On Wolstenholme s theorem and its converse

Transcription

1 Journal of Number Theory On Wolstenholme s theorem and its converse Charles Helou a,, Guy Terjanian b a Pennsylvania State University, 5 Yearsley Mill Road, Media, PA 9063, USA b Université Paul Sabatier, 8 route de Narbonne, 306 Toulouse, France Received 3 May 006; revised 6 June 007 Available online 4 September 007 Communicated by David Goss Abstract For any positive integer n, letw n = n n = nn. Wolstenholme proved that if p is a prime 5, then w p 3. The converse of Wolstenholme s theorem, which has been conjectured to be true, remains an open problem. In this article, we establish several relations and congruences satisfied by the numbers w n, and we deduce that this converse holds for many infinite families of composite integers n. In passing, we obtain a number of congruences satisfied by certain classes of binomial coefficients, and involving the Bernoulli numbers. 007 Elsevier Inc. All rights reserved. MSC: A07; A4; B65; B68. Introduction Wolstenholme s theorem [W,HW,D,Gr] asserts that if p is a prime number 5, then the binomial coefficient w p = p p satisfies the congruence wp 3. It has been conjectured [Gr,Gu,M,Ri] that the converse is true, so that this congruence characterizes prime numbers. In other words, if w n = n n, then wn mod n 3 for composite numbers n. R.J. McIntosh [M] verified that this holds for all composite n<0 9.Hecalledaprimep satisfying w p 4 a Wolstenholme prime, showed that this is equivalent to the condition that p divides the numerator of the Bernoulli number B p 3, found that there are only two Wolstenholme primes < 0 8, and conjectured that there are infinitely many such primes, but that none satisfies the congruence * Corresponding author. address: cxh@psu.edu C. Helou X/$ see front matter 007 Elsevier Inc. All rights reserved. doi:0.06/j.jnt

2 476 C. Helou, G. Terjanian / Journal of Number Theory w p 5. More recently, the limit 0 8 has been extended to 0 9 in [MR], where it is reported that the only Wolstenholme primes < 0 9 are 6843 and On the other hand, A. Granville [Gr] established broader generalizations of Wolstenholme s theorem, as well as many other important congruences involving binomial coefficients. In this article, we show that the converse of Wolstenholme s theorem holds for several infinite classes of composite positive integers n. We also explore the arithmetical properties of the numbers w n, and establish some relations and congruences satisfied by them. Thus, w np w n n 3 p 3 B p 3 p + 3 n3 p 5 B p n5 p 5 B p 5 mod p 6+v p n+v p w n, for all positive integers n and primes p 5, where v p is the p-adic valuation. And for p = 3, we have w 3n w n + 3 n n 3 n n + mod 3 6+v 3n+v 3 w n. We also get a more complicated expression for p =. However, reducing the modulus allows for simpler congruences, namely w np + cn,p w n 4+v pn+v p w n where cn,p = 3 p3 n 3 B p 3 if p 5 and cn,3 = 9n 3, while cn, = 4n 3 n 4. In particular, w np w n 3, for all primes p 5 and all n. It follows that w n mod n 3 for every composite number of the form n = mp h, where m, h are positive integers and p aprime 5such that m and p 3 w m, as well as for all primes p 3 and all integers p+ m p. Furthermore, if n is a multiple of 3 and n is not a sum of an odd number of distinct powers of 3, or if n is an even integer 4, then w n mod n. In addition, w n mod n 3 when n is a power of 3. On the other hand, w p w p 6 for all primes p 5. Moreover, w p p 3 B p 3 p + p5 3 B p B p 5 mod p 6, for all primes p 5. Further congruences, involving lower order Bernoulli numbers but longer expressions, are also given in the text. It follows that w p 4 if and only if B p 3 0. Also,forp 7, we have w p 5 if and only if B p 3 0 and B p B p 3. Moreover, the congruence w p 6 is equivalent to the conjunction of the previous two congruences and the following one B p 3 3 B p B 3p 5 5 pb p p B p We start by establishing some congruences for certain types of products which arise naturally in the context of the present study. We then apply them to obtain congruences for specific classes of binomial coefficients, from which we derive the desired ones for the numbers w n.. Congruences for products The set of natural numbers is denoted by N, the ring of rational integers by Z, and the field of rational numbers by Q. For a prime number p, the ring obtained by localization of Z at p is denoted by Z p, and the p-adic valuation on Q by v p. The congruences considered in what follows generally hold in Z p.

3 C. Helou, G. Terjanian / Journal of Number Theory We here establish some congruences for certain types of products needed in the sequel. Given a Z and m N \{0}, and a prime number p, weset A m a, p = m p i= also written A m when there is no risk of confusion. + ap jp + i Lemma. Let p be an odd prime number, a Z, and m a positive integer. Then mp / mink,m A m a, p = + p k k= r=, 0 j < <j r m k + +k r =k S k j...s kr j r, where the last summation is over all r-tuples k,...,k r in N r satisfying k + +k r = k, and S n j = i < <i n p for all i, j, n N such that i, n p. e i j...e in j, with e i j = Proof. For any j N,letQ j = p ap i= + jp+i. Then Q j = p / i= + ap jp + i p / i= ap + = jp + p i aa + j + jj + p + ip i, p / i= + ei jp. Since the S n j s for n p are the elementary symmetric functions of the e ij s for i p, upon expanding the latter product, we get Q j = + p / n= p n S n j. Therefore A m = m Q j = m + p / n= p n S n j, for which a direct expansion yields m m A m = + p S j + p 4 S j + + p k j + S k j + j <j <j 3 k +k +k 3 =k m + p mp S p j. j <j k +k =k 0 j <j m S k j S k j S k j S k j S k3 j S j S j +

4 478 C. Helou, G. Terjanian / Journal of Number Theory Thus, A m is equal to plus a sum of terms of the type p k C k,for k mp, with C k = k r= 0 j < <j r m k + +k r =k S k j...s kr j r, where if r>m, the sum over j,...,j r is empty, equal to zero, so that the outer summation is in fact for r mink, m. Hence the announced formula. Corollary. We have m m A m a, p + p S j + p 4 S j + 6+3v p a. 0 j <j m S j S j Proof. This results directly from Lemma, since, in the localization Z p of Z at p, each term e i j 0 v pa for i p and j N, and therefore every sum S n j 0 nv pa for n p. Lemma. For any odd prime p, and any a Z and m N \{0}, we have A m + ama + mp T 6 am m a + 3mp 4 T + 3 a m 4m + 6am + 3a p 4 T, + 6 a mm 3m + 6a m + 3a p 4 T 6+v p a, where T n = p i= i n p i n for n N and T, = i <i p i p i i p i. Proof. We apply the previous corollary, after determining m S j modulo p 4+vpa, and both m S j and 0 j <j m S j S j modulo p +vpa. For every j N, note that S j = p e i j = aa + j + i= p i= jj + p + ip i and

5 so that Hence C. Helou, G. Terjanian / Journal of Number Theory jj + p + ip i = ip i 4 for i p, S j aa + j + p i= + jj+ jj + ip i p ip i ip i p jj + ip i ip i p aa + j + T aa + j + j j + p T 4+v p a. m m m S j at a + j + ap T a + j + j j + 4+v p a. Moreover, using the classical formulas for the power sums m j n for n 3 [IR], we get m m a + j + = a + m + mm = ma + m and mm a + j + j j + = a + Therefore m = mm a + 3m. 6 mm m + a m m 6 4 S j ama + mt 6 am m a + 3mp T 4+v p a. Similarly, for every j N,wehaveS j = i <i p S j = a a + j + and it is easily seen that i <i p e i je i j, i.e. j j + p + i p i j j + p + i p i, j j + p + i p i j j + p + i p i i p i i p i for i <i p,

6 480 C. Helou, G. Terjanian / Journal of Number Theory so that S j a a + j + +v p a. i <i p i p i i p i a a + j + T, Hence m m S j a T, a + + 4a + j + 4j +v pa. Moreover, m a + + 4a + j + 4j = a + m + a + mm + mm m 3 Therefore m = m4m + 6am + 3a. 3 S j 3 a m 4m + 6am + 3a T, +v p a. Similarly, for j N, in view of what precedes, S j aa + j + T +vpa. Hence S j S j a T a + j + a + j + +v p a. 0 j <j m 0 j <j m Moreover, using the formulas for the power sums repeatedly, we get 0 j <j m = m j j = j =0 = a + a + j + a + j + a + + a + j + j + 4j j mm + a + mm = mm 3m + 6a m + 3a mm m 3m 6

7 C. Helou, G. Terjanian / Journal of Number Theory Therefore S j S j 6 a mm 3m + 6a m + 3a T 0 j <j m +v p a. Substituting the expressions obtained above into the congruence of the Corollary yields the desired result. Lemma 3. Let p beaprime 7. Let Then T n = p i= i n p i n for n N and T, = T p B p 3 p + p3 6 B p 3 p3 5 B p 5 4, T p 5 B p 5, 3 T, p 5 B p 5, where the B k s are the Bernoulli numbers k N. Proof. For n N,letR n = p i= i n and P n = p First, Then T = p i= T = p i= p i + = p i p i.e. T = p R + p 3 R. Now, T, = which can also be written T, = p i <i p i= in. i <i p p i + = p p i p i = p R. i= p i= i <i p i i + i + p i + ip i i p i i p i. = p p p + +, i p i i p i i p i + + p i i i=. p i p i i + T,

8 48 C. Helou, G. Terjanian / Journal of Number Theory Setting j = p i and j = p i, the latter sum splits as the sum of the following four sums: i <i p i i <i p <p i <j p and i j i < p <j <p i and i j, p <j <j p j j. Added up, these four sums make up the sum of all terms of the type ij j p i. Hence for i<j p and T, = p i<j p p ij. ip i i= Moreover, i<j p ij = p p i i. i= i= So, in view of what precedes, T, = p R R T = p R R p 3 R. By the Fermat Euler theorem [HW], for i p, and h, n N, with h, we have i n i ϕph n h, where ϕp h = p h p is Euler s totient function. So R n P ϕp h n h, provided that n ϕp h. By a classic formula for the power sums in terms of the Bernoulli numbers [IR] or [HW], P m = m + m+ m + r r= p r B m+ r = m+ r= m p r B m+ r, r r for any positive integer m. Moreover, v p B k for any k N, and v p B k 0ifp k [IR]. So m m v p p r B m+ r = r v p r + v p + v p B m+ r r v p r. r r r It follows that P m r v p r h m p r B m+ r h, r r where the summation is over all integers r m + such that r v p r h.

9 C. Helou, G. Terjanian / Journal of Number Theory In particular, R P ϕp 5 5, and taking m = ϕp 5 = p 4 p, we have P m r v p r 5 m p r B m+ r 5. r r For r m +, let s = v p r and r = p s u, with s,u N and p u. Then r v p r 5if and only iff p s u s + 5, which implies that p s s + 5. But p s >s+ 5fors by induction on s. So r v p r 5 if and only if r 5. Moreover, since B k = 0 for all odd k 3, and m + = p 4 p is even and much larger than 3, then B m+ r = 0forr =, 3, 5. Therefore R p 5 p 4 p B p 5 p p 5 p 4 p 5 p 4 p 5 p 4 3 p 4 B 6 p 5 p 4 4 5, that is R p B p 5 p 4 p4 4 B p 5 p Furthermore, by the Kummer congruences [IR], since p p 5 p 4, and p 5 p 4 p 3 p mod ϕp 3, then B p 5 p 4 p5 p 4 p 3 p B p 3 p p + B p 3 p p B p 3 p 3. Hence R p B p 3 p + p4 4 B p 3 p p4 4 B p 5 p But, we similarly have B p 3 p p3 p B p 3 p 3 3 B p 3 and B p 5 p 4 4 p5 p 4 4 B p 5 4 p 5 5 B p 5. Hence R p B p 3 p + p4 6 B p 3 p4 5 B p 5 5, so that T = p R p B p 3 p + p3 6 B p 3 p3 5 B p 5 4.

10 484 C. Helou, G. Terjanian / Journal of Number Theory Similarly, R P ϕp 4 4, and setting m = ϕp 4 = p 4 p 3, we have P m r v p r 4 m p r B m+ r 4. r r Also, the condition r v p r 4, with r m +, amounts to r 4. Moreover, since m + = p 3 p is odd, B m+ r = 0forr even <m.so P m pb p 4 p 3 + p 4 p 3 p 4 p 3 3 p 3 B 3 p 4 p 3 4 4, i.e. R P m pb p 4 p 3 + p3 B p 4 p Also, by the Kummer congruences, B p 4 p 3 p4 p 3 p 3 p B p 3 p p B p 3 p 3 and B p 3 p 3 B p 3, and B p 4 p B p 5. Hence R pb p 3 p p3 3 B p p3 B p 5 4. It follows that p 3 T = pr + R 5 p4 B p 5 5 and p 3 T, = p R R R p R R p4 5 B p 5 5. Therefore T p 5 B p 5 and T, p 5 B p 5. Proposition. Let p be a prime number 7, a Z, m N \{0}, and let A m a, p = ap +. Then m p i= jp+i A m a, p ama + mp3 B p 3 p + 6 ama + mp5 B p 3 5 ama + m a + am + m p 5 B p 5 6+v p a. Proof. This is obtained by substitution of the expressions for T, T and T, from Lemma 3 into the expression for A m = A m a, p from Lemma. Note first that T 0, so that the last term in A m, containing the factors p 4 T and a,is 0 6+v pa and can be dropped. Thus

11 C. Helou, G. Terjanian / Journal of Number Theory A m + ama + mp p B p 3 p + p3 6 B p 3 p3 5 B p 5 5 am m a + 3mp 5 B p 5 5 a m 4m + 6am + 3a p 5 B p 5 ama + mp3 B p 3 p + 6 ama + mp5 B p 3 5 am 3a + m + m a + 3m + a 4m + 6am + 3a p 5 B p 5 ama + mp3 B p 3 p + 6 ama + mp5 B p 3 5 ama + m a + am + m p 5 B p 5 6+v p a. Lemma 4. For the primes p = 5 and p = 3, keeping the above notation A m a, p, for any a Z and m N \{0}, we have A m a, ama + m 54 ama + ma + am + m mod 5 6+v 5a, A m a, ama + m 34 ama + ma m a + m + mod 3 6+v 3a. Proof. We substitute into the expression of A m a, p from Lemma the actual values of T, T and T,, which are easily calculated for p = 5 and p = 3. For p = 5, we have T = 5, T = 3 44 and T, = 4.So i.e. A m a, ama + m am m a + 3m a m 4m + 6am + 3a mod 5 6+v 5a, A m a, ama + m am a 4m + 6am + 3a 3 m mod a + 3m 5 6+v 5 a. The last factor, that of 54 7 am, evaluated modulo 5, can be replaced by a 4m + 6am + 3a + m a + 3m = 3 a 3 + m 3 + ama + m a + m = 3a + m a + am + m.

12 486 C. Helou, G. Terjanian / Journal of Number Theory Therefore A m a, ama + m + 4 ama + m a + am + m mod 5 6+v 5a, which is the desired expression, since 4 is mod 5. For p = 3, we have T =, T = 4 and T, = 0. So A m a, ama + m + 8 amm a 3m + 6a m + 3a m + a + 3m mod 3 6+v 3a. An expansion of the factor of 33 8 amm in this expression yields as a replacement 3 a m + a a m + a 3 a Therefore = 3a m + a + m + a + a = 3a a + ma + m +. A m a, ama + m + 8 amm a a + ma + m + mod 3 6+v 3 a, which is the desired expression, since 8 is mod 3. Remark. The congruence in Proposition is also valid for p = 5, i.e. p 3 A m a, p ama + m B p 3 p p5 6 B p 3 + p5 a + am + m B p v p a, for any prime p 5. Indeed, for p = 5, the right-hand side term in this congruence is R = 5 3 ama + m B B + 5 a + am + m B 0. Since B 0 = and B = 6, while B 98 9 mod 5 3, by the PARI-GP calculator, then B B + 5 a + am + m B 0 5 a + am + m 3 5 a + am + m mod 5 3,

13 C. Helou, G. Terjanian / Journal of Number Theory so that R ama + m 5 a + am + m mod p 6+v p a, which is precisely the expression of A m a, 5 in Lemma 4. Furthermore, on the basis of numerical evidence, we conjecture that the congruence in Proposition holds, not only modulo p 6+v pa, but also modulo p 6+v pa+v p m+v p a+m,forp 5. Remark. In the case of the prime p =, a similar approach allows to obtain an expression for A m a, = m a + j+ modulo 6+va. Indeed, A m a, = + m k= k a k S k, where the S k s for k m are the elementary symmetric functions of the terms j+ for 0 j m. Then A m a, + 5 k= k a k S k mod 6+va with the convention that S k = 0if k>m, and the sums S k are to be determined modulo 6 k for k 5. This is done by replacing each j+ by 5 n=0 n n j n, expanding, and evaluating the resulting expressions modulo 6, as was done in the case of odd primes. The resulting congruence, more complicated than for odd primes, is amm A m a, 5am + 3 5a m + 4 a a + m m 3 4 m m 4 3 a 3 m a a m 5 with the convention that the binomial coefficient m k = 0ifk>m. + m mod 6+v a, Remark 3. Reducing the moduli in the previous results, we get the following simpler congruences, for all a Z and m N \{0}: For any prime p 5, we have A m a, p 3 ama + mp3 B p 3 4+vpa. For p = 3, we have A m a, ama + m mod 34+v3a. 3 For p =, we have A m a, + ama + m am mod 4+va. Indeed, the first congruence follows from Proposition and Remark, taking into account the Kummer congruence encountered in the proof of Lemma 3 B p 3 p 3 B p 3, for p 5. The second one is an immediate consequence of Lemma 4. The third congruence can be verified by induction on m. Indeed, it holds trivially for m =, and assuming that it does for m, we check that it holds for m + too, as follows: A m+ a, = A m a, + a m + + ama + m am + a m + 4m + ama + m am + a + 4am 8am aa + m am + ama + m am + a + 4aa m 8a a m

14 488 C. Helou, G. Terjanian / Journal of Number Theory ama + m am + a 4aa m + am + a + m + am + mod 4+v a. 3. Congruences for binomial coefficients Lemma 5. Let m, n N such that m n, and let p be a prime number. Then np = mp n m m p + i= n mp. jp + i Proof. Starting from the definition of the binomial coefficient np mp as the quotient of two products of consecutive integers, and collecting together, in each of the products, the factors that are divisible by p and those that are not, we get mp np r= n mp + r = = mp mp! nk=n m+ kp n mj= jp m p k=n m i= kp + i p i= jp + i. Clearly, n n kp = k p m and k=n m+ k=n m+ m m jp = j p m, j= j= so that nk=n m+ kp mj= jp nk=n m+ k = = m! n. m Moreover, setting k = n m + j in the double product appearing in the numerator of the expression of np mp, we get n p k=n m i= kp + i = m p i= n mp + jp + i. Substituting these expressions into that of np mp, and writing the quotient of the two double products as the double product of the quotients, we obtain np = mp n m m p i= n mp + jp + i jp + i = n m m p + i= Proposition. Let p be a prime number, and m, n N such that 0 m n. n mp. jp + i

15 C. Helou, G. Terjanian / Journal of Number Theory If p 5, then np mp n p 3 mnn m m B p 3 p p5 6 B p m mn + n p 5 B p 5 mod p 6+v p n m+v p n m. If p = 3, then 3n n 3 + mnn m 3m m 34 m n + n m mod 3 6+v 3 n m+v 3 m n. Proof. First note that the congruences hold trivially in the case m = 0, since the two sides reduce to. So we assume m n. In view of Lemma 5, np mp = n m Am n m, p, where A m a, p = m p ap i= + jp+i for a Z. Then, by Proposition and Remark, for p 5, with a = n m,wehave, A m n m, p mnn m p3 B p 3 p 6 p5 B p 3 + m mn + n p 5 B p v p n m. Similarly, by Lemma 4, for p = 3, we have 3 mod A m n m, 3 + mnn m 34 m n + n m 3 6+v 3 n m. Substituting these expressions into the formula relating np mp and n m yields the desired congruences. Remark 4. For p = 5, we also have the simpler alternative congruence, based on Lemma 4, namely 5n n mnn m 5m m 54 m mn + n mod 5 6+v 5 n m+v 5 m n. Furthermore, using the reduced moduli and the results in Remark 3, we similarly obtain For p 5, np mp n mod m 3 mnn mp3 B p 3 p 4+v p n m+v p m n.

16 490 C. Helou, G. Terjanian / Journal of Number Theory For p = 3, 3 For p =, n m 3n 3m n + 3 mod mnn m 3 4+v 3 n m+v 3 m n. m n + mn m m mn + n mod 4+v n m+v m n. m Remark 5. Since n m = n n m, then, exchanging m and n m, we see that the above congruences are also valid modulo p h+v pm+v p m n, therefore modulo p h+maxv pm,v p n m+v p m n, with h = 6 or 4, according to the context. In fact, as in Remark, we conjecture that these congruences are valid modulo p h+v pm+v p n m+v p n+v p m n. Corollary. Let m, n N be such that 0 m n. For any prime p 5, For p = 3, 3 For p =, np mp 3n 3m n m In particular, n m n m mod. n mod p 3+maxv p m,v p n m+v p m n. m n mod 3 +maxv 3 m,v 3 n m+v 3 m n. m n mod +maxv m,v n m+v m n. m Here, the last congruence follows from the previous one, since if v m = v n m = 0, then m is odd and n is even, and therefore v n m. Remark 6. Using p-adic methods, the modulus in the previous Corollary can be improved to p 3+v pm+v p n+v p n m+v p n m for p 5, and to 3 +v 3m+v 3 n+v 3 n m+v 3 n m for p = 3 [Ro], the Kazandzidis congruences. But here, following a more elementary method, we derived congruences relating np mp and n m to a modulus for which they are not necessarily in the same congruence class, namely modulo p 6+ instead of p 3+.

17 C. Helou, G. Terjanian / Journal of Number Theory Wolstenholme s theorem and generalizations The binomial coefficients related to Wolstenholme s theorem are of the form w n = n n =. We start by considering a slightly broader class consisting of wm,n = mn n = mn m n, n n for any positive integers m, n, so that w n = w,n. They satisfy some congruences that generalize Wolstenholme s theorem. Proposition 3. Let p be a prime number, and m, n be positive integers. If p 5, then w m,np w m,n mm n 3 p 3 6+v p m +v p n+v p w m,n. B p 3 p p5 6 B p m m + n p 5 B p 5 If p = 3, then 3 w m,3n w m,n + mm n 3 34 n mn + mn n mod 3 6+v 3 m +v 3 n+v 3 w m,n. Proof. By Proposition, for p 5, we have mnp mn p mn 3 mn n np n B p 3 p p5 6 B p n mn + m n p 5 B p 5 mod p 6+v p mn n+v p mn n. Similarly, 3mn 3n mn n mod 3 6+v 3 mn n+v 3 mn n. + mn mn n 3 34 n mn + mn n The results now follow upon division of the above congruences by m, noting that v mn p n v p m = v p w m,n. Remark 7. Reducing the moduli, and using Remark 4, we get, for a prime p and for any positive integers m, n, the following congruences: If p 5, then w m,np w m,n 3 mm n3 p 3 B p 3 4+v pm +v p n+v p w m,n. w m,3n w m,n + 3 mm n3 mod 3 4+v 3m +v 3 n+v 3 w m,n. 3 w m,n w m,n + m n 3 m mn + n mod 4+v m +v n+v w m,n.

18 49 C. Helou, G. Terjanian / Journal of Number Theory Corollary. Let p be a prime number, and m a positive integer. If p 5, then p 3 w m,p mm B p 3 p p5 6 B p 3 + m m + p 5 B p v p m. In particular, w m,p 3 mm p3 B p 3 4+vpm, and therefore w m,p 3. If p = 3, then w m,3 = + 3 mm. 3 If p =, then w m, = + m. For p 5, this follows from Proposition 3 by taking n =, since w m, = for all positive integers m.forp = 3 and p =, this follows from the definition. Corollary. For any prime p 5 and any positive integer k, we have w p k +,p k+3. w p k,p k+3. 3 In particular, w p+,p w p,p 4. Specializing to m =, we deduce from Proposition 3 the following results concerning the Wolstenholme binomial coefficients w n = n n. Proposition 4. Let p be a prime number, and n a positive integer. If p 5, then w np w n n 3 p 3 B p 3 p + 3 n3 p 5 B p n5 p 5 B p 5 mod p 6+v p n+v p w n. If p = 3, then W 3n w n + 3 n n 3 n n + mod 3 6+v 3n+v 3 w n. Remark 8. Reducing the moduli, and using Remark 7, we get, for a prime p and any positive integer n: w np w n 3 n3 p 3 B p 3 4+v pn+v p w n,forp 5. w 3n w n + 3 n 3 mod 3 4+v 3n+v 3 w n. 3 w n w n n 3 n mod 4+v n+v w n. Corollary. For any prime p, and any positive integers k,n, we have w np k+ w np k 4+k+v pn+v p w n.

19 C. Helou, G. Terjanian / Journal of Number Theory w np k w np 5+v pn+v p w n. 3 w p k w p 5. 4 If p 5, then w np k w n 3+v pn+v p w n +min,v p n. 5 w 3 k n w n mod 3 +v 3n+v 3 w n + min,v 3 n. 6 w k n w n mod +v n+v w n +3 min,v n. Proof. In view of Proposition 4, if p 5, then w np w n 3+v pn+v p w n +min,v p n ; and w 3n w n mod 3 +v 3n+v 3 w n + min,v 3 n. Also, in view of Remark 8, w n w n mod +v n+v w n +3 min,v n. Moreover, v p w np = v p w n, since v p w np w n + v p w n >v p w n, so that v p w np k = v p w n,fork 0. From these congruences, follows immediately. 6 are then established by induction on k. Corollary. For any prime p 5, we have w p p 3 B p 3 p + 3 p5 B p p5 B p 5 6. In particular, w p 3 p3 B p 3 4, and therefore w p 3. Corollary 3. For any prime p 5 and any integer k, we have w p k w p k 3 p3k B p 3 k+5. If k 3, then w p k w p k k+5. 3 In particular, w p k w p 6. Proof. It follows from Proposition 4 that w p k w p k p 3k B p 3 p k+5. Moreover, by the Kummer congruences, B p 3 p 3 B p 3. Hence the result in. As to and 3, they are immediate consequences of by an induction in 3. Corollary 4. Let p and q be distinct prime numbers 5. We have w pq w q 3 and w pq w p mod q 3. Thus, w pq q 3 if and only if w q 3 and w p mod q 3. Note. Corollary 4 is a direct consequence of formulas and in [M]. Indeed, for distinct primes p,q 5, it follows from these formulas that w pq w p w q 3 q 3, and w p 3 and w q mod q 3, which yields Corollary 4. Corollary 5. For any prime p 5, we have w p w p p 3 B p 3 p + p5 3 B p B p 5 6. w p w p p 3 p B p p 5. 3 w p w p 3 p3 B p w p 6 w p 6 B p 3 p p 3 B p B p w p 5 w p 5 B p p 0. 6 w p 4 w p 4 B p 3 0.

20 494 C. Helou, G. Terjanian / Journal of Number Theory Proof. and 3 follow from Corollaries and 3. follows from by reduction to the modulus p 5, using the Kummer congruences to note that B p 3 p p3 p p p B p p p + B p p 4, 5 and 6 follow from, and 3, respectively. p B p p. Remark 9. The conditions in Corollary 5 can be put in a form more easily verifiable by computation, through the use of the following congruences for Bernoulli numbers, also due to Kummer [K]. Let p be a prime number and m, n be positive integers such that m is even, p m, and n<m. Then n n k Bm+kp k m + kp 0 n. k=0 We also recall the more familiar Kummer congruences [IR] that we have already used in the proofs of Lemma 3 and Corollaries 3 and 5 above. Let p be a prime number and m, m,n be positive integers such that m, m are even, not divisible by p, and both >n.ifm mmod ϕp n, where ϕp n = p n p is Euler s function, then B m m B m m n. Applying to the case m = ϕp n, where the prime p 7 and n is any positive integer, and using to replace B m+kp m+kp by B kp kp, modulo pn,for k n, we get B p n n p n p n p n k= k+ n k Bkp kp n. 3 In particular, for n =, we deduce that for any prime p 7, B p p p + p 4 B p + p 4 p 3 B p 3. 4 Similarly, for n = 3 and any prime p 7, B p 3 p p + 3 p 4 B p 4 3 p 3 B p 3 3p 5 B 3p 5 mod p 3. 5 Moreover, for any integers a,b,n such that p b and n>0, we have ap b n a k p k n b b k. k=0

21 C. Helou, G. Terjanian / Journal of Number Theory Also, by, B p 4 p 4 B 3p 5 3p 5 B p 3 p 3, and, by, B p 3 p 3 B p 4 Using these congruences in 4 and in 5 respectively, we get p 4 + B 3p 5 3p 5 0. B p p 4 3 B p 3 B p pb p 3 6 and B p 3 p B p 3 3 B p B 3p 5 + p 5 B p B p p B p Substituting 6 and 7 into the first two expressions of Corollary 5, we deduce the following additional results. Corollary 6. For any prime p 7, we have w p p B p 3 B p 4 9 p4 B p 3 5. w p p 3 B p 3 3 B p B 3p 5 p B p B p 4 + p 5 90 B p B p The following conditions are equivalent: i w p 5, ii w p 5, iii B p 3 0 and B p B p 3. 4 The following conditions are equivalent: i w p 6, ii w p 6, iii B p 3 0 and B p B p 3 and B p 3 3 B p B 3p 5 5 pb p p B p Remark 0. Let p be a prime number and m, r two positive integers. From the definition, we have w mp+r = c m p, rw mp, where c m p, r = r i= mp + i/ r i= mp + i.in particular, if r p, then the denominator of c m p, r is relatively prime to p and is congruent to r! modulo p, while the numerator is congruent to r! modulo p, so that c m p, r r r. Thus w mp+r w m w r, for r p. It follows that if r is a positive integer and p is a prime factor of w r such that p>r, then p is a prime factor of w n for every positive integer n r. In the special case where r =, we further have c m p, = mp + /mp + mp + h k=0 k m k p k h, for any positive integer h. It follows, in view of Remark 8, that if p 5, then

22 496 C. Helou, G. Terjanian / Journal of Number Theory w mp+ w m + mp m p + m 3 p 3 3 B p 3 mod p 4. A similar, but more complicated, congruence holds modulo p 6, in view of Proposition 4. Remark. For any positive integer n, wehave n+ n = n+ n n+ n, so that wn+ = n+ n+ w n. Since n + and n + are relatively prime, it follows that n + divides w n and n + divides w n+. Thus w n is divisible by n and also by either n + or n+ according to whether n is even or odd. 5. The converse of Wolstenholme s theorem The converse of Wolstenholme s theorem stipulates that if n is a composite positive integer, then w n satisfies the condition CW w n mod n 3. In what follows, we give several infinite families of composite integers n satisfying CW. Proposition 5. For any positive integers m, k and any prime p 5 such that p 3 w m, the integer n = mp k satisfies CW. For any positive integers m, k such that 3 w m, the integer n = 3 k m satisfies w n mod n, and therefore n satisfies CW. 3 For any prime p 5 such that w p 6, and any integer k, the prime power n = p k satisfies CW. 4 For any prime p 3 and any positive integer m such that m<p<m, and any positive integer k, the product n = mp k satisfies w n mod n, and therefore n satisfies CW. Proof. By Corollary to Proposition 4, w mp k w m 3 and w 3 k m w m mod 3,for primes p 5 and integers k. So the condition w m 3 respectively w m mod 3 implies that w mp k 3 hence w mp k mod mp k 3 respectively w 3 k m mod 3 hence w 3 k m mod 3k m. This proves and. By Corollary 3 to Proposition 4, w p k w p 6. So the condition w p 6 implies that w p k 6 hence w p k k 3,fork. This gives 3. If m<p<m, then p m! and p m!, sop does not divide m m = m! m!,i.e. m! w m = m m. If, in addition p 3, then by Corollary to Proposition 4, wmp k w m, which implies the result in 4. Note. In [M, p. 387], it is shown that if n = p satisfies n n mod n 3, then p satisfies p p 6. But then, it easily follows from and in [M] that w p k w p 6 for all positive integers k, which gives the property 3 in Proposition 5 above. Remark. According to the computations reported in [MR], there are only two prime numbers p<0 9 called Wolstenholme primes satisfying w p 4, i.e. satisfying B p 3 0

23 C. Helou, G. Terjanian / Journal of Number Theory , namely p = 6843 and p = Therefore, in view of Corollary to Proposition 4, for all primes 5 p<0 9 other than p and p, and for all positive integers k, we have v p w p k = v p w p = 3. Moreover, we checked with the PARI-GP calculator that B p B p 3, so that in view of Corollary 6 to Proposition 4, w p 5. Therefore v p w p k = v p w p = 4 for all positive integers k. Thus, in view of Proposition 5, for all primes 5 p<0 9 other than p and for all integers k, the prime power n = p k satisfies CW. We also computed the factorization of w m for m 49 and found that, for these values of m, the difference w m has no cubic factors except when m is the power of a prime p 5, for which v p w m = 3; or when m = 39 or m = 7, for which v 3 w m = 3. This raises the question of determining the integers m, that are not prime powers, for which w m has a cubic factor. A related question is that of determining the primes p for which w p has a cubic factor prime to p. A further question is to determine the pairs of distinct primes p,q such that p w q and q w p. In view of Corollary 4 to Proposition 4, the last two conditions together amount to the condition that n = pq satisfies the congruence w n mod n. In [M], three such solutions n to this congruence are given, namely n = and n = and n = , of which the first two are the only solutions n<0 9 which are not prime powers. A simpler question is thus whether the congruence w n mod n has infinitely many solutions that are not prime powers. Proposition 6. Let n be an integer >. If n is not a power of, then w n 0 mod. If n is a power of, then w n 3 mod 3. 3 If n is even and n>, then w n mod n. In particular, all positive even integers n satisfy CW. Proof. By a theorem of Legendre [Gr,Ri], if n = h i=0 i n i, with all n i {0, }, is the -adic development of n, then the -adic valuation of n! is given by v n! = n i 0 n i. Moreover, since n = h i=0 i+ n i has the same non-zero -adic digits as n, then v n! = n i 0 n i, and therefore n v = v n! v n! = n n i n n i = n i. n i 0 i 0 i 0 In other words, v n n is equal to the number of non-zero -adic digits of n. Thus,ifn is not a power of, then it has at least two non-zero -adic digits, and therefore v n n, i.e. w n = n n 0 mod. This proves. On the other hand, by Corollary to Proposition 4, w k w mod 5, for any positive integer k. Since w = 3, this gives. As to 3, it results immediately from and. Note 3. Property 3 of Proposition 6 was already obtained by R.J. McIntosh, who gave a proof of it in the original manuscript [M] of [M].

24 498 C. Helou, G. Terjanian / Journal of Number Theory Lemma 6. Let n be a positive integer, p a prime number 3, and n = h i=0 n i p i be the p-adic development of n, where the integers n i satisfy 0 n i p for 0 i h. We have p w n if and only if n i p+ for at least one i {0,,...,h}. If p w n, then w n hi=0 ni n i, where, by definition, 0 0 =. Proof. By a theorem of Kummer [Gr,Ri], the p-adic valuation v n p n of n n is equal to the number of carries when adding n and n in base p. Since w n = n n and the prime p, it follows that v p w n = v n p n > if and only if there is at least one carry in the addition of n and n in base p, which means that for at least one i, the sum of the p-adic digits n i + n i >p, i.e. n i p+. Hence the first result. If p w n, then, by what precedes, 0 n i p for 0 i h, so that the integers n i are exactly the p-adic digits of n. Then, by a theorem of Lucas [Gr,Ri], we have n n hi=0 ni n i. Hence the second result. Proposition 7. Let n be an integer >. We have w n 0 mod 3 if and only if n is not the sum of distinct powers of 3. If n is the sum of r distinct powers of 3, then w n r mod 3. 3 If 3 n, and either n is not the sum of distinct powers of 3 or n is the sum of an even number of distinct powers of 3, then w n mod n, and therefore n satisfies CW. 4 For any positive integer k, we have w 3 k 0 mod 3 5. Thus, if n is a power of 3, then v 3 w n =, so that w n mod 3 3, and therefore n satisfies CW. Proof. Let n = h i=0 n i 3 i be the 3-adic development of n, where the integers n i satisfy 0 n i for0 i h. By Lemma 6, w n 0 mod 3 if and only if at least one n i, i.e. not all n i = 0or,i.e. n is not of the form r j= 3 i j, with 0 i <i < <i r h. Hence the first result. If n = r j= 3 i j, where 0 i <i < <i r h, i.e.n i = ifi {i,...,i r } and n i = 0 otherwise, then, also by Lemma 6, we have 3 w n and w n rj= = r r mod 3. 3 By what precedes, if n is not the sum of distinct powers of 3, then w n 0 mod 3, and if n is the sum of an even number r of distinct powers of 3, then w n r = mod 3. In either case w n mod 3, and if in addition 3 n, then w n mod n. Hence the third result. 4 By Corollary to Proposition 4, w 3 k+ w 3 k mod 3 4+k, for all k. This congruence holds in particular modulo 3 5, independently of k. Hence, by a simple induction, w 3 k w 3 mod 3 5. Since w 3 = 0, the fourth result follows. Acknowledgment We would like to thank the referee who pointed out some missed or unknown to the authors references.

25 C. Helou, G. Terjanian / Journal of Number Theory References [D] L.E. Dickson, History of the Theory of Numbers, vol., Chelsea, New York, 966. [Gr] A. Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, in: Organic Mathematics, Burnaby, BC, 995, in: CMS Conf. Proc., vol. 0, Amer. Math. Soc., Providence, RI, 997, pp [Gu] R. Guy, Unsolved Problems in Number Theory, third ed., Springer, New York, 004. [HW] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, fifth ed., Oxford University Press, 979. [IR] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, second ed., Springer, New York, 990. [K] E.E. Kummer, Über eine allgemeine Eigenschaft der rationalen Entwicklungscoefficienten einer bestimmten Gattung analytischer Functionen, J. Reine Angew. Math [M] R.J. McIntosh, On the converse of Wolstenholme s theorem, Acta Arith [M] R.J. McIntosh, On primality and Wolstenholme s theorem, preprint, 994. [MR] R.J. McIntosh, E.L. Roettger, A search for Fibonacci Wieferich and Wolstenholme primes, Math. Comp [Ri] P. Ribenboim, The New Book of Prime Number Records, third ed., Springer, New York, 996. [Ro] A.M. Robert, A Course in p-adic Analysis, Springer, New York, 000. [W] J. Wolstenholme, On certain properties of prime numbers, Quart. J. Pure Appl. Math