Arithmetic progressions that consist only of reduced residues

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http://ijmms.hindawi.com © Hindawi Publishing Corp.

ARITHMETIC PROGRESSIONS THAT CONSIST ONLY

OF REDUCED RESIDUES

PAUL A. TANNER III

(Received 13 June 2000 and in revised form 25 February 2001)

Abstract.This paper contains an elementary derivation of formulas for multiplicative functions ofmwhich exactly yield the following numbers: the number of distinct arith-metic progressions ofwreduced residues modulom; the number of the same with first termn; the number of the same with meann; the number of the same with common differ-encen. Withmand oddwfixed, the values of the first two of the last three functions are fixed and equal for allnrelatively prime tom; other similar relations exist among these three functions.

2000 Mathematics Subject Classiﬁcation. 11A07, 11A41.

1. Introduction. Consider this deﬁnition: in modulom, where(aj)wj=1and(bj)wj=1

are arithmetic progressions ofwresidues,(aj)wj=1and(bj)wj=1are distinct if and only

ifaj≡bjfor somej. (To illustrate, (1,2,3) and (−9,7,23) are not distinct in modulo 5.)

What is the number of distinct arithmetic progressions ofwreduced residues modulo m? Forw=1, there areφ(m)such progressions, whereφis the Euler phi function. For m=5 andw=3, there are 12 such progressions:{(1,2,3), (1,4,7), (1,6,11), (2,3,4), (2,4,6), (2,7,12), (3,6,9), (3,7,11), (3,8,13), (4,6,8), (4,8,12), (4,9,14)}. (InSection 2, it is explained how this set is representative of all distinct arithmetic progressions of 3 reduced residues modulo 5.) The answers to this and similar questions are the ﬁnd-ings of this paper. Part of the derivation mentioned in the abstract proceeds similarly to a standard 4-step derivation [1, Theorems 6.2–6.5] of a formula forφ. (Theorems

3.1,3.2,3.3,3.4,3.5,3.6,3.7,3.8,3.9,3.10,3.11,3.12,3.14,3.15,3.16, and3.17 en-compass these 4 steps. This is the motivation for grouping these 16 theorems into 4 collections.) The deﬁnitions are grouped together at the beginning so that after survey-ing the deﬁnitions, the reader can study the main results, Theorems3.14,3.15,3.16,

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2. Deﬁnitions. All variables are positive integers exceptzwhich is an integer,pis a prime. The residue class multiplication table modulopis thep×pmatrix whosexth column forx≤pis the sequence{[jx]}p

j=1(use[z]=[jx]for some nonnegativez < p).

Aw-string modulopis a sequence of the form{[jx]}w

j=1. Thew-string matrix modulo p is thew×p matrix whosexth column forx≤pis thew-string{[jx]}wj=1. (This

matrix is just the firstwrows of the residue class multiplication table modulop. Ifw > pthen we cycle through this table untilwrows are obtained.)|X|is the order of finite setX. The introduction defines distinct arithmetic progressions ofwresidues modulo m. We defineαas an arithmetic progression ofwreduced residues modulom.

Hm,w=

(n, x)|n≤m, x≤m, (n+jx, m)=(m, n)=1,

j=1,2,3, . . . , v, w=v+1.

(2.1)

For eachw >1, we deﬁne a functionρw asρw(m)= |Hm,w|. Deﬁneρ1(m)=φ(m).

Note thatρw(m)is the number of distinctα, since for any arithmetic progression of w >1 integers relatively prime to m, the ﬁrst term and common diﬀerence respec-tively are congruent modulomto somenandxsuch that(n, x)∈Hm,w.

Fm,n,w=

x|x≤m, (n+jx, m)=1, j=1,2,3, . . . , w,

Fm,n,w+ =

z|z≡x(modm)for somex∈Fm,n,w

. (2.2)

For eachn, w, we deﬁne a functionυn,w asυn,w(m)= |Fm,n,w|. Forw >1,υn,w(m)

is the number of distinctαsuch thatnis less than the ﬁrst term of each progres-sion by its common diﬀerence. For instance,υ3,4(7)=3:{(6,9,12,15), (8,13,18,23), (10,17,24,31)}.

FO m,n,w=

x|x≤m, (n−jx, m)=(n+jx, m)=(m, n)=1, j=1,2,3, . . . , v, w=2v+1,

Fm,n,wO+ =

z|z≡x(modm)for somex∈Fm,n,wO

.

(2.3)

For each oddw >1 and by letting(m, n)=1, we deﬁne a functionυO

n,wasυOn,w(m)=

|FO

m,n,w|. We also deﬁne υOn,1(m) = 1. For odd w, υOn,w(m) is the number of

distinct α with mean n. For instance, υO4,5(7) =3: {(2,3,4,5,6), (−8,−2,4,10,16), (−10,−3,4,11,18)}.

FE m,n,w=

x|x≤m,n−(2j−1)x, m=n+(2j−1)x, m=1, j=1,2,3, . . . , v, w=2v,

Fm,n,wE+ =

z|z≡x(modm)for somex∈Fm,n,wE

.

(2.4)

For eachnand even w, we deﬁne a functionυE

n,w asυEn,w(m)= |Fm,n,wE |. For even w, υE

n,w(m) is the number of distinct α with mean n. For instance, υE2,4(7)= 3:

{(−1,1,3,5), (−16,−4,8,20), (−19,−5,9,23)}.

Gm,n,w=

x|x≤m,x+(j−1)n, m=1, j=1,2,3, . . . , w,

G+m,n,w=

z|z≡x(modm)for somex∈Gm,n,w

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Table2.1

[1] [2] [3] [4] [5] [6] [0]

[2] [4] [6] [1] [3] [5] [0]

[3] [6] [2] [5] [1] [4] [0]

[4] [1] [5] [2] [6] [3] [0]

[5] [3] [1] [6] [4] [2] [0]

[6] [5] [4] [3] [2] [1] [0]

[0] [0] [0] [0] [0] [0] [0]

For eachn, w, we deﬁne a functionκn,w asκn,w(m)= |Gm,n,w|. Forw >1,κn,w(m)

is the number of distinct α with common diﬀerence n. For instance, κ5,4(7)=3:

{(1,6,11,16), (3,8,13,18), (5,10,15,20)}.

Fm,n,w∗ =

x|x≤m, (n+jx, m)=(m, n)=1, j=1,2,3, . . . , v, w=v+1. (2.6)

For eachw >1 and by letting(m, n)=1, we deﬁne a functionυ∗n,w asυ∗n,w(m)=

|Fm,n,w∗ |. We deﬁneυ∗n,1(m)=1. Note thatυ∗n,w(m)is the number of distinctαwith

ﬁrst termn. For instance,υ∗1,5(7)=3:{(1,2,3,4,5), (1,5,9,13,17), (1,8,15,22,29)}.

For the sets deﬁned in this paragraph, we select anyn, w. We have{pi}ki=1as the

distinct prime factors ofm=k i=1p

li i,{qa}q

a=1as thosepisuch that(n, pi)=1 and pi> w. We also have{rb}r

b=1as thosepisuch that(n, pi)=1 andpi> wand{sc}s c=1

as thosepisuch thatpi≥w. We have{td}t

d=1as thosepisuch that(n, pi)=1.

The proofs of Theorems 3.1, 3.2, 3.3, and 3.4 give the above instances υ3,4(7), υO4,5(7),υE2,4(7), andκ5,4(7)by use of an example matrix, the residue class

multiplica-tion table modulo 7 as shown inTable 2.1.

3. Discussion

Theorem3.1. For anyn, w,

(i) υn,w(p)=p−wif(n, p)=1andp > w;

(ii) υn,w(p)=1if(n, p)=1andp≤w;

(iii) υn,w(p)=p−1if(n, p)=1andp > w;

(iv) υn,w(p)=0if(n, p)=1andp≤w.

Proof. Letj≤w,x≤p. Consider the following properties of thew-string matrix

modulop. In thejth row, thosexsatisfying[p−n]=[jx]are precisely thosexfor which(jx+n, p)=1, and thosexsatisfying[p−n]=[jx]are precisely thosexfor which(jx+n, p)=1. Since each column’s ﬁrst term is some[x], those columns not containing[p−n]identify exactly thosexfor which(jx+n, p)=1,j=1,2,3, . . . , w. For instance, in the example matrix withn=3 andw=4, those 3 columns not con-taining[4]as one of the ﬁrst 4 entries identify exactly thosexfor which(jx+3,7)=1, j=1,2,3,4. Hence, in the consideredw×pmatrix,υn,w(p)is the number of columns

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implies the formulap−(p−1)=1. Let(n, p)=1 (and thus[0]=[p−n]). Ifp > w, then 1 column contains[0], which implies the formulap−1. Since all entries in the pth row are[0], ifp≤w, then allpcolumns contain[0], which implies the formula p−p=0.

Theorem_3.2. _{For any}_w_with_w=₂_v+1_let_{(n, p)}=₁_{. Then}

(i) υO

n,w(p)=p−w+1ifp≥w;

(ii) υO

n,w(p)=1ifp < w.

Proof. _Let_j≤_v_,_x≤_p_{. Consider the following properties of the}_v_{-string matrix}

modulop. In thejth row, thosexsatisfying[n]=[jx]or[p−n]=[jx]are precisely thosexfor which(jx−n, p)=1 or(jx+n, p)=1, and thosexsatisfying[n]=[jx] and[p−n]=[jx]are precisely thosexfor which(jx−n, p)=(jx+n, p)=1. Since each column’s ﬁrst term is some[x], those columns not containing[n]or[p−n] identify exactly thosex for which(jx−n, p)=(jx+n, p)=1,j=1,2,3, . . . , v. For instance, in the example matrix withn=4 andw=5, those 3 columns not containing [4]or[3]as one of the ﬁrst 2 entries identify exactly thosexfor which(jx−4,7)= (jx+4,7)=1,j=1,2. Hence, in the consideredv×pmatrix,υO

n,w(p)is the number of

columns not containing[n]or[p−n]. Therefore, delete the columns containing[n]or [p−n]and thereby construct formulas for the 2 particular cases ofυO

n,w(p)as follows:

ifp≥w, thenvcolumns contain[n],vcolumns contain[p−n], and no one column contains both[n]and[p−n](since in each of the ﬁrstp−1 columns in the residue class multiplication table modulop, the index of one of these entries exceedsv). Therefore 2v =w−1 columns contain [n] or [p−n] which implies the formula p−(w−1)=p−w+1. Since all entries in thepth column are[0], ifp < w, thenp−1 columns contain[n]or[p−n]which implies the formulap−(p−1)=1.

Theorem3.3. For anyn, wwithw=2v,

(i) υE

n,w(p)=p−wif(n, p)=1andp > w;

(ii) υE

n,w(p)=1ifp=2or ifpis odd and(n, p)=1andp < w;

(iii) υE

n,w(p)=p−1if(n, p)=1andp > w;

(iv) υE

n,w(p)=0ifpis odd and(n, p)=1andp < w.

Proof. Let j ≤v, x ≤p. For the w-string matrix modulo p, to consider only

the odd-indexed entries in the columns, we eliminate the even-indexed rows. Con-sider the following properties of the resultingv×pmatrix whosexth column is the sequence{[(2j−1)x]}v

j=1. In the jth row, thosex satisfying [n]=[(2j−1)x]or [p−n]=[(2j−1)x]are precisely thosexfor which((2j−1)x−n, p)=1 or((2j− 1)x+n, p)=1, and thosex satisfying[n]=[(2j−1)x]and[p−n]=[(2j−1)x] are precisely thosexfor which((2j−1)x−n, p)=((2j−1)x+n, p)=1. Since each column’s ﬁrst term is some[x], those columns not containing[n]or[p−n]identify exactly thosexfor which((2j−1)x−n, p)=((2j−1)x+n, p)=1,j=1,2,3, . . . , v. For instance, in the example matrix with n=2 and w =4, those 3 columns not containing[2]or[5]as one of the ﬁrst 2 odd-indexed entries identify exactly those xfor which((2j−1)x−2,7)=((2j−1)x+2,7)=1,j=1,2. Hence, in the obtained v×pmatrix,υE

n,w(p)is the number of columns not containing[n]or[p−n].

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for the 4 particular cases ofυE

n,w(p)as follows: letp=2. Let(n, p)=1. Ifp > w, then vcolumns contain[n],vcolumns contain[p−n], and no one column contains both [n]and[p−n](since in each of the ﬁrstp−1 columns in the residue class multi-plication table modulop, these entries are not both odd-indexed). Therefore 2v=w columns contain[n]or[p−n], which implies the formulap−w. Since all entries in thepth column are[0], ifp < w, thenp−1 columns contain[n]or[p−n], which implies the formulap−(p−1)=1. Let(n, p)=1 (and thus[0]=[n]=[p−n]). If p > w, then 1 column contains[0], which implies the formulap−1. Since all entries in thepth row are[0], if p < w, then allp columns contain[0], which implies the formulap−p=0. Ifp=2, then 1 column contains[n]or[p−n]for alln, w. This implies the formulap−(p−1)=1.

Theorem3.4. For anyn, w,

(i) κn,w(p)=p−wif(n, p)=1andp > w;

(ii) κn,w(p)=0if(n, p)=1andp≤w;

(iii) κn,w(p)=p−1if(n, p)=1.

Proof. Letj≤w,x≤p,y < p. In the residue class multiplication table modulop,

consider the following properties of the row whose factor is[y]=[n]if(n, p)=1 or the row whose factor is[1]if(n, p)=1. Thosexsatisfying[p−(j−1)n]=[x] for somejare precisely thosexfor which(x+(j−1)n, p)=1 for somej, and those xsatisfying[p−(j−1)n]=[x]forj=1,2,3, . . . , ware precisely thosexfor which (x+(j−1)n, p)=1,j=1,2,3, . . . , w. For instance, in the example matrix withn=5 andw=4, those 3 entries (the ﬁrst 3) in the ﬁfth row not equal to[7−(j−1)5]for j=1,2,3,4 identify exactly thosexfor which(x+(j−1)5,7)=1,j=1,2,3,4. Hence, in the considered row,κn,w(p)is the number of entries[x]not equal to[p−(j−1)n]

forj=1,2,3, . . . , w. Therefore delete those [x]equal to[p−(j−1)n]for some j, and thereby construct formulas for the 3 particular cases ofκn,w(p)as follows: let (n, p)=1 and consider the row whose factor is [y]=[n]. Ifp > w, then the last w entries are equal to[p−(j−1)n]for somej, which implies the formulap−w. Ifp≤wthen allpentries are equal to[p−(j−1)n]for somej, which implies the formulap−p=0. Let(n, p)=1 and consider the row whose factor is[1]. Then for allw,1 entry is equal to[p−(j−1)n]=[0]for somej, which implies the formula p−1.

Theorem_3.5. _{For any}_{n, w}_,

(i) υn,w(pl)=(p−w)pl−1if(n, p)=1andp > w;

(ii) υn,w(pl)=pl−1if(n, p)=1andp≤w;

(iii) υn,w(pl)=(p−1)pl−1if(n, p)=1andp > w;

(iv) υn,w(pl)=0if(n, p)=1andp≤w.

Proof. Letj≤w. The functionυ_n,w(pl)is the number ofx∈ {1,2,3, . . . , pl}

re-maining after deleting thosexwherejx+n≡0(modp)for somej. Asxincreases through the positive integers not exceedingpl_{in their natural order,}_{_[jx]_}w

j=1cycles

through thew-string matrix modulop. ByTheorem 3.1, with each such cycle there arew orp−1 or 1 or pdistinctxwherejx+n≡0(modp)for somej. There are pl−1_{such cycles and therefore the stated formulas for the 4 speciﬁc cases of}_υ

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are immediate:

(i) pl₋_wpl−1₌_(p₋_w)pl−1_;

(ii) pl₋_(p₋₁_)pl−1₌_pl−1_;

(iii) pl₋_pl−1₌_(p₋₁_)pl−1_;

(iv) pl₋_ppl−1₌_0.

Theorem_3.6. _{For any}_w_with_w=₂_v+1_let_{(n, p)}=₁_{. Then}

(i) υO

n,w(pl)=(p−w+1)pl−1ifp≥w;

(ii) υO

n,w(pl)=pl−1ifp < w.

Proof. Letj≤v. The functionυO_n,w(pl)is the number ofx∈ {1,2,3, . . . , pl}

re-maining after deleting thosex where for somej,jx−n≡0(modp)or jx+n≡ 0(modp). Asxincreases through the positive integers not exceedingpl_{in their}

nat-ural order,{[jx]}v

j=1cycles through thev-string matrix modulop. ByTheorem 3.2,

with each such cycle there arew−1 orp−1 distinctxwhere for somej,jx−n≡ 0(modp)orjx+n≡0(modp). There arepl−1_{such cycles and therefore the stated}

formulas for the 2 speciﬁc cases ofυO

n,w(pl)are immediate:

(i) pl₋_(w₋₁_)pl−1₌_(p₋_w₊₁_)pl−1_;

(ii) pl₋_(p₋₁_)pl−1₌_pl−1_.

Theorem_3.7. _{For any}_{n, w}_with_w=₂_v_,

(i) υE

n,w(pl)=(p−w)pl−1if(n, p)=1andp > w;

(ii) υE

n,w(pl)=pl−1ifp=2or ifpis odd and(n, p)=1andp < w;

(iii) υE

n,w(pl)=(p−1)pl−1if(n, p)=1andp > w;

(iv) υE

n,w(pl)=0ifpis odd and(n, p)=1andp < w.

Proof. _Let_j≤_v_{. The function}_υE_n,w_(pl₎_{is the number of}_x∈ {1_,₂_,₃_{, . . . , p}l}

re-maining after deleting those x such that for somej,(2j−1)x−n≡0(modp)or (2j−1)x+n≡0(modp). Asxincreases through the positive integers not exceeding pl_{in their natural order,} _{_[(₂_j₋₁_)x]_}v

j=1 cycles through thev×p matrix

consid-ered in the proof ofTheorem 3.3. By Theorem 3.3, with each such cycle there are w orp−1 or 1 or p distinctx such that for somej,(2j−1)x−n≡0(modp)or (2j−1)x+n≡0(modp). There arepl−1_{such cycles and therefore the stated}

formu-las for the 4 speciﬁc cases ofυE

n,w(pl)are immediate:

(i) pl₋_wpl−1₌_(p₋_w)pl−1_;

(ii) pl₋_(p₋₁_)pl−1₌_pl−1_;

(iii) pl₋_pl−1₌_(p₋₁_)pl−1_;

(iv) pl₋_ppl−1₌_0.

(i) κn,w(pl)=(p−w)pl−1if(n, p)=1andp > w;

(ii) κn,w(pl)=0if(n, p)=1andp≤w;

(iii) κn,w(pl)=(p−1)pl−1if(n, p)=1.

Proof. Letj≤w. The functionκ_n,w(pl)is the number ofx∈ {1,2,3, . . . , pl}

re-maining after deleting thosex such thatx+(j−1)n≡0(modp)for somej. Asx increases through the positive integers not exceedingpl_{in their natural order,}_[x]

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By Theorem 3.4, with each such cycle there are w or p or 1 distinct x such that x+(j−1)n≡0(modp)for somej. There arepl−1_{such cycles; therefore, the stated}

formulas for the 3 speciﬁc cases ofκn,w(pl)are immediate:

(i) pl₋_wpl−1₌_(p₋_w)pl−1_;

(ii) pl₋_ppl−1₌_0;

(iii) pl₋_pl−1₌_(p₋₁_)pl−1_.

Theorem3.9. The functionυ_n,w(m)is multiplicative.

Proof. _{We select any}_{n, w}_{and let}_(m₁_{, m}₂₎=_{1. We consider}_F_m+

1,n,w,Fm+2,n,w, and Fm+1m2,n,wand choose any residue class modulom1containing integers in the comple-ment ofFm+1,n,w. No integer in this residue class is inF

+

m1m2,n,w. There areυn,w(m1) residue classes modulom1containing the integers inFm+1,n,w, and we choose any such residue class. Since(m1, m2)=1, them2least positive integers in this class form a

complete residue system modulom2[1, Theorem 3.6]. There areυn,w(m2)integers

in this residue system that are inFm+2,n,w and thus inF +

m1m2,n,w. Since taking these

υn,w(m2)least positive integers in each of theseυn,w(m1)residue classes modulo m1formsFm1m2,n,w,υn,w(m1m2)=υn,w(m1)υn,w(m2).

Theorem3.10. The functionυO_n,w(m)is multiplicative.

Theorem_3.11. _{The function}_υE_n,w_(m)_{is multiplicative.}

Theorem3.12. The functionκ_n,w(m)is multiplicative.

Proof of Theorems3.10,3.11, and3.12. Employing the relevant restrictions on

the variablesn, w, prove Theorems3.10,3.11, and3.12along lines identical to that of Theorem 3.9’s proof by making the appropriate substitutions with respectively FO

m,n,w, FO +

m,n,w, υOn,w(m);Fm,n,wE , FE +

m,n,w, υEn,w(m);Gm,n,w, Gm,n,w+ , κn,w(m).

Remark3.13. For the following, we recall the convention that empty products have

value 1.

(i) υn,w(m)= q

a=1(qa−1) r

b=1(rb−w) k

i=1p li−1

i if for anyi,(n, pi)=1orpi> w;

(ii) υn,w(m)=0if for somei,(n, pi)=1andpi≤w.

Proof. _{By Theorems}_3.5_and_3.9_,_υ_n,w_(m)=k_i₌₁_υ_n,w_(pli

i). Accordingly, we apply

the appropriate deﬁnitions on the prime factors ofmto obtain the stated formulas: case (i) ofTheorem 3.14covers cases (i), (ii), and (iii) ofTheorem 3.5, and case (ii) of

Theorem 3.14covers case (iv) ofTheorem 3.5.

Theorem 3.15. For any odd w > 1 let (m, n)= 1. Then υO_n,w(m)=s_c₌₁(s_c− w+1)ki=1p

l_i−1 i .

Proof. By Theorems3.6and3.10,υO_n,w(m)=k_i₌₁υO_n,w(p_ili). Accordingly, we ap-ply the appropriate deﬁnitions on the prime factors ofmto obtain the stated formula which covers both cases ofTheorem 3.6.

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(i) υE

n,w(m)= q

a=1(qa−1) r

b=1(rb−w) k

i=1p l_i−1

i if for anyi,pi=2or(n, pi)=1 orpi≥w;

(ii) υE

n,w(m)=0if for somei,piis odd and(n, pi)=1andpi< w.

Proof. By Theorems3.7and3.11,υE_n,w(m)=k_i₌₁υE_n,w(p_ili). Accordingly, we ap-ply the appropriate deﬁnitions on the prime factors ofmto obtain the stated formulas: case (i) ofTheorem 3.16covers cases (i), (ii), and (iii) ofTheorem 3.7, and case (ii) of

Theorem 3.16covers case (iv) ofTheorem 3.7.

(i) κn,w(m)= t

d=1(td−1) r

b=1(rb−w) k

i=1p li−1

i if for anyi,(n, pi)=1orpi> w;

(ii) κn,w(m)=0if for somei,(n, pi)=1andpi≤w.

Proof. By Theorems3.8and3.12,κ_n,w(m)=k_i₌₁κ_n,w(pl_ii). Accordingly, we ap-ply the appropriate deﬁnitions on the prime factors ofmto obtain the stated formu-las: case (i) ofTheorem 3.17covers cases (i) and (iii) ofTheorem 3.8, and case (ii) of

Theorem 3.17covers case (ii) ofTheorem 3.8.

Theorem 3.18. For any w > 1 let (m, n)= 1. Then υ_n,w∗ (m)=s_c₌₁(s_c−w+ 1)ki=1p

l_i−1 i .

Proof. _F_m,n,w∗ =_F_m,n,w₋₁_{; therefore,} _υ∗_n,w_(m)=_υ_n,w₋₁_(m)_{. Therefore, we apply} the appropriate deﬁnitions on the prime factors of m to modify the formula of

Theorem 3.14(i) and thus obtain the formula ofTheorem 3.18.

Corollary_3.19. _{For parts (i), (ii), and (iii), select any}_w_{and let}_{(m, n)}=_{(m, n}₎=₁_.

(i) υ∗n,w(m)=υ∗n,w(m).

(ii) Ifwis even, then

υ∗n,w+1(m)=υ∗n,w+1(m)=υ O

n,w+1(m)=υOn,w+1(m)=υ E

n,w(m)=υEn,w(m). (3.1)

(iii) If each prime factor ofmis greater thanw, then

υ∗n,w+1(m)=υ∗n,w+1(m)=κn,w(m)=κn,w(m). (3.2)

Theorem_3.20. _{For any}_{w >}₁_,_ρ_w_(m)=_φ(m)υ∗_n,w_(m)_.

Proof. _For_{(n, x)}∈_H_m,w_{, there are}_φ(m)_{instances of}_n_{, and by}_{Corollary 3.19}_(i), υ∗n,w(m)instances ofxfor eachn. Therefore,φ(m)υ∗n,w(m)= |Hm,w|.

References

[1] K. H. Rosen,Elementary Number Theory and its Applications, 3rd ed., Addison-Wesley Publishing, Massachusetts, 1993.MR 93i:11002. Zbl 766.11001.

Paul A. Tanner III: Department of Mathematics, University of South Florida,4202 E. Fowler Avenue, Tampa, FL33620, USA