On the periods of some figurate numbers

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Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. pp. 71-80, 2011 Applied Mathematics

On the Periods of Some Figurate Numbers Omur Deveci1_{, Erdal Karaduman}2

1_{Department of Mathematics, Faculty of Science and Letters, Kafkas University, 36100}

Kars, Turkiye

e-mail: o deveci36@ hotm ail.com

Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkiye

e-mail: edum an@ atauni.edu.tr

Received Date:December 7, 2010 Accepted Date: October 19, 2011

Abstract. The number which can be represented by a regular geometrical arrangement of equally spaced point is defined as figurate number. Each of polygonal, centered polygonal and pyramidal numbers is a class of the series of figurate numbers. In this paper, we obtain the periods of polygonal, centered polygonal and pyramidal numbers by reducing each element of these numbers modulo m.

Key words: Period, polygonal number, centered polygonal number, pyramidal number.

2000 Mathematics Subject Classification: 11B75, 11B50. 1. Introduction

The figurate numbers have a very important role to solve some problems in num-ber theory and to determine speciality of some numnum-bers, see for example, [8,9]. The polygonal numbers, the centered polygonal numbers, the pyramidal num-bers and their properties have been studied by some authors, see for example, [1,3,6,15]. The study of Fibonacci numbers by reducing modulo m began with the earlier work of Wall [13] where the periods of Fibonacci numbers according to modulo m were obtained. The theory is expanded to 3-step Fibonacci se-quence by Özkan, Aydin and Dikici [11]. Lü and Wang [10] contributed to study of the Wall number for the k−step Fibonacci sequence. Deveci and Karaduman [5] extended the concept to Pell numbers. Now we extend the concept to the polygonal numbers, the centered polygonal numbers and the pyramidal numbers which are classes of the series of figurate numbers.

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A sequence is periodic if, after a certain point, it consists only of repetitions of a fixed subsequence. The number of elements in the repeating subsequence is called the period of the sequence. For example, the sequence a, b, c, b, c, b, c, · · · is periodic after the initial element a and has period 2. A sequence is simply periodic with period n if the first n elements in the sequence form a repeat-ing subsequence. For example, the sequence a, b, c, a, b, c, a, b, c, , · · · is simply periodic with period 3.

We have the following formulas for the polygonal numbers, the centered polyg-onal numbers and the pyramidal numbers:

Let k be the number of sides in a polygon. The nth_{k−gonal number is obtained} by the formula P (k, n) = (k 2− 1)n 2 − µ k 2− 2 ¶ n.

A polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. We obtain nth _{triangular number, n}th _square number, nth_{pentagonal number, · · · for k = 3, 4, 5, · · · . For more information} on k−gonal numbers, see [12].

The nth _{centered k−gonal number is obtained by the formula} C (k, n) =kn

2 (n − 1) + 1.

A centered polygonal number formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k−gonal number contains k more points than the previous layer. We obtain nth_{centered triangular number, n}th_centered square number, nth _{centered pentagonal number, · · · for k = 3, 4, 5, · · · . For} more information on centered k−gonal numbers, see [2].

The nth_{k−gonal pyramidal number is obtained by formula} P_n(k)= n 2 2 + n 3 µ k 6− 1 3 ¶ − n µ k − 5 6 ¶ .

A pyramidal number represents a pyramid with a base and given number of sides. We obtain nthtriangular pyramidal number, nthsquare pyramidal num-ber, nth _{pentagonal pyramidal number, · · · for k = 3, 4, 5, · · · . For more} infor-mation on k−gonal pyramidal numbers, see [14].

In this paper, the usual notation p is used for a prime number. 2. Polygonal Numbers

In this section, we obtain the lengths of the periods of k−gonal numbers modulo m. The notation LP(m) denote the length of the smallest period which the period is obtained each element of the polygonal numbers by reducing modulo m for k ≥ 3.

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Theorem 2.1. _{Let k ≡ 0 (mod 4). The lengths of the periods of the polygonal} numbers are as follows:

i. _{For u ∈ N, L}P(2u) = ½

2 for u = 1, 2u−1 _{for u > 2.} ii. _{If p 6= 2 and θ ∈ N, then L}P¡pθ¢= pθ. iii. If m = Qt_i=1pei

i (t ≥ 1) where pi’s are distinct primes, then LP(m) = lcm [LP(peii)].

Proof. _{We prove this by direct calculation. Since k ≡ 0 (mod 4) and k ≥ 3,} k = 4 , ( ∈ N).

i. Since

P (4 , n) (mod 2) ≡£(2 − 1) n2_{− (2 − 2) n}¤ _{(mod 2) ≡ 0 (mod 2) for n is even} and

P (4 , n) (mod 2) ≡£(2 − 1) n2_{− (2 − 2) n}¤ _{(mod 2) ≡ 1 (mod 2) for n is odd,} LP(2) = 2. If u > 1, then P¡4 , 2u−1¢ _{(mod 2}u_{) ≡}h_{(2 − 1)}¡₂u−1¢2_{+ (2 − 2) 2}u−1i _{(mod 2}u_{) ≡} ≡£2u ¡₂u−1_{− 1}¢_{+ 2}u¡_{1 − 2}u−2¢¤ _{(mod 2}u_{) ≡ 0 (mod 2}u_{) ,} P¡4 , 2u−1_{+ 1}¢ _{(mod 2}u_{) ≡} ≡h(2 − 1)¡2u−1_{+ 1}¢2_{+ (2 − 2)}¡₂u−1_{+ 1}¢i _{(mod 2}u_{) ≡} ≡£2u¡ _{+ 2}u−1_{− 2}u−2¢_{+ 1}¤ _{(mod 2}u_{) ≡ 1 (mod 2}u_{) , · · · ,} P¡4 , 2u−1_{+ n}¢ _{(mod 2}u_{) ≡} ≡h(2 − 1)¡2u−1_{+ n}¢2_{+ (2 − 2)}¡₂u−1_{+ n}¢i _{(mod 2}u_{) ≡} ≡£2u¡₂u−1 _{+ 2 n − 2}u−2_{− n − + 1}¢_{+ 1}¤ _{(mod 2}u_{) +} +£_{(2 − 1) n}2_{+ (2 − 2) n}¤ _{(mod d 2}u_{) ≡} ≡£(2 − 1) n2_{+ (2 − 2) n}¤ _{(mod 2}u_{) ≡ P (4 , n) (mod 2}u_{) .} So, we get LP(2u) = 2u−1.

ii. The proof is similar to the proof of i. and is omitted. iii. Let lcm [LP(peii)] = β.

P (4 , β) (mod m) ≡£(2 − 1) β2+ (2 − 2) β¤ (mod m) ≡ 0 (mod m) , P (4 , β + 1) (mod m) ≡h(2 − 1) (β + 1)2+ (2 − 2) (β + 1)i (mod m) ≡ ≡£β2_{(2 − 1) + 1}¤ _{(mod m) ≡ 1 (mod m) , · · · ,}

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P (4 , β + n) (mod m) ≡h(2 − 1) (β + n)2+ (2 − 2) (β + n)i (mod m) ≡ ≡ [β (2 β + 4 n − β − 2n − 2 + 2) + 1] (mod m) +

+£_{(2 − 1) n}2_{+ (2 − 2) n}¤ _{(mod m) ≡}

≡£(2 − 1) n2_{+ (2 − 2) n}¤ _{(mod m) ≡ P (4 , n) (mod m) .} So, we get LP(m) = lcm [LP(peii)].

Theorem 2.2:_{Let k ≡ 2 (mod 4). Then L}P(m) = m for m ≥ 2.

Proof: _{We prove this by direct calculation. Since k ≡ 2 (mod 4) and k ≥ 3,} k = 4 + 2, ( ∈ N).

P (4 + 2, m + n) (mod m) ≡h(2 ) (m + n)2_{+ (2 − 1) (m + n)}i _{(mod m) ≡} ≡ m (2 m + 2n − 2 − 1) (mod m) +h(2 ) (n)2_{+ (2 − 1) (n)}i _{(mod m) ≡} ≡ P (4 + 2, n) (mod m) .

So, we get LP(m) = m.

Theorem 2.3. _{Let k ≡ 1 (mod 4) or k ≡ 3 (mod 4). The lengths of the} periods of the polygonal numbers are as follows:

i. _{For u ∈ N, L}P(2u) = 2u+1.

ii. _{If p 6= 2 and θ ∈ N, then L}P¡pθ¢= pθ. iii. If m = Qt_i=1pei

i (t ≥ 1) where pi’s are distinct primes, then LP(m) = lcm [LP(peii)].

Proof. The proof is similar to the proof of Theorem 2.1. and is omitted. 3. Centered Polygonal Numbers

In this section, we obtain the lengths of the periods of centered k−gonal numbers modulo m. The notation LCP(m) denote the length of the smallest period which the period is obtained each element of the centered polygonal numbers by reducing modulo m for k ≥ 3.

Theorem 3.1. _{Let k ≡ 0 (mod 4) that is k = 2}α_.pe1

1 .p e2

2 . · · · .p et

t , α ≥ 2 such that p1, p2, · · · , ptare distinct primes and e1, e2, · · · , etare 0 or positive integers. The lengths of the periods of the centered polygonal numbers are as follows: i. _{For u ∈ N, L}CP(2u) =

½

1 _{for u ≤ α,} 2u−1 for u > α. ii. _{If v ∈ N such that 1 ≤ v ≤ t and λ ∈ N, then L}CP

¡ pλ v ¢ = ½ 1 _{for λ ≤ e}v, pλ−ev v for λ > ev. iii. _{If p 6= 2, p}1, p2, · · · , ptand θ ∈ N, then LCP¡pθ¢= pθ.

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Proof. _{We prove this by direct calculation. Since k ≡ 0 (mod 4) and k ≥ 3,} k = 2α_.pe1 1 .p e2 2 . · · · .p et

t , α ≥ 2 such that p1, p2, · · · , pt are distinct primes and e1, e2, · · · , et are 0 or positive integers.

i. _{If u ≤ α, then} C (k, n) (mod 2u_{) ≡} ∙ k 2n (n − 1) + 1 ¸ (mod 2u_{) ≡ 1 (mod 2}u) . So, we get LCP(2u) = 1. If u > α, then

C¡k, 2u−1¢ (mod 2u_{) ≡}£k₂2u−1¡2u−1_{− 1}¢+ 1¤ (mod 2u_{) ≡ 1 (mod 2}u) , C¡k, 2u−1_{+ 1}¢ _{(mod 2}u_{) ≡}£k 2 ¡ 2u−1_{+ 1}¢₂u−1_{+ 1}¤_{(mod 2}u_{) ≡ 1 (mod 2}u_{) , · · · ,} C¡k, 2u−1+ n¢ (mod 2u_{) ≡}£k₂¡2u−1+ n¢ ¡2u−1_{+ n − 1}¢+ 1¤(mod 2u_{) ≡} ≡£k2 ¡ 2u_{n + 2}u−1¡₂u−1_{− 1}¢¢¤ _{(mod 2}u_{) +}£k 2n (n − 1) + 1 ¤ (mod 2u_{) ≡} ≡£k2n (n − 1) + 1 ¤ (mod 2u_{) ≡ C (k, n) (mod 2}u) . So, we get LCP(2u) = 2u−1.

ii. _{If λ ≤ e}v and 1 ≤ v ≤ t, then C (k, n) ¡mod pλ_v¢_≡ ∙ k 2n (n − 1) + 1 ¸ _¡ mod pλ_v¢_{≡ 1}¡mod pλ_v¢. So, we get LCP ¡ pλ v ¢ = 1. If λ > ev and 1 ≤ v ≤ t, then C¡k, pλ−ev v ¢ ¡ mod pλ v ¢ ≡£k2pλv−ev ¡ pλ−ev v − 1 ¢ + 1¤ ¡mod pλ v ¢ ≡ 1¡mod pλ v ¢ , C¡k, pλ−ev v + 1 ¢ ¡ mod pλ v ¢ ≡ ≡£k2 ¡ pλ−ev v + 1 ¢ pλ−ev v + 1 ¤ ¡ mod pλ_v¢_{≡ 1}¡mod pλ_v¢_{, · · · ,} C¡k, pλ−ev v + n ¢ ¡ mod pλ v ¢ ≡ ≡£k2 ¡ pλ−ev v + n ¢ ¡ pλ−ev v + n − 1 ¢ + 1¤ ¡mod pλv ¢ ≡ ≡£k2 ¡ 2npλ−ev v + pλv−ev ¡ pλ−ev v − 1 ¢¢¤ ¡ mod pλ v ¢ + +£k₂_{n (n − 1) + 1}¤ ¡mod pλv ¢ ≡ ≡£k2n (n − 1) + 1 ¤ ¡ mod pλ v ¢ ≡ C (k, n) ¡mod pλ v ¢ . So, we get LCP¡pλv ¢ = pλ−ev v . iii. _{If p 6= 2, p}1, p2, · · · , pt, then

C¡k, pθ¢ ¡mod pθ¢_≡£k₂pθ¡pθ_{− 1}¢+ 1¤ ¡mod pθ¢_{≡ 1}¡mod pθ¢, C¡k, pθ_{+ 1}¢ ¡_{mod p}θ¢_≡£k 2 ¡ pθ_{+ 1}¢_pθ_{+ 1}¤ ¡_{mod p}θ¢_{≡ 1}¡_{mod p}θ¢_{, · · · ,} C¡k, pθ+ n¢ ¡mod pθ¢_≡£k₂¡pθ+ n¢ ¡pθ_{+ n − 1}¢+ 1¤ ¡mod pθ¢_≡ ≡£k2 ¡ 2npθ_{+ p}θ¡_pθ_{− 1}¢¢¤ ¡_{mod p}θ¢₊£k 2n (n − 1) + 1 ¤ ¡ mod pθ¢_≡ ≡£k2n (n − 1) + 1 ¤ ¡ mod pλ_vpθ¢_{≡ C (k, n)} ¡mod pθ¢.

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So, we get LCP ¡

pθ¢= pθ.

Theorem 3.2. _{Let k ≡ 2 (mod 4) that is k = 2.p}e1

1 .pe22. · · · .pett such that p1, p2, · · · , pt are distinct primes and e1, e2, · · · , et are positive integers. The lengths of the periods of the centered polygonal numbers are as follows: i. _{For u ∈ N, L}CP(2u) =

½

1 for u = 1, 2u _{for u > 1.}

ii. _{If v ∈ N such that 1 ≤ v ≤ t and λ ∈ N, then L}CP¡pλv ¢ = ½ 1 _{for λ ≤ e}v, pλ−ev v for λ > ev. iii. _{If p 6= 2, p}1, p2, · · · , ptand θ ∈ N, then LCP¡pθ¢= pθ.

Proof. We prove this by direct calculation. _{Since k ≡ 2 (mod 4) and} k ≥ 3, k = 2.pe1

1 .p e2

2 . · · · .p et

t such that p1, p2, · · · , pt are distinct primes and e1, e2, · · · , et are positive integers.

i. C (k, n) (mod 2) ≡£k

2n (n − 1) + 1 ¤

(mod 2) ≡ 1 (mod 2). So, we get LCP(2) = 1. If u > 1, then

C (k, 2u_{) (mod 2}u_{) ≡}£k 22u(2u− 1) + 1 ¤ (mod 2u_{) ≡ 1 (mod 2}u_{) ,} C (k, 2u_{+ 1) (mod 2}u_{) ≡}£k 2(2 u_{+ 1) 2}u_{+ 1}¤ _{(mod 2}u_{) ≡ 1 (mod 2}u_{) , · · · ,} C (k, 2u_{+ n) (mod 2}u_{) ≡}£k 2(2u+ n) (2u+ n − 1) + 1 ¤ (mod 2u_{) ≡} ≡£k 2 ¡ 2u+1_{n + 2}u₍₂u_{− 1)}¢¤ _{(mod 2}u_{) +}£k 2n (n − 1) + 1 ¤ (mod 2u_{) ≡} ≡£k2n (n − 1) + 1 ¤ (mod 2u_{) ≡ C (k, n) (mod 2}u_{) .} So, we get LCP(2u) = 2u.

The proofs of ii. and iii. are similar to the proofs of Theorem 3.1. ii. and Theorem 3.1.iii., respectively and are omitted.

Theorem 3.3. _{Let k ≡ 3 (mod 4) that is k = p}e1

1 .p e2

2 . · · · .p et

t such that p1, p2, · · · , pt are distinct primes and e1, e2, · · · , et are positive integers. The lengths of the periods of the centered polygonal numbers are as follows: i. _{For u ∈ N, L}CP(2u) = 2u+1.

ii. _{If v ∈ N such that 1 ≤ v ≤ t and λ ∈ N, then L}CP ¡ pλ v ¢ = ½ 1 _{for λ ≤ e}v, pλ−ev v for λ > ev. iii. _{If p 6= 2, p}1, p2, · · · , ptand θ ∈ N, then LCP¡pθ¢= pθ.

Proof. We prove this by direct calculation. _{Since k ≡ 3 (mod 4), k =} pe1

1 .pe22. · · · .pett such that p1, p2, · · · , ptare distinct primes and e1, e2, · · · , etare positive integers.

C¡k, 2u+1¢ (mod 2u_{) ≡}£k₂2u+1¡2u+1_{− 1}¢+ 1¤ (mod 2u_{) ≡ 1 (mod 2}u) , C¡k, 2u+1_{+ 1}¢ _{(mod 2}u_{) ≡}£k

2 ¡

2u+1_{+ 1}¢₂u+1_{+ 1}¤ _{(mod 2}u_{) ≡ 1 (mod 2}u_{) , · · · ,} C¡k, 2u+1+ n¢ (mod 2u_{) ≡}£k₂¡2u+1+ n¢ ¡2u+1_{+ n − 1}¢+ 1¤ (mod 2u_{) ≡} ≡£k2

¡

2u+2_{n + 2}u+1¡₂u+1_{− 1}¢¢¤ _{(mod 2}u_{) +}£k

2n (n − 1) + 1 ¤ (mod 2u_{) ≡} ≡£k2n (n − 1) + 1 ¤ (mod 2u_{) ≡ C (k, n) (mod 2}u) .

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So, we get LCP(2u) = 2u+1.

The proofs of ii. and iii. are similar to the proofs of Theorem 3.1. ii. and Theorem 3.1.iii., respectively and are omitted.

If k ≡ 1 (mod 4), then the rules are the same the rules of Theorem 3.3. The proof is similar to the proof of Theorem 3.3 and is omitted.

Theorem 3.4. If m =Qt_i=1pei

i (t ≥ 1) where pi’s are distinct primes, then LCP(m) = lcm [LCP(peii)].

Proof. We prove this by direct calculation. Let lcm [LCP(peii)] = β. C (k, β) (mod m) ≡£k2β (β − 1) + 1 ¤ (mod m) ≡ 1 (mod m) , C (k, β + 1) (mod m) ≡£k2(β + 1) β + 1 ¤ (mod m) ≡ 1 (mod m) , · · · , C (k, β + n) (mod m) ≡£k2(β + n) (β + n − 1) + 1 ¤ (mod m) ≡ ≡£k2β (β + 2n − 1) ¤ (mod m) +£k₂_{n (n − 1) + 1}¤ _{(mod m) ≡} ≡£k2n (n − 1) + 1 ¤ (mod m) ≡ C (k, n) (mod 2u) . So, we get LCP(m) = lcm [LCP(peii)]. 4. Pyramidal Numbers

In this section, we obtain the lengths of the periods of k−gonal pyramidal numbers modulo m. The notation LP Y (m) denote the length of the smallest period which the period is obtained each element of the pyramidal numbers by reducing modulo m for k ≥ 3.

Theorem 4.1. _{Let k ≡ 0 (mod 3) or k ≡ 1 (mod 3). The lengths of the} periods of the pyramidal numbers are as follows:

i. If p = 2, 3, then LP Y (pu) = pu+1 for u ∈ N. ii. If p 6= 2, 3 and θ ∈ N, then LP Y

¡

pθ¢_{= p}θ_. iii. If m = Qt_i=1pei

i (t ≥ 1) where pi’s are distinct primes, then LP Y (m) = lcm [LP Y (peii)].

Proof. _{We prove this by direct calculation. Let k ≡ 0 (mod 3) that is k = 3 ,} ( ∈ N).

i. _{If p = 2, then we have for u ∈ N} P₂(3 )u+1₋₁ (mod 2u) ≡ ≡ ∙ (2u+1−1)2 2 + ¡ 2u+1_{− 1}¢3¡3 6 − 1 3 ¢ −¡2u+1_{− 1}¢ ¡3−5 6 ¢¸ (mod 2u_{) ≡} ≡£23u ¡

5 − 22u+3¢¤ _{(mod 2}u_{) ≡ 0 (mod 2}u_{) ,} P₂(3 )u+1 (mod 2u) ≡ ∙ (2u+1)2 2 + ¡ 2u+1¢3¡3 6 − 1 3 ¢ −¡2u+1¢ ¡3−5 6 ¢¸ (mod 2u_{) ≡} ≡£23u ¡

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P₂(3 )u+1₊₁ (mod 2u) ≡ ≡ ∙ (2u+1₊₁₎2 2 + ¡ 2u+1_{+ 1}¢3¡3 6 − 1 3 ¢ −¡2u+1_{+ 1}¢ ¡3−5 6 ¢¸ (mod 2u_{) ≡} ≡£23u ¡

5 − 22u+3¢¤_{+ 1 (mod 2}u_{) ≡ 1 (mod 2}u_{) , · · · ,} P₂(3 )u+1_+n (mod 2u) ≡ ≡ ∙ (2u+1+n)2 2 + ¡ 2u+1_{+ n}¢3¡3 6 − 1 3 ¢ −¡2u+1_{+ n}¢ ¡3−5 6 ¢¸ (mod 2u_{) ≡} ≡£23u ¡

5 − 22u+3¢ (mod 2u)¤+hn₂2 + n3¡3₆ ₋1₃¢_{− n}¡3₆−5¢i (mod 2u_{) ≡} ≡ Pn(3 ) (mod 2u) .

So, we get LP Y (2u) = 2u+1.

If p = 3, then the proof is similar to the proof of the case p = 2 and is omitted. ii. _{If p 6= 2, 3 and θ ∈ N, then we have}

P_p(3 )u₋₁ (mod pu) ≡ ≡h(pu−1)2 2 + (pu− 1) 3¡3 6 − 1 3 ¢ − (pu_{− 1)}¡3−5 6 ¢i (mod pu_{) ≡} ≡hp6u (5 − 2.p u_{) (p}u_{+ 1)}i _{(mod p}u ) ≡ 0 (mod pu) , Pp(3 )u (mod pu) ≡ h_(pu₎2 2 + (p u₎3¡3 6 − 1 3 ¢ − (pu)¡3₆−5¢i (mod pu_{) ≡} ≡hp6u (5 − 2.p u_{) (p}u_{+ 1)}i _{(mod p}u_{) ≡ 0 (mod p}u_{) ,} P_p(3 )u₊₁ (mod pu) ≡ ≡h(pu+1)2 2 + (pu+ 1) 3¡3 6 − 1 3 ¢ − (pu_{+ 1)}¡3−5 6 ¢i (mod pu_{) ≡} ≡hp6u (5 − 2.p u_{) (p}u_{+ 1) + 1}i _{(mod p}u ) ≡ 1 (mod pu) , · · · , P_p(3 )u_+n (mod pu) ≡ ≡h(pu+n)2 2 + (pu+ n) 3¡3 6 − 1 3 ¢ − (pu_{+ n)}¡3−5 6 ¢i (mod pu_{) ≡} ≡hp6u (5 − 2.pu) (pu+ 1) i (mod pu_{) +}

+hn₂2 + n3¡3₆ ₋1₃¢_{− n}¡3₆−5¢i (mod 2u_{) ≡ P}n(3 ) (mod 2u) . So, we get LP Y

¡

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iii. Let lcm [LP Y (peii)] = β. P_β(3 )₋₁ _{(mod m) ≡}h(β−1)₂ 2 _{+ (β − 1)}3¡3 6 − 1 3 ¢ − (β − 1)¡3−5 6 ¢i (mod m) ≡ ≡hβ6(5 − 2.β) (β + 1) i (mod m) ≡ 0 (mod m) ,

P_β(3 ) _{(mod m) ≡}h(β)₂2 + (β)3¡3₆ ₋1₃¢_{− (β)}¡3₆−5¢i _{(mod m) ≡} ≡hβ6(5 − 2.β) (β + 1) i (mod m) ≡ 0 (mod m) , P_β+1(3 ) _{(mod m) ≡}h(β+1)₂ 2 + (β + 1)3¡3 6 − 1 3 ¢ − (β + 1)¡3−5 6 ¢i (mod m) ≡ ≡hβ6(5 − 2.β) (β + 1) + 1 i (mod m) ≡ 1 (mod m) , · · · ,

P_β+n(3 ) _{(mod m) ≡}h(β+n)₂ 2 + (β + n)3¡3₆ ₋1₃¢_{− (β + n)}¡3₆−5¢i _{(mod m) ≡} h β 6(5 − 2.β) (β + 1) i (mod m) +hn₂2 + n3¡3 6 − 1 3 ¢ − n¡36−5 ¢i (mod m) ≡ ≡ Pn(3 ) (mod m) . So, we get LP Y (m) = lcm [LP Y (peii)].

Theorem 4.2. _{Let k ≡ 2 (mod 3). The lengths of the periods of the pyramidal} numbers are as follows:

i. _{For u ∈ N, L}P Y (2u) = 2u+1.

ii. _{If p 6= 2 and θ ∈ N, then L}P Y ¡pθ¢= pθ. iii. If m = Qt_i=1pei

i (t ≥ 1) where pi’s are distinct primes, then LP Y (m) = lcm [LP Y (peii)].

Proof. The proof is similar to the proof of Theorem 4.1. and is omitted. 5. Open Question

Wall in [13] proved that the lengths of the periods of the recurring sequences obtained by reducing a Fibonacci sequences by a modulo m are equal to the lengths of the of ordinary 2−step Fibonacci recurrences in cyclic groups. The theory is expanded to 3-step Fibonacci sequence by Özkan, Aydin and Dikici [11]. Lü and Wang contributed to the study of the Wall number for the k-step Fibonacci sequence [10]. Some works on the concept have been made, for example, [4,7]. Deveci and Karaduman [5] extended the concept to Pell sequences in finite groups. Are there groups such that the lengths of the periods of recurrence sequences obtained by reducing according to a modulo m anyone of polygonal numbers, centered polygonal numbers and pyrimidal numbers are equal to the lengths of the periods of 2−step or k−step Fibonacci recurrences in these groups?

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6. Acknowledgment

The authors thank the referee for her/his valuable suggestions which improved the presentation of the paper.

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