Mod p 3 analogues of theorems of Gauss and Jacobi on binomial coefficients

Transcription

1 ACTA ARITHMETICA 2.2 (200 Mo 3 analogues of theorems of Gauss an Jacobi on binomial coefficients by John B. Cosgrave (Dublin an Karl Dilcher (Halifax. Introuction. One of the most remarkable congruences for binomial coefficients, ue to Gauss (828, is relate to the reresentation of an o rime as a sum of two squares. It is a well-known theorem of Fermat that can be written as a sum of two squares if an only if (mo, an that the reresentation is unique u to sign an orer of the summans. Let us now fix an a such that (. (mo, = a 2 + b 2, a (mo. The theorem of Gauss can now be state as follows. Theorem (Gauss. Let the rime an the integer a be as in (.. Then ( ( /2 (.2 2a (mo. ( / For a roof an generalizations of this result see, e.g., [2,. 268]. Beukers [3] first conecture an extension to a congruence (mo 2, an this was first rove by Chowla, Dwork, an Evans []. Theorem 2 (Chowla, Dwork, Evans. Let an a be as in (.. Then ( ( /2 (.3 ( + ( / 2 q (2 ( 2a (mo 2. 2a Here q (m is the Fermat quotient to base m ( m, efine for o rimes by (. q (m := m. 200 Mathematics Subect Classification: Primary B65; Seconary S80, 05A0. Key wors an hrases: Wilson s theorem, Gauss theorem, factorials, congruences, -aic gamma function, Catalan numbers. DOI: 0.06/aa2-2- [03] c Instytut Matematyczny PAN, 200

2 0 J. B. Cosgrave an K. Dilcher Congruences such as (.3 have been very useful in large-scale comutations to search for Wilson rimes; see [6] or [7]. While the congruences (.2 an (.3 have been extene to numerous other binomial coefficients (see [2], it is one of the uroses of this aer to exten them to a congruence moulo 3. Theorem 3. Let an a be as in (.. Then ( ( ( /2 2a (.5 ( / 2a 2 8a 3 ( + 2 q ( (2E 3 q (2 2 (mo 3. Here E n enotes the nth Euler number, efine in (.. The secon main result, Theorem 7, of which Theorem 3 is a consequence, concerns quotients of what we call Gauss factorials. These quotients resemble binomial coefficients, an we will rove congruences moulo arbitrarily high owers of. This will be one in Sections 2 an 3, an in Section we erive Theorem 3 from the results of Section 2. In much the same way as ust outline one can erive (mo 3 congruences also for numerous other binomial coefficients. Here we will restrict ourselves to the following imortant classical case. In analogy to (. we fix an o rime an integers r, s such that (.6 (mo 6, = r 2 + 3s 2, r (mo 3, s 0 (mo 3. The integer r is then uniquely etermine. The following congruence, analogous to Gauss Theorem, is ue to Jacobi (837; see [2,. 29] for remarks an references. Theorem (Jacobi. Let an r be as in (.6. Then ( 2( /3 (.7 r (mo. ( /3 In analogy to Theorem 2, this congruence has also been extene, aarently ineenently by Evans an Yeung; see [2,. 293] for remarks an references. Theorem 5 (Evans; Yeung. Let an r be as in (.6. Then ( 2( /3 (.8 r + ( /3 r (mo 2. For the usefulness of this congruence, see [6] or [7]. We are now reay to state the following extension. Theorem 6. Let an r be as in (.6. Then ( 2( /3 r 2 ( ( (.9 ( r ( /3 r B 2 3 (mo 3.

3 Theorems of Gauss an Jacobi 05 Here B n (x is the nth Bernoulli olynomial; for a efinition see (5.5. The roof of this result, in Section 5, is analogous to the eveloment in Sections 2. We conclue this aer with several remarks in Section Gauss factorials an the -aic gamma function. We foun it convenient to introuce the following notation. For ositive integers N an n let N n! enote the rouct of all integers u to N that are relatively rime to n, i.e., (2. N n! =. N gc(,n= In a revious aer [5] we calle these roucts Gauss factorials, a terminology suggeste by the theorem of Gauss which states that for any integer n 2 we have { (mo n for n = 2,, α, or 2 α, (2.2 (n n! (mo n otherwise, where is an o rime an α is a ositive integer. Note that the first case in (2.2 inicates exactly those n that have rimitive roots. For references, see [8,. 65]. Dearting from (.3 we were able to rove the congruence (2.3 ( 2 2! (( 2! 2 2a 2a (mo 2, with an a as in (.. The roof is similar to (but easier than that of Theorem 3. Base on numerical exeriments, using the comuter algebra system Male [], it was easy to conecture (2. ( 3 2! (( 3! 2 2a 2a 2 8a 3 (mo 3. In this section an the next we are going to rove this, an in fact the following general congruence. Theorem 7. Let an a be as in (. an let α 2 be an integer. Then ( α 2!! 2 2a C 0 2a C 2 8a 3 C α (2.5 α 2 (2a 2α (( α α = 2a 2a = ( ( 2 2 a 2 (mo α, where C n := n+( 2n n is the nth Catalan number, which is always an integer.

4 06 J. B. Cosgrave an K. Dilcher If the summation on the right is consiere 0 for α =, then Gauss Theorem can also be seen as a secial case of (2.5. As in the roofs of Theorem 2 an its generalizations in [] an [2], the -aic gamma function an its connection with Jacobi sums will be useful here. Following the exosition in [2,. 277], we fix an o rime an efine a function F on the nonnegative integers by F (0 := an (2.6 F (n := ( n (n, 0<<n with (2.6 interrete so that F ( =. Let now Q enote the -aic comletion of Q, an Z the ring of -aic integers in Q. The -aic gamma function is then efine by (2.7 Γ (z = lim n z F (n (z Z, where n runs through any sequence of ositive integers -aically aroaching z. For the existence of this limit an for other roerties see, e.g., [2,. 277] or [2,. 0 ff.]. Among the roerties we require here is the fact that for any ositive integer n, (2.8 z z 2 (mo n imlies Γ (z Γ (z 2 (mo n. The Jacobi sum over the finite fiel F is efine as follows. If χ an ψ are characters on F, then the Jacobi sum J(χ, ψ is efine by J(χ, ψ = a χ(aψ( a, where a runs through the elements of F. See, e.g., [2, Sect. 2.] for a somewhat more general efinition. The following roerties are use in this aer. First, let = f + be a rime an let g be a rimitive root moulo. Define the integers a an b by ( 2 (2.9 = a 2 + b 2, a (mo, b a g ( / (mo, where ( 2 is the Legenre symbol. For a fixe g the integers a, b are uniquely etermine an iffer from a an b in (. only (ossibly in sign. Next, let χ be a character (mo of orer such that χ(g = i. Then from Table 3.2. in [2] we have (2.0 (2. J(χ, χ = ( f (a + ib, J(χ 3, χ 3 = ( f (a ib. Furthermore, let P be a rime ieal in the ring of integers Z[i] iviing the

5 Theorems of Gauss an Jacobi 07 rime. Then it follows from Theorem 2.. in [2,. 66] that (2.2 J(χ, χ 0 (mo P. Finally, the connection between Jacobi sums an the -aic gamma function is a consequence of the ee Gross Koblitz formula for Gauss sums. There is no nee to go into etails here; instea we ust quote a secial case of ientity (9.3.7 in [2,. 278], namely (2.3 J(χ 3, χ 3 = Γ ( /2 Γ ( / Proof of Theorem 7. With the above efinitions an rearations we are now in a osition to rove Theorem 7. Continuing to use the ieas in [2, Ch. 9], we aly (2.8 to (2.3 an obtain (3. J(χ 3, χ 3 Γ ( + ( α /2 Γ ( + ( α / 2 (mo α. Since the arguments of Γ are now integers, we have J(χ 3, χ 3 F ( + (α /2 F ( + ( α / 2 (mo α, an finally, comaring (2.6 with (2., (3.2 J(χ 3, χ 3 ( α 2! (( α! 2 (mo α. Here we have use the fact that + ( α /2 (mo 2, which accounts for the minus sign in (3.2. With a view to evaluating the right-han sie of (2., we note that (2.0 with (2.2 gives (3.3 (a + ib α 0 (mo P α. Since this hols for any rime ieal P iviing the rime, we may conclue that this congruence hols also moulo α. We now exan the left-han sie of (3.3 an searate real an imaginary arts, to obtain (α /2 (3. ib =0 ( α ( a α 2 b α/2 =0 ( α ( a α 2 b 2 2 (mo α.

6 08 J. B. Cosgrave an K. Dilcher Because of the relationshi b 2 = a2, the first sum, S, in (3. becomes (α /2 S = a α =0 ( ( ( α k. 2 + k k=0 Setting := k an noting that ( ( k = ( k =, we get S = a α (α /2 ( a 2 (α /2 = a 2 ( α 2 + ( The inner sum has an exlicit evaluation as 2 α 2( α ; see ientity (3.2 in [0]. Hence (α /2 (3.5 S = (2a α ( ( α. Similarly, if S 2 is the secon sum in (3., we get S 2 = a α = a α α/2 =0 k=0 (α /2 ( ( ( α 2 k ( a 2 α/2 = a 2 ( α 2 k a 2 (. Using the ientity (3.20 in [0] to evaluate the inner sum, we obtain (3.6 S 2 = α/2 2 (2a α To simlify notation, we set We now claim that (3.7 ib S α 2 a + 2a S = ( α y := a 2. By (3.5 an (3.6 this is equivalent to α/2 ( α α α y (α /2 ( α ( α + 2 ( y = α ( α a 2.. ( 2 2 y (mo α. ( 2 2 y (mo α

7 or α/2 [( α α α ( α 2 = α 2 = ( (α /2 Theorems of Gauss an Jacobi 09 ( α ( 2 2 ( ] y y ( (α /2 ( 2 2 ( α ( α y It is easy to verify that ( ( ( α α α α α = 2, y + (mo α. which simlifies the left-han term in the above congruence. For the rightmost term we set k := + an change the orer of summation. Then the congruence above is equivalent to (3.8 ( α y α ( k ( α/2 k= = ( 2 2 ( α + k k y k (mo α. We have therefore rove our claim (3.7 if we can show that the coefficients of the owers of y on both sies of (3.8 are ientical u to ower α. But this is an immeiate consequence of the ientity (3.9 0 ( + ( 2 ( n + n m = ( n n m as can be seen by setting n = α k an n m = k in (3.9. This ientity can be foun, in slightly change form, in [,. 83 ff.]. To comlete the roof of Theorem 7, we note that (3.7 with (2. an (3.2 immeiately gives (2.5. It only remains to verify that ( f a = a. But this follows from (2.9 an the 2n comlementary law of quaratic recirocity. Inee, recall that f = ( /; then (as is also shown in [2,. 08] ( 2 a ( 2 8 = ( + 2 = ( f (mo, where we have use the fact that ( + /2 (mo 2. Hence ( f a (mo, an thus ( f a = a by (.. The roof is now comlete.

8 0 J. B. Cosgrave an K. Dilcher. Proof of Theorem 3. In orer to erive Theorem 3 from Theorem 7 we nee a number of auxiliary results on congruences for certain finite sums. We begin by listing three easy congruences. (. (.2 (.3 Lemma. For all rimes 5 we have = ( /2 = ( / = 0 (mo 2, 2q (2 (mo, 3q (2 (mo. The congruence (.2 also hols for = 3. Congruences of this tye were obtaine by several authors in the early 900s, with the most extensive an general treatment in a aer by Emma Lehmer [3]. The congruences ( an (3 in that aer, which are given moulo 2, immeiately reuce to (.2 an (.3, resectively, when taken moulo, an (. follows as a secial case from a congruence in [3,. 353]. While Lemma woul be sufficient to obtain (2.3 from (.3 an vice versa, for the roof of Theorem 3 we nee to exten these congruences. For the following lemma we nee the Euler numbers E n which can be efine by the generating function (. 2 e t + e t = n=0 E n n! tn ( t < π. The Euler numbers are integers, an the first few are E 0 =, E 2 =, E = 5, E 6 = 6, an E 2+ = 0 for 0. For further roerties see, e.g., [, Ch. 23]. (.5 (.6 Lemma 2. For all rimes 5 we have = ( /2 = an for (mo, (.7 ( / = 0 (mo, 2 2q (2 + q (2 2 (mo 2, 3q ( q (2 2 E 3 (mo 2.

9 Theorems of Gauss an Jacobi The congruence (.5 is a secial case of a more general one in [3,. 353]. (.6 an (.7 follow from congruences in [6,. 290]. We will also nee congruences for a number of ouble sums: Lemma 3. For all rimes 5 we have (.8 0 (mo, k (.9 an for (mo, (.0 <k <k ( /2 <k ( / k 2q (2 2 (mo, k 9 2 q (2 2 2E 3 (mo. Proof. As secial cases of congruences in [6,. 296] we have, for 5, (. an for (mo, ( /2 = 0 (mo, 2 (.2 ( / = Now note that for =, 2, or we have (.3 <k ( / 2 E 3 (mo. k = (( / 2 = 2 ( / 2 = We then see that for =, the congruences (. an (.5 imly (.8. Likewise, for = 2, the congruences (.2 an (. give (.9. Finally, in the case =, the congruences (.3 an (.2 imly (.0. We are now reay to rove Theorem 3. Proof of Theorem 3. We begin with the simle ientity 3 = 2 + ( = 2 or = ; this shows that with s := ( 2 / we have ( 3 s [ ( (.! = [( + ( + ] (s + s + ]. 2.

10 2 J. B. Cosgrave an K. Dilcher Now for each = 0,,..., s we have (.5 ( + ( + [ (! = (! (mo 3, <k ] k where the secon congruence follows from (. an (.8. Similarly, ( (.6 (s + s + ( [ ( /! + s + ] s2 2 (mo 3. k = <k ( / When = 2, we use (.6 an (.9 to obtain ( (s + s + (! [ + s( 2q (2 + q (2 2 + s 2 2 2q (2 2 ] (mo 3. Uon simlifying an using the fact that s /2 (mo 2 we get, together with (.5 an (., ( 3 ( (.7! (! (2 /2! ( + q (2 (mo In the case = we use (.7 an (.0 an the fact that now s / (mo 2. Then in comlete analogy to the case = 2, from (.6 we obtain ( 3 ( (.8! (! (2 / ( + 3 q ( q ( E 3 (mo 3. Next we note that + q (2 q (2 + 2 q (2 2 (mo 3, an therefore uon multilying an simlifying we get ( + 3 q ( q ( E 3 + q (2! + 2 q ( (2E 3 q (2 2 (mo 3.

11 Theorems of Gauss an Jacobi 3 Finally, if we ivie (.7 by the square of (.8, this last congruence together with (2. gives the esire congruence ( Proof of Theorem 6. In orer to rove Theorem 6, we follow the same eveloment as in Sections 2. In articular, we first rove the following result which is of ineenent interest. Theorem 8. Let an r be as in (.6 an let α 2 be an integer. Then ( 2( α 3! (( α 3! 2 r + C 0 r + C 2 r C α (5. α 2 r 2α α ( 2 2 = r + r 2 (mo α, = where C n is again the nth Catalan number. Proof. Fix a rimitive root g moulo, an let χ be a cubic character moulo such that χ(g = e 2πi/3 = ( + i 3/2. Then from Table 3.. in [2,. 06] we have (5.2 (5.3 J(χ, χ = 2 (r + is 3, J(χ 2, χ 2 = 2 (r is 3, where r an s are as in (.6, with the sign of s fixe by the congruence 3s (2g ( /3 + r (mo. Furthermore, for any rime ieal P in the ring of integers of Q( 3 iviing we fin, again by Theorem 2.. in [2,. 66], that J(χ, χ 0 (mo P, an with (5.2 we get (r +is 3 α 0 (mo P α. As before, we may conclue that this last congruence also hols moulo α. We exan the left-han sie an searate real an imaginary arts: i 3 s (α /2 =0 ( α ( 3 r α 2 s α/2 =0 ( α ( 3 r α 2 s 2 (mo α. 2 Using the relationshi 3s 2 = r 2, the left-han sum, S 3, becomes (α /2 ( ( ( α k S 3 = r α 2 + k r 2 =0 k=0 (α /2 ( ( α = (2r α r 2,

12 J. B. Cosgrave an K. Dilcher where we have use ientity (3.2 in [0]. Similarly, for the right-han sum, S, we get (see also (3.6 S = α/2 ( α 2 (2rα α α ( r 2. In the same way as in Section 3 we now obtain, with z := /r 2, i 3 s S α ( ( 2 2 r + 2r z (mo α ; S 3 = see also (3.7. Now, with (5.3 this gives α (5. J(χ 2, χ 2 r + r = ( ( 2 2 ( r 2 (mo α. Next, in analogy to (2.3 an (3., (3.2 we have ( J(χ 2, χ 2 = Γ 2( ( 2/3 α Γ ( /3 2 3! (( α 3! 2 (mo α. Finally, this combine with (5. immeiately gives (5.. The roof of Theorem 6 is analogous to that of Theorem 3 in Section. For the next lemma, which sulements Lemmas 3, we nee the Bernoulli olynomials B n (x, efine by the generating function (5.5 te tx e t = n=0 B n (x tn n! ( t < 2π. See Section 6 for some further remarks on the numbers B 2 (/3 that aear in all the congruences in the following lemma. (5.6 (5.7 (5.8 (5.9 Lemma. For all rimes (mo 3 we have ( /3 = 2( /3 = ( /3 = 2( /3 = 2 2 B 2( 3 (mo, 2 2 B 2( 3 (mo, 3 2 q (3 + 3 q (3 2 6 B 2( 3 (mo 2, 3 2 q (3 + 3 q ( B 2( 3 (mo 2,

13 Theorems of Gauss an Jacobi 5 (5.0 (5. <k ( /3 <k 2( /3 k 9 8 q (3 2 B 2( 3 (mo, k 9 8 q (3 2 + B 2( 3 (mo. Proof. The congruence (5.6 can be foun in [6,. 302]. Then (5.7 follows from the observation that 2( /3 = 2 = = ( /3 2 = ( /3 ( 2 = (mo, 2 where we have use (.5. The congruence (5.8 was rove in [6,. 30]. To obtain (5.9, we rewrite (5.2 2( /3 = = = ( /3 = where we have use (.. Using ( /3 = 2 (mo 2 (mo 2, (see [3,. 359] an (5.6, we see that (5.2 gives (5.9. Finally, (5.0 follows from (.3 with = 3, together with (5.6 an (5.8. Similarly, (5. follows from (.3 with = 3/2, together with (5.7 an (5.9. Proof of Theorem 6. The roof is very similar to that of Theorem 3 in Section. First, using (. (.6 with = 3/2 an noting that s 2/3 (mo 2 in this case, we obtain, with (5.9 an (5., (5.3 ( 2( 3 3! (! 2(2 /3 ( 2( 3! ( + q (3 9 2 B 2 ( 3 (mo 3. Similarly, with = 3 an thus s /2 (mo 2, we obtain, with (5.8 an (5.0, ( 3 ( (5.! (! (2 /3! 3 3 ( + 2 q (3 8 2 q (3 2 + ( 36 2 B 2 3 (mo 3. Next we note that + q (3 ( 9 2 B q (3 + 2 q (3 2 + ( 9 2 B 2 3 (mo 3, 2 3

14 6 J. B. Cosgrave an K. Dilcher an therefore uon multilying an simlifying we get ( + 2 q (3 8 2 q (3 2 + ( 36 2 B q (3 ( 9 2 B + ( 6 2 B 2 3 (mo Finally, if we ivie (5.3 by the square of (5. an use this last congruence, then with (5. we get the esire congruence ( Further remarks. Numerous other congruences of the tye (.2, (.7 an extensions of the tye (.3, (.8 have been obtaine; see [2, Ch. 9]. For all these cases our metho can be use to erive analogues of Theorems 7 an 8, an of Theorems 3 an Further extensions of Theorems 3 an 6 to congruences (mo woul also be ossible. However, these congruences woul be increasingly comlicate an woul require ifferent values of Bernoulli olynomials which woul arise from higher analogues of Lemmas. 3. We can obtain the following irect consequence of Theorem 7. Corollary. We have the -aic exansion J(χ 3, χ 3 = Γ ( /2 Γ ( / 2 = 2a + 2a = ( ( 2 2 a 2, where χ is the character moulo of orer as use in (2., an an a are as in (.. This follows irectly from (3.2 an (2.5. A similar exression exists for J(χ, χ, via (2.0 an (3.7. Similarly, from Theorem 8 we obtain Corollary 2. We have the -aic exansion J(χ 2, χ 2 = Γ ( 2/3 Γ ( /3 2 = r r = ( 2 2 ( r 2 where χ is the cubic character moulo, an an r are as in (.6. (. The numbers B 2 3 that occur in Theorem 6 an Lemma are interesting in their own right. To simlify notation, set b n := 3 n ( B n 3. In (5.5 we set x = 3 ; then we relace t by 3t an 3t resectively, an subtract the two resulting ientities from each other. Uon simlifying the left-han sie we then obtain t (6. e t + + e t = 2 t 2n+ b 2n+ 3 (2n +!. n=0,

15 Theorems of Gauss an Jacobi 7 The sequence of numbers generate by the left-han sie of (6. has been stuie by Glaisher [9] as analogues to the Euler numbers efine in (.. In articular, it turns out that his so-calle G-numbers G n are relate to the numbers b n by b 2n+ = ( n+ G n (n, an in aition we have b = 2. Thus we have, for o rimes 5, B 2 ( 3 = 3 2 ( ( /2 G ( 3/2, where Glaisher s G-numbers are integers, as was shown in [9], along with numerous other roerties. This connection with Glaisher s work is also mentione in [3,. 352]. Finally, see [5, A002] for some roerties, references, an values for these numbers. Acknowlegments. Research suorte by the Claue Shannon Institute, Science Founation Irelan Grant 06/MI/006, an by the Natural Sciences an Engineering Research Council of Canaa. References [] M. Abramowitz an I. A. Stegun, Hanbook of Mathematical Functions, Nat. Bureau of Stanars, Washington, 96. [2] B. C. Bernt, R. J. Evans, an K. S. Williams, Gauss an Jacobi Sums, Wiley, New York, 998. [3] F. Beukers, Arithmetical roerties of Picar Fuchs equations, Séminaire e théorie es nombres, Paris , M.-J. Bertin an C. Golstein (es., Progr. Math. 5, Birkhäuser, Boston, 98, [] S. Chowla, B. Dwork, an R. Evans, On the mo 2 etermination of `( /2 ( /, J. Number Theory 2 (986, [5] J. B. Cosgrave an K. Dilcher, Extensions of the Gauss Wilson theorem, Integers 8 (2008, #A39. [6] R. Cranall, Toics in Avance Scientific Comutation, Sringer, New York, 996. [7] R. Cranall, K. Dilcher, an C. Pomerance, A search for Wieferich an Wilson rimes, Math. Com. 66 (997, [8] L. E. Dickson, History of the Theory of Numbers. Volume I: Divisibility an Primality, Chelsea, New York, 97. [9] J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. Lonon Math. Soc. 3 (899, [0] H. W. Goul, Combinatorial Ientities, rev. e., Goul Publications, Morgantown, WV, 972. [] R. L. Graham, D. E. Knuth, an O. Patashnik, Concrete Mathematics, 2n e., Aison-Wesley, Reaing, MA, 99. [2] N. Koblitz, -Aic Analysis: A Short Course on Recent Work, Lonon Math. Soc. Lecture Note Ser. 6, Cambrige Univ. Press, Cambrige, 980. [3] E. Lehmer, On congruences involving Bernoulli numbers an the quotients of Fermat an Wilson, Ann. of Math. 39 (938, [] Male, htt:// [5] N. J. A. Sloane, On-Line Encycloeia of Integer Sequences, htt:// att.com/ nas/sequences/.

16 8 J. B. Cosgrave an K. Dilcher [6] Z. H. Sun, Congruences involving Bernoulli an Euler numbers, J. Number Theory 28 (2008, John B. Cosgrave 79 Rowanbyrn Blackrock, County Dublin Irelan Karl Dilcher Deartment of Mathematics an Statistics Dalhousie University Halifax, Nova Scotia, B3H 3J5, Canaa Receive on an in revise form on (588