PARTIAL SEMIGROUPS AND PRIMALITY INDICATORS IN THE FRACTAL GENERATION OF BINOMIAL COEFFICIENTS TO A PRIME SQUARE MODULUS

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1 PARTIAL SEMIGROUPS AND PRIMALITY INDICATORS IN THE FRACTAL GENERATION OF BINOMIAL COEFFICIENTS TO A PRIME SQUARE MODULUS D. DOAN, B. KIVUNGE 2, J.J. POOLE 3, J.D.H SMITH 4, T. SYKES, AND M. TEPLITSKIY 5 Abstract. This paper, resulting from two summer programs of Research Experience for Undergraduates, examines the congruence classes of binomial coefficients to a prime square modulus as given b a fractal generation process for lattice path counts. The process depends on the isomorphism of partial semigroup structures associated with each iteration. We also consider integralit properties of certain critical coefficients that arise in the generation process. Generalizing the application of these coefficients to arbitrar arguments, instead of just to the prime arguments appearing in their original function, it transpires that integralit of the coefficients is indicative of the primalit of the argument.. Introduction The general topic of this paper is the investigation of a fractal generation process for modular binomial coefficients. Previous work in the area, more recentl from a dnamical sstems viewpoint, has most often focussed on the distinction between zero and non-zero congruences [] [2] [5] [7] [8], connecting back to Kummer s classical results on the divisibilit of binomial coefficients b prime powers [4]. Our concern is rather with an algebraic fractal generation process for each modulus, exhibiting isomorphisms of total or partial semigroup structures defined on sets of digits and on sets of squares under the Pascal addition or tile sum of Definition 2.3. Throughout the paper, p will denote a given prime number. Section 2 reviews the case of modulus p. (Although this case is alread well understood, our algebraic approach will serve as a useful model for the more complex prime square case.) Theorem 2.4 gives an isomorphism from the (total) additive group C p of integers modulo the prime p to a set of p p tiles appearing in Pascal s square modulo p. The main theorem of the paper, proved in the final Section 6, is the corresponding result for modulus p 2 (Theorem 4.5). This theorem gives an isomorphism to a set of p p tiles appearing in Pascal s square modulo p 2 from a partial semigroup structure D p on an indexed set of digits modulo p 2 (Definition 3.). The isomorphism, which also functions as the ke iterative step in the fractal generation process (Corollar 4.6), is defined in terms of certain production coefficients (Definition 4.) that ma be viewed as modular harmonic sums, or discrete modular versions of the logarithmic 99 Mathematics Subject Classification. Primar B65; Secondar A07, A5, 20M99. Ke words and phrases. binomial coefficient, Pascal s triangle, lattice paths, fractal, semigroup, partial algebra, rectangular band, primalit test. During the second summer of work on this project, Kivunge, Poole, Smith and Teplitski were partiall supported b NSF grant DMS Poole and Skes received additional funding from the Alliance and the George Washington Carver Program.

2 2 DOAN, KIVUNGE, POOLE, SMITH, SYKES, AND TEPLITSKIY integral r dt/t = log r. Section 5 generalizes the application of these coefficients to arbitrar arguments, instead of just to the prime arguments appearing in their original function. It transpires that integralit of the coefficients is indicative of the primalit of the argument. Problems 5.5 and 5.6 ask for a determination of exact conditions for this integralit, and for a combinatorial interpretation of the coefficients in those cases where the are integral. A distinguishing feature of our approach is the wa we address binomial coefficients, using Pascal s square as partiall displaed in Table. Thus the binomial coefficient ( ) x+ appears in the location with coordinates (x, ). We consider the square as the result of the iterative construction process initialized b placing an entr of at each location having at least one zero coordinate, and then filling in b the linear assembl rule ( ) ( x + x + (.) = ) + at each location with both coordinates positive. ( ) x + x\ : Table. Pascal s Square. Displaing binomial coefficients in this form, rather than in the more customar Pascal s triangle, is well known to identif ( ) x+ directl as the number of geodesics or minimal-length paths through points of the square lattice from (0, 0) to (x, ). (Each such path arriving at (x, ) previousl passed through either (x, ) or (x, ), while points on the border have a unique geodesic from the origin.) In the fractal generation process embodied in Corollar 4.6, the expansion of each digit of Pascal s square modulo p 2 depends on the residues of its addressing coordinates x, modulo p. 2. Prime moduli We begin b considering an algebraic fractal construction of Pascal s square to the prime modulus p. Lemma 2.. There is a p p block p 0 p :... :

3 FRACTAL BINOMIAL COEFFICIENTS 3 appearing in Pascal s square modulo p. In particular, all except the first elements of the bottom row and rightmost column are zero. Proof. A p p block bordered on the left and the top b ones appears in the NW corner {(x, ) 0 x, < p} of the ordinar, non-modular Pascal s square. Consider the diagonal {(x, ) x + = p} just below the diagonal from the SW to the NE corner of the block. All the binomial coefficients appearing on that diagonal are of the form ( p ) with 0 < < p. Now ( p ) = p!!(p )!. In this fraction, all the numbers multiplied together in the denominator are strictl less than p, so do not cancel the p appearing in the numerator. This implies that ( ) p with 0 < < p is divisible b p. Thus there are zeroes in the corresponding places of the modular square, and the rest of the block is completed b zeroes according to the assembl rule (.). Lemma 2.2. For each 0 r < p, a p p block of the form r r r... r r r 2r... (p )r 0 r (p )r :... : 0 r is assembled according to the rule (.) of Pascal s square modulo p. Proof. The assembl rule is linear, so the blocks of Lemma 2.2 are obtained as the multiples b r of the block of Lemma 2.. Definition 2.3. Given p p blocks x x 2... x p x 2... x 2p and x p x p2... x pp 2... p p, p p2... pp their Pascal sum or tile sum is defined to be the p p block obtained b filling in the bottom p p right hand corner of the scheme z z 2... z p z 2... z 2p z p z p2... z pp 2... p p p p2... pp x x 2... x p z z 2... z p x 2... x 2p z 2... z 2p x p x p2... x pp z p z p2... z pp according to the assembl rule (.) of Pascal s square.

4 4 DOAN, KIVUNGE, POOLE, SMITH, SYKES, AND TEPLITSKIY Theorem 2.4. For 0 r < p, let [r] denote the p p block r r... r r... 0 r from Lemma 2.2. Then there is an isomorphism r [r] from the additive group C p of integers modulo p to the set of p p blocks under Pascal addition modulo p. Proof. For each 0 r, s < p, consider the modular Pascal addition s s... s s... 0 s r r... r * r... 0 r of [r] to [s]. The square marked b is filled in as r + s (modulo p). Because of the adjoining zeroes, the remaining squares in the same row and the same column as the marked square are also filled in as r + s. This creates [r + s] as the Pascal sum of [r] and [s] modulo p. Corollar 2.5. The Pascal square to a prime modulus p is generated b the following fractal process: () Start with an initial configuration of ; (2) For each iterative step, the output configuration is obtained b appling the production rule r [r] to each entr of the input configuration. For each natural number x, let... x 2 x x 0 be the base p expansion of x, so that (2.) x = x i p i i=0 with integers 0 x i < p. The following immediate consequence of Corollar 2.5 is a well-known instance of Kummer s criterion [2] [4]. Corollar 2.6. If there is a natural number i such that x i + i p, then ( ) x + 0 (mod p). Proof. Under the stated condition, the (x, )-entr of Pascal s square mod p includes an (x i, i )-entr of a tile [r] in its ancestr according to the fractal process of Corollar 2.5. B Lemma 2.2, this entr is 0, which expands to an all-zero tile at each step. 3. The partial semigroup Definition 3. of this section specifies the partial semigroup structure D p, involving residues modulo p 2, that for the prime square case plas a role analogous to that plaed b the cclic group C p of residues modulo p in Theorem 2.4. The modular locations of the definition will correspond to the modulo p residues x 0, 0 of

5 FRACTAL BINOMIAL COEFFICIENTS 5 coordinates x, of absolute locations in Pascal s square, according to the notation of (2.). This modular addressing is a ke feature of our fractal generation process. Definition 3.. The algebra of located residues modulo p is defined to be the set D p of all elements r x with r Z/p 2 Z, x, C p such that ( x (3.) x x (mod p). + ) (mod p). r (mod p 2 ). The residues x, modulo p are known as the modular locations. The partial addition on D p is defined b (3.2) r x( ) + s (x ) = (r + s) x if and onl if x x (mod p). (mod p). ( x + ) ( x r (mod p 2 + ) ), s (mod p 2 ). From the discussion of Remark 3.3 below, it will transpire that the algebra structure defined on D p b (3.2) is a partial semigroup. Example 3.2. The partial addition table for D 2 is exhibited as follows. Note that the columns have been labelled in a different order to the rows, so that transposition of Pascal s square modulo 4 corresponds to transposition of the table Table 2. Partial addition on D 2. Remark 3.3. One might choose to extend the partial addition (3.2) on the set D p to a total addition (3.3) r xz + s t = (r + s) x on the set T p = Z/p 2 Z Z/pZ Z/pZ = {r x r Z/p 2 Z, x, Z/pZ}.

6 6 DOAN, KIVUNGE, POOLE, SMITH, SYKES, AND TEPLITSKIY Note that the subset (3.4) {0 x x, Z/pZ} is a subalgebra of T p that forms a so-called rectangular band [6,.3]. It is apparent that the operation (3.3) is associative, making T p a semigroup, namel the product of the cclic group Z/p 2 Z with the rectangular band (3.4). However, T p is certainl not a group, since for example = 0 00 = The fractal process Just as in the modulo p case, the fractal generation process for Pascal s square modulo p 2 expands digits into p p blocks. The expansion process involves multiplication of the modular locations of the digit b certain coefficients that ma be viewed as modular harmonic sums, or discrete modular versions of r dt/t = log r. Definition 4.. For each positive integer r less than p, define the production coefficient (4.) λ r = r as a residue modulo p (recalling that the non-zero residues, 2,..., r are invertible modulo p). B convention for r = 0, the production coefficient λ 0 is defined to be zero. For odd primes, the production coefficients are smmetrical. Lemma 4.2. For odd p and 0 r < p/2, one has λ r = λ p r. Proof. For 0 < s < p, there is a congruence ( s s + ) = s p s s (p s) p s = = 0 modulo p, so that s + p s = 0 [3, 7.8]. The statement of the lemma is proved b downward induction: it is triviall true for r = (p )/2. Suppose λ s = λ p s. Then ( ) ( ) λ s λ p s λs λ p (s ) = s + p s = 0, so λ s λ p (s ) = 0 as required. Corollar 4.3. If p is odd, then λ p = λ 0 = 0. Note that λ p = for p = 2. The ke role of the production coefficients appears in the following: Definition 4.4. Suppose r Z/p 2 Z, x, C p. Then the located block [r] x is defined to be the p p arra of Z/p 2 Z-elements (4.2) r r( + pλ x) r( + pλ 2 x)... r( + pλ p x) r( + pλ )... : r( + pλ 2 )... : :... : r( + pλ p ) completed according to the assembl rule (.) modulo p 2. (Since λ 0 = 0, the top left-hand entr ma also be written in the equivalent forms r( + pλ 0 x) = r(+pλ 0 ), consistent with the remaining first row and column entries respectivel.)

7 FRACTAL BINOMIAL COEFFICIENTS 7 The main theorem ma now be stated as follows, along with its immediate corollar ielding the fractal generation process for Pascal s square modulo p 2. Theorem 4.5. There is a homomorphism (4.3) r x [r] x from the partial semigroup D p of located residues modulo p 2 to the algebra of located p p blocks under Pascal addition modulo p 2. The proof of Theorem 4.5 is given in Section 6. Corollar 4.6. The Pascal square to a prime square modulus p 2 is generated b the following fractal process: () Start with an initial configuration of 00 ; (2) For each iterative step, the output configuration is obtained b appling the production rule r x [r] x to each modularl located entr of the input configuration. Remark 4.7. The homomorphism of Theorem 4.5 cannot extend to the total semigroup T p of Remark 3.3, since it would take associative additions of T p to non-associative unlocated tile additions. 5. Generalized production coefficients In this section, we digress from the context of Theorem 4.5 to consider the generalized production coefficients (5.) λ r (q) = { ( ) q + r } = { } (q + r)(q + r )... (q + ) r! q q q r! for arbitrar positive integers q and 0 < r < q. For q prime, the generalized production coefficients reduce to the modular production coefficients of Definition 4. (see Corollar 5.2 below). Our concern is the question of when the generalized production coefficients take integral values. The following propositions suggest that integralit of the coefficients λ r (q) is an indicator of the primalit of q. We use Landau s big O notation in an algebraic sense, to identif a certain integral multiple O(n) of an integer n (contrast with [3,.6]). Proposition 5.. Suppose that r is a prime divisor of a composite positive integer q. Then the generalized production coefficient λ r (q) is not integral. Proof. The generalized production coefficient (5.) expands as λ r (q) = { [ O(q 2 ) + q r! q r! ]} (5.2) r r! O(q) + r! + 2 = + + r! r + (r )!, r! a fraction in which all the terms in the numerator and denominator are positive integers. Recalling that r divides q, it is apparent that the prime r is a divisor of each summand in the numerator except the last. Thus the numerator, not being congruent to 0 modulo r, does not contain a factor of r that would cancel the prime factor r of the denominator. In other words, the coefficient λ r (q) is not integral in this case.

8 8 DOAN, KIVUNGE, POOLE, SMITH, SYKES, AND TEPLITSKIY Corollar 5.2. If q is a prime number, then the generalized production coefficients reduce modulo q to the production coefficients of Definition 4.. Proof. Equation (5.2) shows that (5.) is congruent to (4.) when q is the prime p of Section 4. Remark 5.3. For a prime r, the non-integralit condition of Proposition 5. is necessar for r to divide q, but not sufficient. For example, λ 5 (27) is not integral (although λ 7 (27) is). Proposition 5.4. The generalized production coefficient λ r (q) is integral for all positive integers r that are less than the smallest prime divisor p of q. Proof. For r < p, there is an integer ( ) q + r (q + r)... (q + ) (5.3) = = q r! qp (q), r! where P (q) is a polnomial in q with integer coefficients. For an positive integer m r < p, the number m does not divide q, and so r! is coprime to q. Thus r! cancels with P (q) in the final term of (5.3), and is also integral. λ r (q) = q { ( q + r q ) } Propositions 5. and 5.4 suggest the following: = P (q) r! Problem 5.5. For each positive integer q, determine exactl which values of r make the generalized production coefficient λ r (q) integral. Problem 5.6. Is there a combinatorial interpretation of the coefficient λ r (q) in those cases for which it is integral? 6. Proof of the main theorem This section is devoted to the proof of Theorem 4.5. The proof demonstrates the preservation of the located partial addition (6.) r x( ) + s (x ) = (r + s) x from D p under the production rule (4.3). It depends on a local version of the transposition smmetr of the modulo p Pascal square. Lemma 6.. In the context of (6.), there is a congruence (6.2) rx s (mod p). Proof. For natural numbers x,, one has ( ) ( ) x + (x + )! x + (6.3) x = x (x )!( )! =. The desired result (6.2) is then just the modulo p reduction of (6.3). In view of the anomalous behavior of λ p for p = 2, it is convenient to treat that case separatel.

9 FRACTAL BINOMIAL COEFFICIENTS 9 Proposition 6.2. There is a homomorphism r x [r] x from the algebra D 2 of located residues modulo 4 to the algebra of located 2 2 blocks under Pascal addition modulo 4. Proof. For p = 2, the block (4.2) completes to [r] x = r r( + 2x) r( + 2) 2r( + x + ) Corresponding to the partial addition (6.) in D 2, one then has the tile sum s s[ + 2(x )] s( + 2) 2s(x + ) r r( + 2x) r + s + 2(rx + s) (r + s)( + 2x) r[ + 2( )] 2r(x + ) (r + s)( + 2) 2(r + s)( + x + ) To verif the homomorphic propert, it remains to establish that rx + s 0 (mod 2). But this follows immediatel b the case p = 2 of Lemma 6.. Example 6.3. The case p = 3 of Theorem 4.5 is also sufficientl direct that it is worth exhibiting explicitl. The block (4.2) now completes to [r] x = r r( + 3x) r r( + 3) 2r + 3r(x + ) 3r( + x + ) r 3r( + x + ) 6r( + x + ) The tile sum orresponding to the partial addition (6.) in D 3 is : s 3s(x + ) 6s(x + )... r r + s (r + s) + 3s(x + ) r + s... 3r(x + ) r + s + 3r(x + ) r(x + ) r + s To verif the homomorphic propert, it remains to establish 3s(x + ) = 3(r + s)x and 3r(x + ) = 3(r + s). These equations follow immediatel b the case p = 3 of Lemma 6., namel rx s (mod 3). For the proof of Theorem 4.5 in the general odd prime case, a fuller description of [r] x is provided b Proposition 6.7 below. The proposition depends on three lemmas. The reciprocals on the right hand sides of the equations (6.4), (6.5) and (6.6) in the statements of the lemmas are interpreted as in Definition 4.. Lemma 6.4. For an odd prime p and 0 < < p, there is a congruence ( ) p + (6.4) p (mod p 2 ). Proof. ( ) p + = (p + )(p + 2)... (p + )p ( )... 2 ( )( 2)... 2 p ( ) p (mod p 2 )..

10 0 DOAN, KIVUNGE, POOLE, SMITH, SYKES, AND TEPLITSKIY Lemma 6.5. For an odd prime p and 0 < < p, there is a congruence ( ) ( ) 2p + p + (6.5) p (mod p 2 ). Proof. ( ) ( ) 2p + p + (2p + )(2p + 2)... (2p + )2p (p + )(p + 2)... (p + )p = ( )... 2 ( )... 2 ( )( 2) ( )( 2)... 2 p p ( )... 2 ( )... 2 ( )( 2)... 2 p p (mod p 2 ). ( )... 2 Lemma 6.6. For an odd prime p and 0 < < p, there is a congruence ( ) ( ) p + + p p + (6.6) p (mod p 2 ). p p Proof. ( ) p + + p p (2p + )(2p + 2)... (2p + )2p(2p )... (p + )p = (p + )(p + )... (p + )p(p )... 2 (2p + )(2p + 2)... (2p + )2p ( )... 2 ( ) 2p + (mod p 2 ). The desired result then follows b (6.5). Proposition 6.7. If p is odd, then the located block [r] x of (4.2) completes to (6.7) r... r( + pλ j x)... r r( + pλ i ) rpi ( + x + ) r... rpj ( + x + )... rp( + x + ) Proof. B the linearit of the assembl rule (.), it suffices to prove that (6.8)... + pλ j x... + pλ i pi ( + x + )... pj ( + x + )... p( + x + )

11 FRACTAL BINOMIAL COEFFICIENTS is correctl completed from its left hand column and top row according to (.) modulo p 2. B linearit and the smmetr of Pascal s square, it suffices in turn to prove that (6.9) and (6.0) pi... pj pλ j pi 0... pj... p are correctl completed from their left hand columns and top rows according to (.) modulo p 2. Now the form of (6.9) in the top left hand corner of Pascal s square modulo p 2 follows b Lemma 6.4. On the other hand, the tile (6.0) is bordered on the left hand column and top row b the difference (6.)... + pλ j... : : : :. B (5.) with q = p and r = j, it is apparent that the completion of the left hand term of (6.) occupies the locations {(x, ) p x < 2p, 0 < p} in the modulo p 2 Pascal square. The completion of the right hand term occupies the locations {(x, ) 0 x, < p} in the modulo p 2 Pascal square. That (6.0) completes as indicated then follows b Lemmas 6.5 and 6.6. Remark 6.8. On dividing the tile (6.0) b p, one obtains a curious natural example of the emergence of a smmetrical output (the right hand column and bottom row) from an asmmetrical input (the left hand column and top row) under the assembl rule (.) modulo p. For instance, the p = 5 case ields The proof of the main theorem is now readil concluded along the lines exhibited for p = 3 b Example 6.3. Proposition 6.9. For an odd prime p, there is a homomorphism r x [r] x from the algebra D p of located residues modulo p 2 to the algebra of located p p blocks under Pascal addition modulo p 2.

12 2 DOAN, KIVUNGE, POOLE, SMITH, SYKES, AND TEPLITSKIY Proof. Using Proposition 6.7, the top row of the tile sum [r] x( ) + [s] (x ) is computed as follows: : s... p(j ) s(x + ) pj s(x + )... r r + s... (r + s) + pλ j s(x + ) (r + s) + pλ j s(x + )... (Recall λ =.) B Lemma 6. (local smmetr), the tpical entr (r + s) + pλ j s(x + ) of the top row of the tile sum reduces to (r + s)( + pλ j x), since the s term ma be replaced b rx. The top row of the tile sum is thus of the required form. B smmetr, the left hand column also appears in the required form, so that [r] x( ) + [s] (x ) is indeed given b [r + s] x. References [] Bondarenko, B.A., Generalized Triangles and Pramids of Pascal (Russian), Uzbek Academ of Sciences, Tashkent, 990. [2] v. Haeseler, F., Peitgen, H.-O., and Skorder, G., Pascal s triangle, dnamical sstems and attractors, Ergodic Th. and Dnamical Sstems, 2 (992), [3] Hard, G.H., and Wright, E.M., An Introduction to the Theor of Numbers, Clarendon Press, Oxford, 968. [4] Kummer, E.E., Über Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (852), [5] Peitgen, H.-O., Jürgens, H., and Saupe, D., Fractals for the Classroom, Part II, Springer, New York, NY, 992. [6] Romanowska, A.B. and Smith, J.D.H., Modes, World Scientific, River Edge, NJ, [7] Sved, M., The geometr of the binomial arra modulo p 2 and p 3, Discrete Math. 92 (99), [8] Sved, M. and Pitman, J., Divisibilit of binomials b prime powers, Ars Combinatoria 26 (988), Department of Mathematics, Iowa State Universit, Ames, Iowa , USA address: dddoan@iastate.edu address: 2 bkivunge@iastate.edu address: 3 jespoo@sbcglobal.net address: 4 jdhsmith@math.iastate.edu address: 5 mishat@rice.edu URL: