1 p\_2 \dots p\_r \right| \\ &= (p\_1 - 1) \left( p\_2 - 1 \right) \dots \left( p\_r - 1 \right) .\end{aligned}$$

This is completed the proof of Lemma 1.

**Corollary 1.** *If <sup>p</sup> is a prime integer and <sup>α</sup> is a positive integer, then <sup>U</sup>*(*p*) = *<sup>p</sup>* <sup>−</sup> <sup>1</sup> *and <sup>U</sup>*(*pα*) = *<sup>U</sup>*(*p*)*. In particular, U*(2*α*) = 1*.*

**Proof.** It is trivial by Lemma 1.

**Corollary 2.** *Let n* > 1 *be a positive integer and let Ord*2(*n*) = *m. Then,*

$$\mathcal{U}\_0(n) > \mathcal{U}\_1(n) > \mathcal{U}\_2(n) > \cdots > \mathcal{U}\_m(n) \,. \tag{1}$$

**Proof.** It is trivial by Lemma 1.

**Remark 3.** *We compare U*(*n*) *with ϕ*(*n*) *as follow on Table 4.*


**Lemma 2.** *The function U is multiplicative function. That is, U*(*mn*) = *U*(*m*)*U*(*n*) *with* (*m*, *n*) = 1*. Furthermore, if m is a multiple of n, then U*(*mn*) = *U*(*m*)*.*

**Proof.** Let *m* = *pe*<sup>1</sup> <sup>1</sup> *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> ...*pei <sup>i</sup>* and *<sup>n</sup>* <sup>=</sup> *<sup>q</sup> <sup>f</sup>*<sup>1</sup> <sup>1</sup> *<sup>q</sup> <sup>f</sup>*<sup>2</sup> <sup>2</sup> ...*<sup>q</sup> fs <sup>s</sup>* be positive integers. Then *<sup>p</sup>e*<sup>1</sup> <sup>1</sup> , *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> , ..., *<sup>p</sup>ei <sup>i</sup>* and *q f*1 <sup>1</sup> , *<sup>q</sup> <sup>f</sup>*<sup>2</sup> <sup>2</sup> , ..., *<sup>q</sup> fs <sup>s</sup>* are distinct primes. If (*m*, *n*) = 1 and also *p*|*m*, *pn*, *q*|*n*, and *qm* by Lemma 1, we note that

$$\begin{aligned} \mathcal{U}(mm) &= \prod\_{t\_k|mn} (t\_k - 1) \\ &= \prod\_{p\_i|m} (p\_i - 1) \prod\_{q\_s|n} (q\_s - 1) \\ &= \mathcal{U}(m) \mathcal{U}(n) .\end{aligned}$$

Let *m* be a multiple of *n*. If *pi*|*n* then *pi*|*m*. Thus, by Lemma 1, *U*(*mn*) = *U*(*m*). This is completed the proof of Lemma 2.

**Remark 4.** *Two functions U*(*n*) *and ϕ*(*n*) *have similar results as follows on Table 5. Here, n*|*m means that m is a multiple of n.*


**Theorem 2.** *For all n* <sup>∈</sup> <sup>N</sup> <sup>−</sup> {1}*, there exists m* <sup>∈</sup> <sup>N</sup> *satisfying Um*(*n*) = <sup>1</sup>*.*

**Proof.** Let *n* = *pe*<sup>1</sup> <sup>1</sup> *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> ...*per <sup>r</sup>* , where *p*1, ..., *pr* be distinct prime integers with *p*<sup>1</sup> < *p*<sup>2</sup> < ... < *pr*. We note that *<sup>U</sup>*(*n*) = *<sup>r</sup>* ∏ *i*=1 (*pi* − 1) by Lemma 1.

If *r* = 1 and *p*<sup>1</sup> = 2, then *U*(*n*) = 1 by Corollary 1. If *pi* is an odd positive prime integer, then *<sup>U</sup>*(*pei <sup>i</sup>* ) = *pi* − 1 by Corollary 1. We note that *pi* − 1 is an even integer. Then there exist distinct prime integers *qi*<sup>1</sup> , ..., *qis* satisfying

$$p\_i - 1 = 2^{l\_i} q\_{i\_1}^{f\_{i\_1}} \cdots q\_{i\_s}^{f\_{i\_s}} \, .$$

where *fis* <sup>≥</sup> 1, *li* <sup>≥</sup> 1 and *qi*<sup>1</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *qis*. It is well known that *qis* <sup>≤</sup> *pi*−<sup>1</sup> <sup>2</sup> <sup>≤</sup> *pr*−<sup>1</sup> <sup>2</sup> .

By Lemma 2, we get

$$\mathcal{U}l\_2(p\_i^{c\_i}) = \mathcal{U}(p\_i - 1) = \mathcal{U}(q\_{i\_1}) \cdots \mathcal{U}(q\_{i\_s}).\tag{2}$$

By using the same method in (2) for 1 ≤ *ij* ≤ *s*, we get

$$\begin{aligned} \mathcal{U}\_{\mathcal{U}}(n) &= \mathcal{U}(\prod\_{i=1}^{r} (p\_i - 1)) \\ &= \mathcal{U}\left(2^{l\_1 + l\_2 + \dots + l\_r} \prod\_{i=1}^{r} q\_{i\_1}^{c\_{i\_1}} \dots q\_{i\_w}^{c\_{i\_w}}\right) \\ &= \mathcal{U}\left(\prod\_{i=1}^{r} q\_{i\_1}^{c\_{i\_1}} \dots q\_{i\_w}^{c\_{i\_w}}\right) \\ &= \prod\_{i=1}^{r} (q\_{i\_1} - 1) \cdot \dots (q\_{i\_w} - 1) \\ &= q\_{j\_1}^{(2)} \cdot \dots \cdot q\_{j\_k}^{(2)} \end{aligned}$$

with *q* (2) *<sup>j</sup>*<sup>1</sup> < *<sup>q</sup>* (2) *<sup>j</sup>*<sup>2</sup> < ... < *<sup>q</sup>* (2) *jk* . It is easily checked that *q* (2) *jk* <sup>≤</sup> max{ *<sup>q</sup>*1*u*−<sup>1</sup> <sup>2</sup> , ..., *qru*−<sup>1</sup> <sup>2</sup> }.

Using this technique, we can find *l* satisfying

$$\mathcal{U}\_{l-1}(n) = \left(q\_{j\_1}^{(l-1)} - 1\right) \cdots \left(q\_{j\_u}^{(l-1)} - 1\right) = 2^h \prod\_{u=1}^{s'} q\_{j\_u}^{(l)}$$

with *q* (*l*) *ju* < 100.

By Appendix A ( Values of *U*(*n*) (1 ≤ *n* ≤ 100)), we easily find a positive integer *v* that *Uv*(*n* ) = 1 for 1 <sup>≤</sup> *<sup>n</sup>* <sup>≤</sup> 100. Thus, we get *Uv*(*Ul*−1(*n*)) = 1. Therefore, we can find *<sup>m</sup>* <sup>=</sup> *<sup>v</sup>* <sup>+</sup> *<sup>l</sup>* <sup>−</sup> <sup>1</sup> <sup>∈</sup> <sup>N</sup> satisfying *Um*(*n*) = 1.

**Corollary 3.** *For all n* <sup>∈</sup> <sup>N</sup> <sup>−</sup> {1}*, there exists m* <sup>∈</sup> <sup>N</sup> *satisfying Ord*(*n*) = *m.*

**Proof.** It is trivial by Theorem 2.

**Remark 5.** *Kim and Bayad [3] considered iterated functions of odd divisor functions Sm*(*n*) *and order of n. For order of divisor functions, we do not know Ord*(*n*) = ∞ *or not. But, functions Um*(*n*) *(resp., ϕl*(*n*)*), we know Ord*(*n*) < ∞ *by Corollary 3 (resp., [15]).*

**Theorem 3.** *Let n* <sup>&</sup>gt; <sup>1</sup> *be a positive integer. Then Ord*<sup>2</sup> (*n*) <sup>=</sup> <sup>1</sup> *if and only if n* <sup>=</sup> <sup>2</sup>*<sup>k</sup> for some k* <sup>∈</sup> <sup>N</sup>*.*

**Proof.** (⇐) Let *<sup>n</sup>* = <sup>2</sup>*k*. It is easy to see that *<sup>U</sup>* (*n*) = *<sup>U</sup>*<sup>1</sup> (*n*) = 1.

(⇒) Let *<sup>n</sup>* <sup>=</sup> *<sup>p</sup>e*<sup>1</sup> <sup>1</sup> *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> ...*per <sup>r</sup>* be a factorization of *n*, and all *pr* are distinct prime integers. If *Ord*<sup>2</sup> (*n*) = 1, then by using Lemma 1 we can note that,

$$1 = \left(p\_1 - 1\right)\left(p\_2 - 1\right)\dots\left(p\_r - 1\right). \tag{3}$$

According to all *pr* are distinct prime integers, then it is easy to see that there is only exist *p*<sup>1</sup> and that is *<sup>p</sup>*<sup>1</sup> <sup>=</sup> 2. Hereby *<sup>n</sup>* <sup>=</sup> <sup>2</sup>*<sup>k</sup>* for some *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>.

This is completed the proof of Theorem 3.

**Remark 6.** *If k* > 0 *then* 2*<sup>k</sup> is an absolute Möbius* 3*-gonal (triangular) shape number with A* 2*k* = 1 2 <sup>2</sup>*<sup>k</sup>* − <sup>1</sup> *by Theorem 3.*

**Theorem 4.** *Let n, m and m be positive integers with greater than* 1 *and let Ord*<sup>2</sup> (*n*) = *m and C*(*n*) + 1 = *m . Then, A*(*n*), *<sup>B</sup>*(*n*) <sup>∈</sup> <sup>Z</sup> *if and only if n* <sup>≡</sup> <sup>1</sup> (mod 2)*. Furthermore,*

$$A(n) = \sum\_{k=1}^{m-1} \mathcal{U}\_k(n) + \frac{1}{2}(1+n) - m \tag{4}$$

*and*

$$B(n) = \sum\_{k=1}^{m'-1} \varphi\_k(n) + \frac{1}{2}(1+n) - m'. \tag{5}$$

**Proof.** First, we consider *A*(*n*). We find the set {(0, *U*<sup>0</sup> (*n*)),(1, *U*<sup>1</sup> (*n*)),...,(*m*, *Um* (*n*))}. Thus, we have

$$\begin{aligned} A(n) &= \frac{1}{2} \left( \mathcal{U}l\_0(n) + \mathcal{U}l\_1(n) \right) + \frac{1}{2} \left( \mathcal{U}l\_1(n) + \mathcal{U}l\_2(n) \right) + \dots + \frac{1}{2} \left( \mathcal{U}l\_{m-1}(n) + \mathcal{U}l\_m(n) \right) - m \\ &= \mathcal{U}l\_1(n) + \dots + \mathcal{U}l\_{m-1}(n) + \frac{1}{2} \left( 1 + n \right) - m . \end{aligned}$$
 
$$\equiv \frac{1}{2} \left( 1 + n \right) \pmod{1}.$$

Similarly, we get (5). These complete the proof of Theorem 4.

#### **3. Classification of the Absolute Möbius Divisor Function** *U***(***n***) with** *Ord***2(***n***) = 2**

In this section, we study integers *n* when *Ord*2(*n*) = 2. If *Ord*2(*n*) = 2, then *n* has three cases which are 3-gonal (triangular) shape number, 4-gonal convex shape number, and 4-gonal non-convex shape numbers in Figure 5.

**Figure 5.** 3-gonal (triangular), 4-gonal convex, 4-gonal non-convex shapes.

**Theorem 5.** *Let <sup>p</sup>*1, ..., *pr be Fermat primes and <sup>e</sup>*1, ...,*er be positive integers. If <sup>n</sup>* = <sup>2</sup>*<sup>k</sup> <sup>p</sup>e*<sup>1</sup> <sup>1</sup> *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> ...*per <sup>r</sup> , then Ord*2(*n*) = 2*.*

**Proof.** Let

$$p\_i = 2^{2^{m\_i}} + 1 \quad (1 \le i \le r) \tag{6}$$

be Fermat primes. By Corollary 1 and Lemma 2 we have

$$\begin{aligned} \mathcal{U}\left(n\right) &= \mathcal{U}\left(2^k p\_1^{\epsilon\_1} p\_2^{\epsilon\_2} \dots p\_r^{\epsilon\_r}\right) \\ &= \mathcal{U}\left(2^k\right) \mathcal{U}\left(p\_1^{\epsilon\_1}\right) \mathcal{U}\left(p\_2^{\epsilon\_2}\right) \dots \mathcal{U}\left(p\_r^{\epsilon\_r}\right) \\ &= \left(p\_1 - 1\right) \left(p\_2 - 1\right) \dots \left(p\_r - 1\right) \\ &= 2^{2^{m\_1}} 2^{2^{m\_2}} \dots 2^{2^{m\_r}} \\ &= 2^t. \end{aligned}$$

Thus, we can see that *U* (*n*) = *U*<sup>1</sup> (*n*) = 2*<sup>t</sup>* and *U*<sup>2</sup> (*n*) = *U* (*U*<sup>1</sup> (*n*)) = *U* 2*t* = 1. Therefore, we get Theorem 5.

The First 32 values of *U*(*n*) and *ϕ*(*n*) for *n* = 2*<sup>k</sup> pe*<sup>1</sup> <sup>1</sup> *<sup>p</sup>e*<sup>2</sup> <sup>2</sup> ...*per <sup>r</sup>* are given by Table A2 (see Appendix B).

**Remark 7.** *Iterations of the odd divisor function S*(*n*)*, the absolute Möbius divisor function U*(*n*)*, and Euler totient function ϕ*(*n*) *have small different properties. Table 6. gives an example of differences of ϕk*(*n*), *Uk*(*n*)*, and Sk*(*n*) *with k* = 1, 2*.*



**Lemma 3.** *Let n* = *pi be Fermat primes. Then* 3 *is an absolute Möbius* 3*-gonal (triangular) shape number and pi* (= 3) *are absolute Möbius* 4*-gonal convex numbers.*

**Proof.** The set {(0, 3),(1, 2),(2, 1),(1, 0)} makes a triangle. Let *pi* <sup>=</sup> <sup>2</sup>2*mi* <sup>+</sup> 1 be a Fermat primes except 3 . We get *<sup>U</sup>* (*pi*) <sup>=</sup> <sup>2</sup>2*mi* . So, we get

$$\mathbf{A} = \left\{ \left( 0, 2^{2^{m\_i}} + 1 \right), \left( 1, 2^{2^{m\_i}} \right), \left( 2, 1 \right), \left( 0, 1 \right) \right\}.$$

Because of  <sup>2</sup>2*mi* <sup>+</sup> <sup>1</sup> <sup>−</sup> 22*mi* < 22*mi* <sup>−</sup> <sup>1</sup> , the set **A** gives a convex shape. This completes the proof Lemma 3.

**Lemma 4.** *Let pi be Fermat primes. Then* <sup>2</sup>*m*<sup>1</sup> *pi and <sup>p</sup>m*<sup>2</sup> *<sup>i</sup> are absolute Möbius* 4*-gonal non-convex shape numbers with m*1, *m*2(≥ 2) *positive integers.*

**Proof.** Let *pi* <sup>=</sup> <sup>2</sup>2*mi* <sup>+</sup> 1 be a Fermat primes. Consider

$$2^{m\_1}p\_i - (p\_i - 1) = 2^{m\_1} \cdot 2^{2^{m\_i}} - 2^{2^{m\_i}} \text{ and } (p\_i - 1) - 1 = 2^{2^{m\_i}} - 1.$$

So, <sup>2</sup>*m*<sup>1</sup> *pi* − (*pi* − <sup>1</sup>) > (*pi* − <sup>1</sup>) − 1. Thus, <sup>2</sup>*m*<sup>1</sup> *pi* are absolute Möbius 4-gonal non-convex shape numbers. Similarly, we get *pi <sup>m</sup>*<sup>1</sup> − (*pi* − <sup>1</sup>) > (*pi* − <sup>1</sup>) − 1.

Thus, these complete the proof Lemma 4.

**Lemma 5.** *Let p*1, ... , *pr be Fermat primes. Then* 2*p*<sup>1</sup> ... *pr are absolute Möbius* 4*-gonal non-convex shape numbers.*

*Furthermore, if <sup>m</sup>*,*e*1, ...*er are positive integers then* <sup>2</sup>*<sup>m</sup> <sup>p</sup>e*<sup>1</sup> <sup>1</sup> ... *<sup>p</sup>er <sup>r</sup> are absolute Möbius* 4*-gonal non-convex shape numbers.*

**Proof.** The proof is similar to Lemma 4.

**Lemma 6.** *Let r be a positive integer. Then*

$$\prod\_{i=0}^{r} \left( 2^{2^i} + 1 \right) - 2 \prod\_{i=0}^{r} 2^{2^i} + 1 = 0.1$$

**Proof.** We note that

$$\prod\_{i=0}^{r} \left(\mathbf{x}^{2^i} + 1\right) = \frac{\mathbf{x}^{2^{r+1}} - 1}{\mathbf{x} - 1} \text{ and } \prod\_{i=0}^{r} \mathbf{x}^{2^i} = \mathbf{x}^{2^{r+1} - 1}.$$

Let *f*(*x*) := ∏*<sup>r</sup> i*=0 22*i* + 1 <sup>−</sup> <sup>2</sup> <sup>∏</sup>*<sup>r</sup> <sup>i</sup>*=<sup>0</sup> 22*<sup>i</sup>* + 1. Thus *f*(2) = 0. This is completed the proof of Lemma 6.

**Corollary 4.** *Let fi* ∈ *F*1*. Then fi is an absolute Möbius* 3*-gonal (triangular) shape number.*

**Proof.** It is trivial by Lemma 6.

**Remark 8.** *Fermat first conjectured that all the numbers in the form of fn* = <sup>2</sup>2*<sup>n</sup>* + 1 *are primes [16]. Up-to-date there are only five known Fermat primes. That is, f*<sup>0</sup> = 3*, f*<sup>1</sup> = 5*, f*<sup>2</sup> = 17*, f*<sup>3</sup> = 257*, and f*<sup>4</sup> = 65537*.*

*Though we find a new Fermat prime p*6*, 6th Fermat primes, we cannot find a new absolute Möbius* 3*-gonal (triangular) number by*

$$\prod\_{i=0}^{4} \left( 2^{2^i} + 1 \right) \times \left( 2^{2^{i'}} + 1 \right) - 2 \left( \prod\_{i=0}^{4} 2^{2^i} \right) 2^{2^{i'}} + 1 > 0. \tag{7}$$

**Lemma 7.** *Let <sup>p</sup>*1, *<sup>p</sup>*2, ... , *pr*, *pt be Fermat primes with <sup>p</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; ... <sup>&</sup>lt; *pr* <sup>&</sup>lt; *pt and <sup>t</sup>* <sup>&</sup>gt; <sup>5</sup>*. If <sup>n</sup>* <sup>=</sup> *<sup>r</sup>* ∏ *i*=1 *pi* ∈ *F*<sup>1</sup> *then n* × *pt are absolute Möbius* 4*-gonal convex shape numbers.*

**Proof.** Let *pt* = 22*<sup>k</sup>* + 1 be a Fermat prime, where *k* is a positive integer. We note that *r* ≤ 5 and *pt* = 22*<sup>k</sup>* + 1 > 22<sup>6</sup> + 1. In a similar way in (7), we obtain

$$\begin{aligned} &p\_1 \dots p\_r p\_t - 2\left(p\_1 - 1\right) \dots \left(p\_r - 1\right) \left(p\_t - 1\right) + 1 = \\ &\left\{ \prod\_{i=0}^{r-1} \left( 2^{2^i} + 1 \right) \right\} \left( 2^{2^t} - 1 \right) - 2^{1 + 2^0 + 2^1 + \dots + 2^{r-1} + 2^k} + 1 > 0. \end{aligned} \tag{8}$$

By Theorem 5, *Ord*2(*n* × *pt*) = 2. By (8), *n* × *pt* is an absolute Möbius 4-gonal convex shape number. This completes the proof of Lemma 7.

**Lemma 8.** *Let p*1, *p*2,..., *pr*, *pt be Fermat primes with p*<sup>1</sup> < *p*<sup>2</sup> < ... < *pr* < *pt.*

*Then m* = *pf*<sup>1</sup> <sup>1</sup> ··· *<sup>p</sup>fu <sup>u</sup> are absolute Möbius* 4*-gonal non-convex shape numbers except m* ∈ *F*<sup>0</sup> ∪ *F*<sup>1</sup> ∪ *F*2*.* **Proof.** Similar to Lemmas 5 and 7.

**Proof of Theorem 1 (Main Theorem).** It is completed by Remark 6, Theorem 5, Lemmas 3 and 4, Corollary 4, Remark 8, Lemmas 7 and 8.

**Remark 9.** *If n are absolute Möbius* 3*-gonal (triangular) or* 4*-gonal convex shape numbers then n is the regular n-gon by Gauss Theorem.*

**Example 2.** *The set V*<sup>3</sup> *is* {(0, 3),(1, 2),(2, 1),(0, 1)}*. Thus, a positive integer* 3 *is an absolute Möbius* 3*-gonal convex shape number.*

*Similarly,* 15*,* 255*,* 65535*,* 4294967295 *are absolute Möbius* 3*-gonal convex numbers derived from*

*V*<sup>15</sup> = {(0, 15),(1, 8),(2, 1),(0, 1)}*, V*<sup>255</sup> = {(0, 255),(1, 128),(2, 1),(0, 1)}*, V*<sup>65535</sup> = {(0, 65535),(1, 32768),(2, 1),(0, 1)}*, V*<sup>4294967295</sup> = {(0, 4294967295),(1, 2147483648),(2, 1),(0, 1)}*.*

**Remark 10.** *Let Min*(*m*) *denote the minimal number of m-gonal number. By using Maple 13 Program, Table 7 shows us minimal numbers Min*(*m*) *about from* 3*-gonal (triangular) to* 14*-gonal shape number.*


**Table 7.** Values of Min(m).

**Conjecture 1.** *For any positive integer m* (≥ 3)*, Min*(*m*) *is a prime integer.*

**Author Contributions:** The definitions, lemmas, theorems and remarks within the paper are contributed by D.K., U.S. and S.I. Also, the introduction, body and conclusion sections are written by D.K., U.S. and S.I. The authors read and approved the final manuscript.

**Funding:** This work was funded by "Research Base Construction Fund Support Program" Jeonbuk National University in 2019. Supported by Balikesir University Research, G. No: 2017/20.

**Conflicts of Interest:** The authors declare that there is no conflict of interest.

#### **Appendix A. Values of** *U***(***n***)**


**Table A1.** Values of *U*(*n*) (1 ≤ *n* ≤ 100).

### **Appendix B. Values of** *<sup>n</sup>* **<sup>=</sup> <sup>2</sup>***<sup>k</sup> <sup>p</sup>***1***p***2...***pi***,** *<sup>U</sup>* **(***n***),** *<sup>ϕ</sup>***(***n***)**


**Table A2.** Values of *n* = 2*<sup>k</sup> p*<sup>1</sup> *p*2...*pi*, *U* (*n*), *ϕ*(*n*) with *Ord*2(*n*) = 2.

#### **References**


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