SIZE OF THE SET OF RESIDUES OF INTEGER POWERS OF FIXED EXPONENT

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1 SIZE OF THE SET OF RESIDUES OF INTEGER POWERS OF FIXED EXPONENT RICHARD J. MATHAR Abstract. The ositive integers corime to some integer m generate the abelian grou (Z/nZ) of multilication modulo m. Admitting only integers of the form n k of a common fixed exonent k defines a subgrou of this. We consider the order of this subgrou which is divisor of Euler s multilicative totient function ϕ(m). If the generating integers n do not need to be corime to m, semigrous of the integers of the format n k (mod m) are defined in an equivalent manner by multilication modulo m. We show that the orders of these semigrous are multilicative functions of m for fixed exonent k, and derive linear recurrences and (conjectural) closed forms for the orders of these multilicative functions if m is a rime ower. 1. Grou of Multilication Modulo m 1.1. Set of Totients. The number of integers corime to some ositive integer m is counted by Euler s totient ϕ(m) [9, A000010]: Definition 1. (Euler s totient) (1) ϕ(m) = #n : (n, m) = 1, 1 n m}, Definition 2. (.,.) denotes the greatest common divisor of the two arguments. Theorem 1. [1, thm. 2.5] ϕ is a multilicative function of its argument Multilicative Grou with Totients. For some fixed modulus m, the elements of this set of totients n i with (n i, m) = 1 define an abelian grou (Z/mZ) of multilication modulo m with the roduct rule (2) (n i mod m)(n j mod m) = (n i n j ) mod m. 1 is the unit element. The order of an element n i of this grou is denoted by ord m (n i ). From Euler s theorem of number theory [1, Thm. 5.17] (or the theory of grous of finite order ϕ(m)) it follows that the order is a divisor of ϕ(m) [8, 3.2.2]. Examle 1. The multilication table of the grou of multilication modulo m = 10 is Table 1 with ord 10 (1) = 1, ord 10 (3) = ord 10 (7) = 4, ord 10 (9) = 2. Examle 2. The multilication table of the grou of multilication modulo m = 20 is Table 2 with ord 20 (1) = 1, ord 20 (3) = ord 20 (7) = ord 20 (13) = ord 20 (17) = 4, ord 20 (9) = ord 20 (11) = ord 20 (19) = 2. Date: November 10, Mathematics Subject Classification. Primary 05C30; Secondary 05C20, 05C75. Key words and hrases. Grah Enumeration, Combinatorics. 1

2 2 RICHARD J. MATHAR Table 1. Grou of multilication modulo 10 (Cayley table) Table 2. Grou of multilication modulo 20 (Cayley table). i\k n 1 n 2 1 mod m n 3 1 mod m n 4 1 mod m... 2 n 2 n 2 2 mod m n 3 2 mod m n 4 2 mod m... 3 n 3 n 2 3 mod m n 3 3 mod m n 4 3 mod m ϕ(m) n ϕ(m) n 2 ϕ(m) mod m n 3 ϕ(m) mod m... Table 3. Structure of the table of residues of k th owers in the multilicative grou modulo m Table 4. Table 3 for the modulus m = Number of residues of n k i (mod m). The owers of the elements n i of the grou of multilication modulo m, n k i (mod m), can be arranged as in Table 3. The table has ϕ(m) rows labeled uniquely by the elements n i, and columns labeled by the owers k 1. Examle 3. The table of the k-owers of the elements of the multilicative grou modulo m = 5 is shown in Table 4. The table of the k-owers of the elements of the multilicative grou modulo m = 10 is shown in Table 5. Remark 1. As a result of the abelian structure, the sequences of integers in two rows of Table 3 which are inverses of each other (in the grou) are given by inverting

3 SIZE OF THE SET OF RESIDUES OF n k mod m Table 5. Table 3 for the modulus m = 10. the left-right order or reading. This is equivalent to transversing the cycle grah of the grou in the two oosite directions Grou of residues of n k mod m. Theorem 2. The residues n k i (mod m), the set of numbers in the k-th column of Table 3, are a subgrou (Z/n k Z) of the multilicative grou modulo m: (3) (Z/n k Z) (Z/nZ) Proof. For two elements n k i (mod m) and n k j (mod m) in the same column k of the table, the roduct (defined with the multilication in the grou) is (n k i mod m)(n k j mod m) = (n i n j ) k mod m according to (2). Because n i n j is also an element in the grou of multilication modulo m, the (n i n j ) k mod m is also somewhere in the k-th column (closure roerty [1, Thm. 6.4 ]). Furthermore the unit element of the grou is in all columns of the table because 1 k mod m = 1 mod m. The finite set of numbers in some fixed k-th column of Table 3 defines: Definition 3. N m (k) is the number of distinct residues of the k-th owers of the elements in the multilicative grou modulo m, the order of (Z/n k Z). (4) N m (k) = #n k mod m : (n, m) = 1 n Z} = (Z/n k Z). A row in Table 3 is eriodic with eriod length ord m (n i ). The least common multile (l.c.m.) of these orders of the rows, [ord m (n 1 ), ord m (n 2 ),..., ord m (n ϕ(m) )], gives the eriod length of the entire table, and is known as Carmichael s λ-function [2, 6][9, A002322][5, Defn 2.20]. In consequence, N m (k) is a eriodic function of k with a eriod length given by λ(m). Examle 4. In Table 5 the number of distinct elements in column k are N 10 (1) = 4, N 10 (2) = 2, N 10 (3) = 4, N 10 (4) = 1, and eriodic with N 10 (k + 4) = N 10 (k) thereafter. λ(10) = 4. Because the order of the subgrou in the k-th column of the table has been defined as N m (k), and because subgrou orders divide grou orders by Lagrange s Theorem: (5) N m (1) = ϕ(m); N m (k) ϕ(m). If the exonent k is comosite with a roer divisor d k, 1 < d < k, one can interrut the comutation of the set of residues by comuting first the residues n d mod m, and then comuting their k/d-th owers modulo m. Because in the latter ste the number of residues cannot increase beyond the N m (k/d), we have: (6) N m (k) N m (d), d k. Even stronger, the residues n k mod m are a subgrou of the residues of n d mod m: The two subgrous of Z/mZ share the unit element. Each k-th ower can be written

4 4 RICHARD J. MATHAR k\m OEIS A A A A A A A A Table 6. To corner of the table of N m (k), orders of the subgrous of the k-th owers of the elements of the multilicative grou modulo m, and their sequence numbers in the Online Encycloedia of Integer Sequences [9]. as a d-th ower, so n n k/d is a homomorhism between the two subgrous. Again by Lagrange s theorem Theorem 3. (7) N m (k) N m (d), d k. So far this characterizes the residue sets for fixed modulus m and variable exonents k Multilicative grou order of m. For fixed k and variable m, scanning Table 6 along rows, we find: Theorem 4. For fixed exonent k the orders N m (k) are multilicative functions of m. Proof. We need to show that for corime (m 1, m 2 ) the size (order of the set of elements) of column k of two tables of the format of table 3 fulfills N m1 (k)n m2 (k) = N m1m 2 (k). We do this by settling the numbers corime to m 1 along rows and the numbers corime to m 2 along columns of a two-dimensional N m1 (k) N m2 (k) maing array, and by showing that there is a 1-to-1 ma of the roducts m 1 m 2 to airs of m 1 and m 2, which means, each m 1 m 2 can be laced at exactly one entry of the array according to that ma. If that association m 1 m 2 m 1, m 2 is indeed unique, the number of distinct elements in the grou of order N m1m 2 (k) is obviously the roduct of the grou orders N m1 (k) and N m2 (k), because in a rectangular array the number of entries is the roduct of the number of rows by the number of columns. The forward and backward mas are: (1) Given an element n k 1 mod m 1 and an element n k 2 mod m 2 with (n 1, m 1 ) = 1 and (n 2, m 2 ) = 1, fixing rows and columns in the maing table, the n 1 and n 2 are not necessarily unique in the multilicative grous modulo

5 SIZE OF THE SET OF RESIDUES OF n k mod m 5 m 1 and modulo m 2. (See for examle Table 5 where the 9 aears twice in column k = 2.) Select one reresentative air n 1 and n 2 and derive the unique solution x of the generalized Chinese Remainder Theorem for corime (m 1, m 2 ) = 1 [4, 7]: x n 1 (mod m 1 ); x n 2 (mod m 2 ). Note that x = n 1 + αm 1 and x = n 2 + βm 2 imly that x k (n 1 + αm 1 ) k n k 1 (mod m 1 ) and x k (n 2 +βm 2 ) k n k 2 (mod m 2 ) by binomial exansion: although n 1 and n 2 may not be unique reresentatives, the algorithm finds a unique x which has a unique ower x k with the fitting exonent to be ut as n 12 = x k (mod m 1 m 2 ) into the maing table. (2) Each element n k 12 in the residue set of k-owers modulo m 1 m 2 is maed uniquely onto a air of residues modulo m 1 and modulo m 2 as follows: because n 12 is corime to m 1 m 2 by definition, it is individually corime to m 1 and to m 2, so n k 12 aears in the row of n k 12 mod m 1 and in the column of n k 12 mod m 2. Finally note that the assembly of the elements in the table with the Chinese Remainder Theorem followed by the disassociation into row and column indices leads to the same row and columns that we started with. In summary, the maing table is full and all its entries are unique. Theorem 4 reduces the comutation of N m (k) to the cases where m = e are owers of rimes, and to finding the integer ratios imlied by Theorem 3. For the quartic and octic characters Finch derived the orders of the kernel of the grous [3]. Dividing the grou order ϕ( e ) = ( 1) e 1 through these orders yields Theorem 5. (8) N 2 e(4) = (9) N e(4) = Theorem 6. 1 (10) N 2 e(8) = (11) N e(8) = 1, e 3; 2 e 4, e 4. 2 e 1, 3 (mod 4), e 1; 1 4 e 1, 1 (mod 4), e , e 5; 2 e 5, e 5. 2 e 1, 3, 7} (mod 8), e 1; 1 4 e 1, 5 (mod 8), e 1; 1 8 e 1, 1 (mod 8), e 1. Based on a numerical search for linear recurrences and ordinary generating functions e 0 N e(k)xe we arrive at the following analysis for small k: Conjecture 1. For k = 5 1, e < 1; (12) N 5 e(5) = 4, e = 1; 4 5 e 2, e > 1.

6 6 RICHARD J. MATHAR n\k mod m 0 3 mod m 0 4 mod m mod m 1 3 mod m 1 4 mod m mod m 2 3 mod m 2 4 mod m mod m 3 3 mod m 3 4 mod m m 1 m 1 (m 1) 2 mod m (m 1) 3 mod m (m 1) 4 mod m... Table 7. Structure of the table of residues of k th owers of the integers < m for some fixed m. Conjecture 2. If k = 6 (13) N 2 e(6) = 1, e 3; 2 e 3, e 3. (14) N 3 e(6) = 1, e 2; 3 e 2, e 2. Conjecture 3. If k = 7 1, e < 1; (15) N 7 e(7) = 6, e = 1; 6 7 e 2, e 2. Conjecture 4. If k = 9 1, e = 0; (16) N 3 e(9) = 2, 1 e 3; 2 3 e 3, e 3. If the bases are corime to the exonents: Conjecture 5. If (, k) = 1, (17) N e(k) = 1 k e 1, 1 (mod k) 1 ( 1,k) e 1, 1 (mod k). 2. Number of k-th owers modulo m 2.1. Sets of k-th owers modulo m. The revious section considered owers of bases corime to m. In this section we consider the sets of owers of any numbers modulo m, totients or non-totients. A table of the k th owers modulo m is defined by inserting lines to Table 3 handling all 0 n < m leading to Table 7. Examle 5. The extension of Table 5 by rows including numbers (n, m) 1 is shown in Table 8. The grou roerty of Section 1.3 is lost, which means for examle that the unit element of multilication modulo m, the 1, may be missing in some of the rows of Table 7. These elements do not have inverses. The m m table for multilications modulo m creates only a semigrou. For the same reason that led to Equ. (6) we have:

7 SIZE OF THE SET OF RESIDUES OF n k mod m Table 8. Examle of Table 7 for modulus m = 10. Theorem 7. (18) R m (k) R m (d), d k Periodicity with resect to the exonent. There still is a eriodicity of n k mod m along each row of Table 7 which generalizes the grou orders of Section 1.3. Let x be the eriod length associated with n. Here is a constructive roof of its existence. First there are cases where n m. (19) n k+x n k (mod m) n k (n x 1) 0 (mod m) Let d be the greatest divisor of m which is divisible through all rime factors of n. By removing all owers of rimes for all rimes which m and n have in common, d = n, m ν(m) and (n, m/d) = 1. For examle d = 9 if m = 9 and n = 6. Definition 4. ν a (m) denotes the greatest exonent such that the ower of a divides m: a νa(m) m. The eriod length x is found by dividing the revious equation through d: n j (20) d nk j (n x 1) 0 (mod m d ). The exonent j is the smallest exonent such that n j /d is integer. To be valid for general exonents k we require (21) n x 1 0 (mod m d ) nx 1 (mod m d ). n and m/d are corime. x = ord m/d (n) is well defined, and equals the order of n in the multilicative grou modulo m/d. If m/d = 1, set x = 1. At that ste, the exonent k must be sufficiently large to suort this integer division, k j 0, which means, the eriodicity may ste in after some set of transient k. Examle 6. Column m = 16 in Table 9 starts with 3 transient terms and is λ(16) = 4-eriodic: 16, 4, 10, 2, 9, 3, 9. The reduction for cases (n, m) > 1 has removed rime owers from m; these x are divisors of those derived from the cases (n, m) = 1 (because the Carmichael function is essentially equal to ϕ and this is multilicative). Eventually the least common multile of these x is again Carmichael s λ-function, so inclusion of the n

8 8 RICHARD J. MATHAR k\m OEIS A A A A A A A A A A A Table 9. To corner of the table of R m (k), size of the set of the k-th owers of integers modulo m. that are not corime to m does not change the eriodicity of the number of residues as a function of k Cardinality of the Set of k-th Power Residues. Similar to Definition 3, droing the requirement of co-rimality, we count how many different integers aear in a column of Table 7: Definition 5. R m (k) is the number of distinct residues of the k-th owers of the integers modulo m. (22) R m (k) = #n k mod m : n Z}. The array of the R m (k) is illustrated in Table 9 for small m and k. The eriodicity of the residues evaluated in Section 2.2 for the individual n induces that, for fixed m, R m (k) is a eriodic function of k with a eriod length defined by the l.c.m. of the eriod lengths x for the individual n. Remark 2. As long as the n k are smaller than the modulus, all of them are counted by R m (k), so we have the simle bounds (23) k m 1 R m (k) m. The following theorem was claimed in the Online Encyloedia of Integer Sequences [9] for all sequences cited in Table 9 without roof; the observation deemed to be interesting enough to write u this manuscrit: Theorem 8. For fixed exonent k, the cardinality R m (k) of the set of residues is a multilicative function of m. The roof is essentially the same as the roof for Theorem 4. Since R m (k) is multilicative, the evaluation is reduced to derive R m (k) for rime owers m = e.

9 SIZE OF THE SET OF RESIDUES OF n k mod m 9 k\e Table 10. The number of weak comositions of k into e arts, ( k+e 1 e 1 ), a tilted version of Pascal s triangle [9, A007318] Inheritance Structure in Prime Power Bases. We can rehrase the roblem of folding the n k into the interval [0, e ) as reresenting n k for 0 n m in the basis : e (24) n = α j,n, j, 0 < α k j,n k, <. j=0 The ower has the multinomial exansion (25) n k = α j,nk, j j 0 = k 0 +k k e 1 =k k j 0 k k! k 0!k 1!k 2! k e 1! αk0 k 0,n k, (α k 1,n k,) k1 (α k2,n k, 2 ) k2 (α ke 1,n k, e 1 ) ke 1. Each term of the sum on the right hand side has a structure defined by a weak comosition (comosition into non-negative arts) of k into e arts, k e 1 j=0 k j. The number of weak comositions of k in e arts defines Table 10. Examle 7. For k = 3 and e = 4 the structure of the right hand side contains 20 weak comositions of 3; 16 of them are:

10 10 RICHARD J. MATHAR (26) \j Reducing n k modulo e means that only terms where e 1 jk j < e j=0 (the first moment of the row sums of the structure is less than e) contribute. Examle 8. If the constraint (26) is imosed to Examle 7, only the following 7 of the 20 structures remain: \j j jk j This reduces the number of contributing structures to those in Table 11. The ordinary generating function of row k in this table is x (27) (1 x) 2 k j=1 (1 xj ). So row k is obtained from row k 1 by convolution with an aerated sequence with generating function 1 (28) 1 x k = x ik. i 0 A combinatorial interretation of that convolution is: a weak comosition for (k, e) is constructed by (i) adding a 1 to the left-most art of a weak comosition for (k 1, e), or (ii) by adding a 1 to the left-most art of a weak comosition of (k 1, e k), and inserting a leading zero and k 1 trailing zeros, (iii) by adding

11 SIZE OF THE SET OF RESIDUES OF n k mod m 11 k\e OEIS A A A A A A A A A A A Table 11. To corner of the number of weak comositions of k into e arts under the restriction (26) [9, A092905]. a 1 to the left-most art of a weak comosition of (k 1, e 2k) and adding this with 2k zeros (2 leading, 2k 2 trailing), and so on Conjectured Terms for Prime Power Moduli. By numerical exerimentation with large set of different rimes and exonents e we arrive at the following set of conjectures. Conjecture 6. R e(k) obeys a linear recurrence with constant coefficients: (29) R e(k) = R e 1(k) + R e k(k) R e k 1(k), e k 1. Conjecture 7. If the modulus e and the exonent k are corime, so (, k) = 1, (30) R e(k) = 1 ( 1,k) e+k 1 k, (, k) = 1, e 0 1 This leaves the cases where (, k) > 1. If = k we summarize exerience based on 7 by: Conjecture 8. For = k, ( 1) e+k 2 (31) R e() = 1, e 1 (mod ), k =, e 0 + ( 1)e+k 2 1, e 1 (mod ) Conjectural equations for comosite k 12, (, k) > 1, are gathered in Equations (32) (40). e e 0 (mod 4) (32) R 2 e(4) = 2 + e e 1, 2, 3} (mod 4), k = 4, = 2, e 0 (33) e+k 3, e + 3 0, 1, 2, 3} (mod 6) R 2 e(6) = 2 + e+k 3, e + 3 4, 5} (mod 6), k = 6, = 2.

12 12 RICHARD J. MATHAR (34) e+k 2, e + 4 0, 1, 2, 3, 4} (mod 6) R 3 e(6) = 2 + e+k 2, e (mod 6), k = 6, = 3. e+3, e + 3 0, 1, 2, 3} (mod 8) (35) R 2 e(8) = 2 + e+3, e + 3 4, 5, 6, 7} (mod 8), k = 8, = 2. (36) 2 e+6, e + 6 0, 1, 2,..., 6} (mod 9) R 3 e(9) = 3 + 2e+6, e + 6 7, 8} (mod 9), k = 9, = 3. (37) e+7, e + 7 0, 1, 2, 3,..., 7} (mod 10) R 2 e(10) = 2 + e+7, e + 7 8, 9} (mod 10), k = 10, = 2. (38) 2 e+8, e + 8 0, 1, 2,..., 8} (mod 10) R 5 e(10) = 3 + 2e+8, e (mod 10), k = 10, = 5. (39) e+8, e + 8 0, 1, 2,..., 8} (mod 12) R 2 e(12) = 2 + e+8, e + 8 9, 10, 11} (mod 12), k = 12, = 2. (40) e+10, e , 1, 2,..., 10} (mod 12) R 3 e(12) = 2 + e+10, e (mod 12), k = 12, = 3. In all these cases the factor in the numerator of the floor-functions aears to be, the same as in (30). References 1 ( 1,k) 1. Tom M. Aostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Sringer, MR R. D. Carmichael, On comosite numbers which satisfy the fermat congrues a 1 1mod, Am. Math. Monthly 19 (1912), no. 2, Steven Finch, Quartic and octic characters modulo n, arxiv: (2016). 4. Aviezri S. Fraenkel, New roof of the general chinese remainder theorem, Proc. Amer. Math. Soc. 14 (1963), no. 5, MR Michal Křížek, Florian Luca, and Lawrence Somer, 17 lectures on fermat numbers, CMS Books in Mathematics, vol. 9th, Sringer, New York, MR Romeo Meštrović, Generalizations of carmichael numbers i, arxiv: [math.nt] (2013). 7. Oystein Ore, The general chinese remainder theorem, Am. Math. Monthly 59 (1952), no. 6, MR József Sándor and B. Crstici, Handbook of number theory ii, Kluwer, MR Neil J. A. Sloane, The On-Line Encycloedia Of Integer Sequences, Notices Am. Math. Soc. 50 (2003), no. 8, , htt://oeis.org/. MR URL: htt:// Max-Planck Institute of Astronomy, Königstuhl 17, Heidelberg, Germany