SOME CONGRUENCES INVOLVING INVERSE OF BINOMIAL COEFFICIENTS Текст научной статьи по специальности «Математика»

Chelyabinsk Physical and Mathematical Journal. 2023. Vol. 8, iss. 1. P. 59-71.

DOI: 10.47475/2500-0101-2023-18105

SOME CONGRUENCES INVOLVING INVERSE OF BINOMIAL COEFFICIENTS

L. Khaldi1", R. Boumahdi2b

1 University of Bouira, Bouira, Algeria

2 University of Science and Technology Houari Boumediene, Bab-Ezzouar, Algeria al.khaldi@univ-bouira.dz, br_boumehdi@esi.dz

Let p be an odd prime number. In this paper, among other results, we establish some congruences involving inverse of binomial coefficients. These congruences are mainly determined modulo p, p2, p3 and p4 in the p-integers ring in terms of Fermat quotients, harmonic numbers and Bernoulli numbers in a simple way. Furthermore, we extend an interesting theorem of E. Lehmer to the class of inverse binomial coefficients.

Keywords: congruence, binomial coefficient, Fermat quotient, gamma function.

1. Introduction

For any non-negative integers n, k the binomial coefficients are defined by

^ = J kl^, if 0 ^ k ^ n;

0, otherwise,

and the harmonic numbers Hn are the rational numbers defined by

n 1

Ho = 0 and Hn = V —, n g N*.

k

k=l

In many mathematics domains, such as combinatorics, graph theory and number theory, binomial coefficients often appear naturally and play an important role. However, it is well-known that it is difficult to compute the values of combinatorial sums involving inverses of binomial coefficients. The gamma function is defined by

r

r(x) = tx~le~tdt, for x g R+,

Jo

and it is connected to the inverse binomial coefficient via the following relation

r l

1 _ k!(n - k)! _ r(k + 1)r(n - k + 1)

Ï) = n = r(n +1) .,0

(n +1W tk(1 - t)n-kdt. (1)

J 0

The Bernoulli polynomials (Bn(x))n^0 may be defined by means of the exponential generating function as follows:

text ^ Bn(x) n

et — 1 n!

n=o

tn

and the Bernoulli numbers by Bn = Bn(0), n ^ 0. The Fermat quotient of a positive integer m with respect to an odd prime p not dividing m is defined by

/ , mp-1 — 1

qm = qp(m) =-. (2)

p

The concept of congruence is very old. Although its origin goes to a distant past, it was not until the eighteenth century, especially with Gauss [1], that it was formulated with a rigorous mathematical language and it was Babbage [2], who in 1819, initiated the congruences for the binomial coefficients by establishing that

^ P ^ ^ = 1 (modp2) for p ^ 3. (3)

In fact, Babbage thought that he had found a modulo p2 analogue of the theorem of Wilson concerning the characterization of the prime numbers. So Babbage thought that

p is a prime number if and only if ^ p ^ ^ = 1 (modp2) for p ^ 3. In fact it is not, since if we consider N = 283686649 = 168432, we have

2N - 1 N1

^ = 1 (mod N2) .

It has been proven that N = 283686649 is the only composed number less than 109 satisfying the previous congruence [3]. In 1862, Wolstenholme [4] improved Babbage's results by proving the following two historical congruences

Hp-1 = 0 (modp2) for p ^ 5, (4)

/2p — 1\

( 1 = 1 (modp3) for p ^ 5. (5)

2p - 1 2 ( ) ( — 1 \ = 1 — 3p3Bp-3 (modp4) for p ^ 5,

The congruence

1 . - - _ ------------ , ------- (6)

p — 1 3

has been proven in 1900 by Glaisher [5] as an improvement modulo p4 of Wolstenholme's congruence.

In turn, in 1895, Morley [6] proved that the following congruence holds for any prime p ^ 5

p — ^ _ p-

(—1) £^(pp—= 4p-1 (modp3) .

Note that this congruence implies the following one which is easy to prove directly

pp—1 ) = (—1) 2-1 (mod p) . 2 /

This congruence is not a criterion of primality. Ayad and Kihel [7] studied the odd composed numbers n satisfying

nn—11 = (—1)(mod n). (7)

They found that n = 5907 = 3 x 11 x 179 is the smallest composed integer verifying (7). They conjectured that this congruence does not admit solution n such that n is the product of two prime odd numbers. Morley's congruence was generalized to the following result by Carlitz [8]:

(-1)2-1 (P-i1) = 4p-1 + -2p3Bp_3 (modp4) , for p > 5. (8)

Congruences concerning binomial coefficients and harmonic numbers have had extensive literature in the last three decades for more details see Z.W. Sun [9] or Mestrovic's rich survey [3] and the references therein.

Let us also recall that if p is a prime number, we denote by Z(p) the set of rational numbers having denominators relatively prime with p. It is easy to prove that Z(p) is a ring (called p-integers ring). For two reduced rational numbers a = a/6 and ^ = c/d g Z(p) we write a = ^ (mod p), if ad — 6c is divisible by p and the denominators 6, d are relatively prime with p.

In the present work, we exploit some properties of the inverse binomial coefficients to establish congruences for sums involving these numbers. This paper is organized as follows. Section 2 is dealing with some preliminary lemmas needed in the proofs of the main results. Section 3 is concerned with stating results concerning congruences for

p-i s 1

etit , s g i1, 2}.

k=1 \m)

Section 4 is devoted to present congruences involving the sums

LrJ (_1)fc_1

y^^-1-)-, r = 2, 3, 4 and 6

k=1 V k

and Section 5 is about further remarks.

2. Auxiliary lemmas

In this section, we state some basic facts which will be used in the proofs of our theorems.

Lemma 1. Let p be an odd prime, then

=(—1)m-i(1—pHm-i) (mod p2),

holds true for each m = 1, 2,... ,p. Proof. We have

( -1'r-1(m— 1) = n (! - p) = 1 —pHm-1 (modp2

' i=1

m— 1

which immediately gives the congruence (9). □

Lemma 2. For any positive integers n,m with m ^ n we have

£

k=m \m.

Hn

k

m-l

1

(m-1)

if m = 1; if m = 1.

Proof. If m =1, it is clear that

n1 (k

k=

i (Î)

If m = 1, in one hand, using the identities

—D = Hn .

C+

m

k ^ and (k\ - k (k - 1 m— 1/ \m) m\m — 1

we obtain

m

k k-l m Km-1) \m-1

ik-l

m \m-2

m - 1 I (к-l) ( k ) m - 1 (k-lV k ) m - 1 (k-lV k )

m l m l m l m l m l m l

m

ik-l m2

1

(10)

m — 1 ffc-M ( k A(k-1\

\m-y Vm-1 \m—2)) m\m-1J \m.

In the other hand, let f be an arbitrary real function, we define the difference operator by vf (x) = f (x + 1) - f (x). For f (t) = we have

£

k=m m

m

m1

£

k=m

k-l ml

k

ml

m

m - 1

k=m

v/(k) = -

m

m1

(/(n) - /(m - 1))

m

m1

1

n ml

Which gives the desired result. □

Lemma 3. For any odd prime integer p and any m E {1,..., P+1} we have

2-1 \ (-1)m-1 (2m - 2\ ( m- 1 ) f A 2, m - 1) m -1) I1 - 2i-l) ^.

In particular,

S-r \ = (-1)m-V2m - 2

m- 1 ~ 22m-2 \m - 1

(modp) .

Proof. For m g {1, 2,..., }, the definition of the binomial coefficient yields to

p-i 2

1 p - 1/ p - 1

m - 1 (m - 1)! 2 1

1

p - 1

- (m - 1) + 1 =

(-1)

2m-l(m - 1)!

m l m- l m- l

n (2k-1) n I1

22 (p - 1)(p - 3)(p - 5) ■ ■ ■ (p - (2(m - 1) - 1)) =

2m-1(m - 1)!

k=1

p

2k 1

(-1)m-V2m - 2

22m-2 m 1

m1

1

k=1

p

2k 1

1

î

m

1

1

1

1

1

1

Therefore it easy to show that

m— 1 / \ / m— 1

n(! - ^ - 1 - p 5:1

fc=i v 7 \ ¿=1

2k - 1 / \ " ^ 2i - 1

fc=i v ' "

Then

^ - (zmrv2™ - (1 - p (mod p

m - V 22m-2 \m - 2 M ^^ 2i - 1'V

□

Lemma 4. [10]. Let p be a prime such that 4p = a2 + 3b2 = 1 (mod 3) with a, b g Z and a = 1 (mod 3). Then

p ^____j

(¿) --a + a (mod p2) . (11)

Lemma 5. For any positive integer n we have

In particular,

^ (-1))-1 = 1 , (-1)""1 (12) k (2T) 2n + 2+ 2(2n) . ( j

n- (-1)fc-1 = 1 , (-1)n (13)

(2n) 2n + 2 + 4(2n-1). ( )

k=1 U/ Vn-1/

Proof. Let us consider In = k=1 ( (L- , then using the relation (1) relating gamma

( k )

function and binomial coefficient one obtain

\k-1 n p1

(_1) k

In = £ = £(-1)k_1 (2n + 1) / tk(1 - t)2n_kdt

k=1 UJ k=i

=-(2n+1) i1(1 -t)2n( t (1-t)'

n

therefore by the identity ^^xk = x(X_-1) and after simplifying, it follows

x-

k=1

In = (—1)n-1(2n + 1) / tn+1(1 - + (2n + 1) / t(1 - t)2ndt =

./0 JO

= (_ i)n-i 2n +1 1 + 2n + 1 1 = (-1)n-1 + 1 ( 1) 2n + 2 (2nn+11) + 2n + 2 (2n+1) 2 (2nn) + 2n + 2"

□

Note that identities (10) and (12) can be found without proof in [11].

Lemma 6. [12]. Let p be a prime of the form 4q + 1 such that p = a2 + b2 with a, b g Z and a = 1 (mod 4). Then

= (1 + ^2) (2a - 2a) (modp2) . (14)

1

1

Lemma 7. [13]. Let be a prime such that p = 1 (mod 6) and p = a2 + 3b2 with a,b g Z and a = — 1(mod3). Then

,p-1x f 2(—1)P-1+1a (modp) , if b = 0(mod3); (p-J = < (—1)2-1 (a + 3b) (modp), if b = 1(mod3);

[(—1)2-1 (a — 3b) (modp), if b = 2(mod3).

Lemma 8. [10]. Let p be a prime of the form 6q +1 such that 4p = a2 + 3b2 with a,b g Z and a = 1 (mod 3). Then

t) = (—1)+1 (a — p) f 1 + ^) (mod p2

s-lJ v V aJ ' 3 ,

6 / V /

3. Congruences for sums of the inverse of binomial coefficients

p-i s

In this section we determine ^^(k) modulo p and p2 for s g {1, 2} in terms of

, (m)

k=m

harmonic numbers, inverse binomial coefficients and Fermat quotient. Theorem 5. Let p ^ 5 be a prime and let m g {1, 2,... ,p}. Then we have

p-1 1 _ JO (mod p2) , if m = 1;

tm 15 ( m-1 (1 - (-1)m-1(1 + pHm-1)) (modp2), if m = 1.

Proof. Taking n = p — 1 in the congruence (10) of Lemma 2, we get

p-1 1 iHp-i, if m = 1;

E

Ik

1 — TP-n , if m =1.

k=m\m) ^ m—1 ^ (p-^)

In virtue of the congruence (9) of Lemma 1 and using the identity

1 1 + x + x2 + x3 +----, (15)

1 - x

which is valid for any x E pZ(p, and also by the congruence (4), we conclude

p—1 1 _ JO (mod p2) , if m = 1;

{ m-ï(1 — (—1)m-1(1+ pHm-i))(modp2), if m =1.

k=m \mj \ m

□

Corollary 1. Let p ^ 5 be a prime and let m G {1, 2,... ,p}, we have

p-ï _1_ = | 0 (modp), if m = 1;

tm O {m-ï (1 + (—1)m)(modp), if m =L ( )

Note that when p ^ 5 is a prime and m g {1, 2,... ,p}, we also have

p-1 |0(modp) , if m = 1;

= < mm (modp), if m =1 m is even; k=m W I o (modp), if m =1 m is odd.

m

Theorem 7. Let p = 3 be an odd prime number and let m g |1,..., }. Then

2-1 ^ [ —2q2 + pq2 (modp2) , if m =1

22

1 I —2q2 + pq

k=m (m- ={ mmr (1 — (—4)m-1(2m__12)-1 (1+ pXW) (mod p2), if m = 1 (17)

Proof. Taking n = p-1 in the identity (10) of Lemma 2, we obtain

p-i [ Hp-i, if m =1;

2 ^ 1 I 2

k=m (1- = I mm- (1 — , if m = 1. (18)

k=m \mj I m_1 I (2—1 ) J ' '

p-i 2

If m =1, by the congruence k = — 2q2 + pq| (modp2) (see [14, Congruence (45)]) we

k=1

deduce the first congruence of (17). If m = 1, from Lemma 3 and identity (15), we get

1 2m 2 _1 1 1 = (—1)m_141-m/2m 2» 1

cmtù vm—^ 1—p ^'¿r

=(—1)m_1 ^(m—2) _1 p £ ¿r) H p2-. (K9)

Substituting (19) in the relation (18) and after simplifying, we deduce the second part of the congruence (17). □

The reduction modulo p of the congruence (17) gives the following corollary.

Corollary 2. Let p = 3 be an odd prime number and let m g {1,..., p--r}. Then

p-i (

^ 1 _ J—2q2(mod p), if m = 1;

ktm (m- I mmr (1 — (—4)m_1(2,m__12)_0 (modp) , if m =1. (20)

It should be noted that when m =1, the congruence (17) is equivalent to the congruence mentioned above [14, Congruence (45)] and if m = 1, the congruence (20) can be found in [15].

4. Congruences concerning Fermat quotient

1p 1

This section is devoted to present the congruences ( _>-- , r = 2, 3,4, 6, modulo

k=1 (

p, p2 and p4 in terms of Fermat quotient and Bernoulli numbers. Theorem 9. Let p ^ 5 be a prime, we have

p-i

Y^-ly- = 1 - p(1 - q2) + p2 (1 - 3- p3 ( 1 - 2q| - ^Bp _3) (modp4) .

Proof. Taking n = 2-1 in the identity (12) of Lemma 5, we write

p-i

fi (zi£1 = ^_iz1)j51 (21)

Z^ (P-1) 1 + p ofp-1, . (21)

By the identity (15), we have

1

1 + p

Thereafter by the congruence (8), we obtain

- 1 - p + p2 - p3 (modp1) . (22)

/ \ P— 1

^gf - 42-1 + 1 p3Bp-3 fm0dp1) . <23)

2

The equality (2) which define the Fermat quotient implies 4p-1 = (2p-1)2 = 1 + 2pq2 + p2q2, replacing in (23) with identity (15), we obtain

( ) _ fmod p1) —

(P—) 1 + 2pq2 + p2 q2 + 12 p3Bp-3

- 1 - 2pq2 + 3p2q2 - 4p3q3 - -2p3Bp-3 fmodp1) . (24)

Combining the congruences (22), (24) and identity (21), it comes

p—i

E- 1 - p(1 - q2) + p2 ( 1 - 3fc2) - p3 ( 1 - 2q3 - -4Bp-3) fmodp1)

□

Theorem 11. Let p be a prime such that 4p = a2 + 3b2 — 1 (mod3) with a,b e Z and a — 1 (mod 3). Then

V (-1)k-^ 3 (J 1 + (-1)131 p

h TW " 411 - 2 p) + +*

In particular,

LfJ / 1 r, / ^p—1

(-1) 3 (-1) 3

= 4 + (mod p).

k=1 ( k ) 4 2a

Proof. Taking n = P31 in the identity (12) of Lemma 5, we obtain

Lfi (-!)*-1 _ 3_1__(-1)fj1

¿1 (T) =41 + 1 p - 2( 2f—f)

3

In view of the congruence (11) of Lemma 4, we write

, 2if—i1, - 4 1 + 1 - 2(f—1), lmod p ) — ) 41 + ~1p

— 4(1 — 1 p)—^ —a+p (mod p2) — |(1 — 1 p)(1 + 02

□

Theorem 13. Let p a be prime of the form 4q + 1 and p = a2 + 62 with a, 6 g Z and a — 1 (mod 4). Then

i p i -1 ( pf) = 3^ 3 ) 4a

+ 412p — §p) (modp2)

Proof. Taking n = 2-1 in identity (12) of Lemma 5, we get

^pi (-1)k-1 _ 2_1__(-1)pri

¿1 (Pf) = « + 3P - 2(Pg)

4

then using the congruence (14) of Lemma 6, we have

I p I

H (-1)k-' 2/ 1\ (—1)1-1 , ,2) £ Iff = H1 — spJ— 2(1 + (M2)(2 a—2a )(mod p ' =

2 ( 1 N (—1)p-1 ( 1 q2 \ ( n 2)

= ai1 — — 4a y1 + 402p — TpJ (mod p )

□

Corollary 3. Let p be a prime of the form 4q + 1 and p = a2 + b2 with a, b g Z and a = 1 (mod 4). Then

^ (-1)k-1 _ 2 (-1)P-i ( d ) £ ^^ = 3--(mod p) .

k=1 i 2 1 3 4a

Theorem 15. Let p = 1 (mod 6) be a prime and p = a2 + 362 with a, 6 g Z and a — — 1 (mod3). Then

l p j / 1)k_1 [ 5 + 4a(mod p) , if 6 = 0(mod3);

£ —i-i— = 11 — 2at3b)(modp) , if 6 = 1(mod3);

k=1 (J - 2(a-3b) (mod p), if b = 2 (mod 3).

Proof. Taking n = 2-1 in the identity (12) of Lemma 5, we have

(—1)k_1 3 1 (—1f

6

S (?) 51 + 5p 2(p—)

6

In virtue of Lemma 7, we get

l6j (_ 1)k_1 [ 3 + 4a (modp) , if 6 = 0(mod3)

£ M"" — I I — ^a+ay (modp) , if 6 — 1 (mod 3) k=1 ( k ) I I

5 2(a-

(p-1 ) — I 5 2(a+3b) V^^FJ > a- u = ±

^ 11 — 2(a_a) (modp) , if 6 — 2 (mod 3)

□

Theorem 17. Let p be a prime of the form 6q +1 such that 4p = a2 + 3b2 with a,b e Z and a — 1 (mod3). Then

l f I

V- (-1)k-1 3 1 /3 1 1 \ f J 2) Y — 5 + 2a - H25 - 203 + 3^ (modp).

Proof. Taking n = in the identity (12) of Lemma 5, it comes

E (-1)k-1 _ 3 1 (-1)f—1

k=i (f) 51 + 5p 2(g)

Thanks to Lemma 8, we get

k=1 v k

I p I

x6- (—1)k-1 3 A M 1 ( i 2)

= 5 \ — 5p) + 2(a — a)(1 + ^) Hdp)

- 5 o—1 ow™

- 3 + 2a—p{ 25— 203 + èq2)(mod p2).

□

5. Remarks

Remark 1. Recall that the falling factorial zm is the polynomial in z defined by z- := 1

m-1

and zm : = Y\ (z - j) for any integer m ^ 1. Since Q) = mm, we easily may rewrite the

j=o m m-

congruence (16) in the form

P-i 1 _ l0 (modp) , if m =1;

, km I ( l+)-l)ml)2 (modp), if m = 1.

k=m \ (m-2)!(m-l)2 \ fJi

Remark 2. If we consider the case of infinite power sums involving inverse of binomial coefficients

œ 1

Zm(n) = ^ _fcyn, for n, m ^ 1,

k=m m

then by taking m = 1, we have

œ 1

Z (n) = Un) = Y, 1, for n > 2

k=1

where ((n) is the Riemann zeta function. The special case for n =1 and by Lemma 2, we obtain

^ 1

Zm(1) = ^TfcT = -^T, for m > 2. ( k) m — 1

k=m \mj

For any integer n ^ 1, then

1 (2?V)2n

Z (2n) = - ^^nyr B2„,n = 12,...

So we can calculate (m(n) according to Z (n) for example

ro i ro 1

Z2(2) = £7^72 = -12 + 8Z(2) and Z2(3) = £ —3 = 80 - 48Z(2)

k=2 (2) k=2 (2)

Remark 3. Setting n = p in the identity (13), and by using the congruence (3) of Babbage we obtain

p-1 (-1)k_1 11 11 11 7 n 2) ,

£ - - 4 p-) - 4 - 2p (mod p) ' for p ^ 3

From congruence (5) we deduce

£ =4- ^p+2p2 7mod p3)'for p ^5-

fc=i

Then (6) yields to

p-1 (-1)k-1 11 1 2 /1 1 \ 3 / , „

= ^ - + / -{ 2 + P3 HdP4) ' for P > 3

k=i ^ /

References

1. Gauss C.F. Recherches arithmétiques, traduction française de Disquisitiones Arithmeticae. Paris, Blanchard, 1953.

2. Babbage C. Demonstration of a theorem relating to prime numbers. Edinburgh Philosophical Journal, 1819, vol. 1, pp. 46-49.

3. Mestrovic R. Wolstenholme's theorem: its generalizations and extensions in the last hundred and fifty years (1862-2012). ArXiv :1111.3057, 2011.

4. Wolstenholme J. On certain properties of prime numbers. The Quarterly Journal of Pure and Applied Mathematics, 1862, vol. 5, pp. 35-39.

5. Glaisher J.W.L. On the residues of the sums of products of the first p — 1 numbers, and their powers, to modulus p2 or p3. The Quarterly Journal of Mathematics, 1900, vol. 31, pp. 321-353.

6. Morley F. Note on the congruence 24n = (—1)n , where 2n + 1 is a prime. Annals of Mathematics, 1895, vol. 9, pp. 168-170.

7. Ayad M., Kihel O. Recognizing the primes using permutations. International Journal of Number Theory, 2012, vol. 8, no. 8, pp. 2045-2057.

8. Carlitz L. A theorem of Glaisher. Canadian Journal of Mathematics, 1953, vol. 5, pp. 306316.

9. SunZ.W. On congruences related to central binomial coefficients. Journal of Number Theory, 2011, vol. 131, pp. 2219-2238.

10. Berndt B.C., Evans R.J., Williams K.S. Gauss and Jacobi Sums. John Wiley and Sons Inc., 1998.

k

11. Gould H.W. Combinatorial Identities. New York, Morgantown Printing and Binding Co, 1972.

p-i

12. ChowlaS., DworkB., Evans R. On the mod p2 determination of . Journal of Number Theory, 1986, vol. 24, no. 2, pp. 188-196. 4

13. Hudson R.H., Williams K.S. Binomial coefficients and Jacobi sums. Transactions of the American Mathematical Society, 1984, vol. 281, no. 2, pp. 431-505.

14. Lehmer E. On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Annals of Mathematics, 1938, vol. 39, pp. 350-360.

15. Eisenstein F.G. Eine neue Gattung zahlentheoretischer Funktionnen, welche von zwei Elementen ahhangen und durch gewisse linear Funktional-Gleichungen definirt werden. Berichte Knigl. Preuss. Akad. Wiss. Berlin, 1850, vol. 15, pp. 36-42.

Article received: 18.04.2022.

Corrections received: 13.11.2022.

Челябинский физико-математический журнал. 2023. Т. 8, вып. 1. С. 59-71.

УДК 511.1 DOI: 10.47475/2500-0101-2022-18105

НЕКОТОРЫЕ СООТВЕТСТВИЯ, ВКЛЮЧАЮЩИЕ ОБРАТНЫЕ БИНОМИАЛЬНЫЕ КОЭФФИЦИЕНТЫ

Л. Халди, Р. Боумахди

1 Университет Буйры, Буйра, Алжир

2 Университет науки и технологии Хуари Бумедьен, Баб-Эззуар, Алжир al.khaldi@univ-bouira.dz, br_boumehdi@esi.dz

Пусть p — нечётное простое число. В этой статье, среди прочих результатов, мы доказываем некоторые соответствия, включающие обратные биномиальные коэффициенты. Эти соответствия в основном определяются по модулю p, p2, p3 и p4 в кольце p-целых чисел в терминах коэффициентов Ферма, гармонических чисел и чисел Бернулли простым способом. Кроме того, мы распространяем интересную теорему Э. Лемера на класс обратных биномиальных коэффициентов.

Ключевые слова: конгруэнтность, биномиальный коэффициент, коэффициент Ферма, гамма-функция.

Поступила в редакцию 18.04.2022. После переработки 13.11.2022.

Сведения об авторах

Халди Лаала, ассистент лаборатории компьютерных наук, математики и физики для сельского и лесного хозяйства, кафедра математики, Университет Буйры, Буйра, Алжир; e-mail: l.khaldi@univ-bouira.dz.

Боумахди Рашид, доцент Национальной высшей школы математики и лаборатории арифметики, кодирования, комбинаторики и криптографии, кафедра алгебры и теории чисел, Университет науки и технологии Хуари Бумедьен, Баб-Эззуар, Алжир; e-mail: hml@ciencias.unam.mx.