ARITHMETIC PROGRESSIONS THAT CONSIST ONLY OF REDUCED RESIDUES

This paper contains an elementary derivation of formulas for multiplicative functions of m which exactly yield the following numbers: the number of distinct arithmetic progressions of w reduced residues modulo m; the number of the same with first term n; the number of the same with mean n; the number of the same with common difference n. With m and odd w fixed, the values of the first two of the last three functions are fixed and equal for all n relatively prime to m; other similar relations exist among these three functions. 2000 Mathematics Subject Classification. 11A07, 11A41.


Discussion
Theorem 3.1.For any n, w,  Theorem 3.4.For any n, w, Proof.Let j ≤ w, x ≤ p, y < p.In the residue class multiplication table modulo p, consider the following properties of the row whose factor is for some j are precisely those x for which (x + (j − 1)n, p) = 1 for some j, and those x satisfying [p − (j − 1)n] = [x] for j = 1, 2, 3,...,w are precisely those x for which (x + (j − 1)n, p) = 1, j = 1, 2, 3,...,w.For instance, in the example matrix with n = 5 and w = 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 those x for which (x +(j −1)5, 7) = 1, j = 1, 2, 3, 4. Hence, in the considered row, κ n,w (p) is the number of entries [x] not equal to [p −(j −1)n] for j = 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 of κ n,w (p) as follows: let (n, p) = 1 and consider the row whose factor is [y] = [n].If p > w, then the last w entries are equal to [p − (j − 1)n] for some j, which implies the formula p − w.If p ≤ w then all p entries are equal to [p − (j − 1)n] for some j, which implies the formula p − p = 0. Let (n, p) = 1 and consider the row whose factor is [1].Then for all w, 1 entry is equal to [p − (j − 1)n] = [0] for some j, which implies the formula p − 1.
Theorem 3.5.For any n, w, Proof.Let j ≤ w.The function υ n,w (p l ) is the number of x ∈ {1, 2, 3,...,p l } remaining after deleting those x where jx + n ≡ 0 (mod p) for some j.As x increases through the positive integers not exceeding p l in their natural order, {[jx]} w j=1 cycles through the w-string matrix modulo p.By Theorem 3.1, with each such cycle there are w or p − 1 or 1 or p distinct x where jx + n ≡ 0 (mod p) for some j.There are p l−1 such cycles and therefore the stated formulas for the 4 specific cases of υ n,w (p l ) Theorem 3.6.For any w with ) is the number of x ∈ {1, 2, 3,...,p l } remaining after deleting those x where for some j, jx − n ≡ 0 (mod p) or jx + n ≡ 0 (mod p).As x increases through the positive integers not exceeding p l in their natural order, {[jx]} v j=1 cycles through the v-string matrix modulo p.By Theorem 3.2, with each such cycle there are w − 1 or p − 1 distinct x where for some j, jx − n ≡ 0 (mod p) or jx + n ≡ 0 (mod p).There are p l−1 such cycles and therefore the stated formulas for the 2 specific cases of υ O n,w (p l ) are immediate: n,w (p l ) = 0 if p is odd and (n, p) = 1 and p < w.
Proof.Let j ≤ v.The function υ E n,w (p l ) is the number of x ∈ {1, 2, 3,...,p l } remaining after deleting those x such that for some j, (2j − 1)x − n ≡ 0 (mod p) or (2j −1)x +n ≡ 0 (mod p).As x increases through the positive integers not exceeding p l in their natural order, {[(2j − 1)x]} v j=1 cycles through the v × p matrix considered in the proof of Theorem 3.3.By Theorem 3.3, with each such cycle there are w or p − 1 or 1 or p distinct x such that for some j, (2j − 1)x − n ≡ 0 (mod p) or (2j −1)x +n ≡ 0 (mod p).There are p l−1 such cycles and therefore the stated formulas for the 4 specific cases of υ E n,w (p l ) are immediate: Proof.Let j ≤ w.The function κ n,w (p l ) is the number of x ∈ {1, 2, 3,...,p l } remaining after deleting those x such that x + (j − 1)n ≡ 0 (mod p) for some j.As x increases through the positive integers not exceeding p l in their natural order, [x] cycles through the multiplication table row considered in the proof of Theorem 3.4.By Theorem 3.4, with each such cycle there are w or p or 1 distinct x such that x + (j − 1)n ≡ 0 (mod p) for some j.There are p l−1 such cycles; therefore, the stated formulas for the 3 specific cases of κ n,w (p l ) are immediate: Theorem 3.9.The function υ n,w (m) is multiplicative.
Proof.We select any n, w and let (m 1 ,m 2 ) = 1.We consider F + m 1 ,n,w , F + m 2 ,n,w , and F + m 1 m 2 ,n,w and choose any residue class modulo m 1 containing integers in the complement of F + m 1 ,n,w .No integer in this residue class is in F + m 1 m 2 ,n,w .There are υ n,w (m 1 ) residue classes modulo m 1 containing the integers in F + m 1 ,n,w , and we choose any such residue class.Since (m 1 ,m 2 ) = 1, the m 2 least positive integers in this class form a complete residue system modulo m 2 [1, Theorem 3.6].There are υ n,w (m 2 ) integers in this residue system that are in F + m 2 ,n,w and thus in F + m 1 m 2 ,n,w .Since taking these υ n,w (m 2 ) least positive integers in each of these υ n,w (m 1 ) residue classes modulo Theorem 3.12.The function κ n,w (m) is multiplicative.
Proof of Theorems 3.10, 3.11, and 3.12.Employing the relevant restrictions on the variables n, w, prove Theorems 3.10, 3.11, and 3.12 along lines identical to that of Theorem 3.9's proof by making the appropriate substitutions with respectively Proof.By Theorems 3.5 and 3.9, υ n,w (m) = k i=1 υ n,w (p l i i ).Accordingly, we apply the appropriate definitions on the prime factors of m to obtain the stated formulas: case (i) of Theorem 3.14 covers cases (i), (ii), and (iii) of Theorem 3.5, and case (ii) of Theorem 3.14 covers case (iv) of Theorem 3.5.).Accordingly, we apply the appropriate definitions on the prime factors of m to obtain the stated formula which covers both cases of Theorem 3.6.Theorem 3.16.For any n and even w, ).Accordingly, we apply the appropriate definitions on the prime factors of m to obtain the stated formulas: case (i) of Theorem 3.16 covers cases (i), (ii), and (iii) of Theorem 3.7, and case (ii) of Theorem 3.16 covers case (iv) of Theorem 3.7.
Proof.By Theorems 3.8 and 3.12, κ n,w (m) = k i=1 κ n,w (p l i i ).Accordingly, we apply the appropriate definitions on the prime factors of m to obtain the stated formulas: case (i) of Theorem 3.17 covers cases (i) and (iii) of Theorem 3.8, and case (ii) of Theorem 3.17 covers case (ii) of Theorem 3.8.
Consider the following properties of the w-string matrix modulo p.In the jth row, those x satisfying [p − n] = [jx] are precisely those x for which (jx + n, p) = 1, and those x satisfying [p − n] = [jx] are precisely those x for which (jx + n, p) = 1.Since each column's first term is some [x], those columns not containing [p − n] identify exactly those x for which (jx + n, p) = 1, j = 1, 2, 3,...,w.For instance, in the example matrix with n = 3 and w = 4, those 3 columns not containing [4] as one of the first 4 entries identify exactly those x for which (jx +3, 7) = 1, j = 1, 2, 3, 4. Hence, in the considered w ×p matrix, υ n,w (p) is the number of columns not containing [p − n].Therefore, delete the columns containing [p − n] and thereby construct formulas for the 4 particular cases of υ n,w (p) as follows.Let (n, p) = 1.If p > w, then w columns contain [p − n], which implies the formula p − w.Since all entries in the pth column are [0], if p ≤ w, then p −1 columns contain [p −n], which implies the formula p − (p − 1) = 1.Let (n, p) = 1 (and thus [0] = [p − n]).If p > w, then 1 column contains [0], which implies the formula p − 1.Since all entries in the pth row are [0], if p ≤ w, then all p columns contain [0], which implies the formula p − p = 0. Theorem 3.2.For any w with w Consider the following properties of the v-string matrix modulo p.In the jth row, those x satisfying [n] = [jx] or [p −n] = [jx] are precisely those x for which (jx −n, p) = 1 or (jx +n, p) = 1, and those x satisfying [n] = [jx] and [p − n] = [jx] are precisely those x for 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 those x for which (jx − n, p) = (jx + n, p) = 1, j = 1, 2, 3,...,v.For instance, in the example matrix with n = 4 and w = 5, those 3 columns not containing [4] or [3] as one of the first 2 entries identify exactly those x for which (jx − 4, 7) = (jx +4, 7) = 1, j = 1, 2. Hence, in the considered v ×p matrix, υ O 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 of υ O n,w (p) as follows: if p ≥ w, then v columns contain [n], v columns contain [p −n], and no one column contains both [n] and [p − n] (since in each of the first p − 1 columns in the residue class multiplication table modulo p, the index of one of these entries exceeds v).Therefore 2v = w − 1 columns contain [n] or [p − n] which implies the formula p −(w −1) = p −w +1.Since all entries in the pth column are [0], if p < w, then p −1 columns contain [n] or [p − n] which implies the formula p − (p − 1) = 1.Theorem 3.3.For any n, w with w = 2v, (i) υ E n,w (p) = p − w if (n, p) = 1 and p > w; (ii) υ E n,w (p) = 1 if p = 2 or if p is odd and (n, p) = 1 and p < w; (iii) υ E n,w (p) = p − 1 if (n, p) = 1 and p > w; (iv) υ E n,w (p) = 0 if p is odd and (n, p) = 1 and p < 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.Consider the following properties of the resulting v × p matrix whose xth column is the sequence {[(2j − 1)x]} v j=1 .In the jth row, those x satisfying [n] = [(2j − 1)x] or [p − n] = [(2j − 1)x] are precisely those x for which ((2j − 1)x − n, p) = 1 or ((2j − 1)x + n, p) = 1, and those x satisfying [n] = [(2j − 1)x] and [p − n] = [(2j − 1)x] are precisely those x for 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 those x for 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 x for which ((2j − 1)x − 2, 7) = ((2j − 1)x + 2, 7) = 1, j = 1, 2. Hence, in the obtained v × p matrix, υ E n,w (p) is the number of columns not containing [n] or [p − n].Therefore, we delete the columns containing [n] or [p −n] and thereby construct formulas for the 4 particular cases of υ E n,w (p) as follows: let p = 2. Let (n, p) = 1.If p > w, then v columns contain [n], v columns contain [p −n], and no one column contains both [n] and [p − n] (since in each of the first p − 1 columns in the residue class multiplication table modulo p, these entries are not both odd-indexed).Therefore 2v = w columns contain [n] or [p − n], which implies the formula p − w.Since all entries in the pth column are [0], if p < w, then p − 1 columns contain [n] or [p − n], which implies the formula p − (p − 1) = 1.Let (n, p) = 1 (and thus [0] = [n] = [p − n]).If p > w, then 1 column contains [0], which implies the formula p − 1.Since all entries in the pth row are [0], if p < w, then all p columns contain [0], which implies the formula p − p = 0.If p = 2, then 1 column contains [n] or [p − n] for all n, w.This implies the formula p − (p − 1) = 1.

2. Definitions.
All variables are positive integers except z which is an integer, p is a prime.The residue class multiplication table modulo p is the p ×p matrix whose xth column for x ≤ p is the sequence {[jx]} p j=1 (use [z] = [jx] for some nonnegative z <p).A w-string modulo p is a sequence of the form {[jx]} w j=1 .The w-string matrix modulo p is the w × p matrix whose xth column for x ≤ p is the w-string {[jx]} w j=1 .(This matrix is just the first w rows of the residue class multiplication table modulo p.If w > p then we cycle through this table until w rows are obtained. for some i, p i is odd and (n, p i ) = 1 and p i < w.