On the Divisibility of Fermat Quotients

Transcription

1 Michigan Math. J. 59 (010), On the Divisibility of Fermat Quotients Jean Bourgain, Kevin Ford, Sergei V. Konyagin, & Igor E. Shparlinski 1. Introduction For a prime p and an integer a the Fermat quotient is defined as q p (a) = ap 1 1. p It is well known that divisibility of Fermat quotients q p (a) by p has numerous applications, which include Fermat s last theorem and squarefreeness testing; see [6; 7; 8; 16]. In particular, the smallest value l p of a for which q p (a) 0 (mod p) plays a prominent role in these applications. In this direction, Lenstra [16, Thm. 3] has shown that { 4(log p) if p 3, l p (1) (4e + o(1))(log p) if p ; see also [7]. Granville [9, Thm. 5] has shown that in fact l p (log p) () for p 5. A very different proof of a slightly weaker bound l p (4 + o(1))(log p) has been obtained by Ihara [1] as a by-product of the estimate log l log log p + + o(1) (3) l k l k <p l W(p) as p, where the summation is taken over all prime powers up to p of primes l from the set W(p) ={l prime : l<p, q p (l) 0 (mod p)}. However, the proof of (3) given in [1] is conditional on the extended Riemann hypothesis. It has been conjectured by Granville [8, Conj. 10] that l p = o((log p) 1/4 ). (4) Received December, 008. Revision received August 14,

2 314 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski It is quite reasonable to expect a much stronger bound on l p. For example, Lenstra [16] conjectures that l p 3; this has been supported by extensive computation (see [5; 14]). The motivation for the conjecture (4) comes from the fact that this has some interesting applications to Fermat s last theorem [8, Cor. 1]. Although this motivation relating l p to Fermat s last theorem does not exist anymore, improving the bounds (1) and () is still of interest and may have some other applications. Theorem 1. as p. We have l p (log p) 463/5+o(1) Following the arguments of [16], we derive the following improvement of [16, Thm. ]. Corollary. For every ε>0 and a sufficiently large integer n, if a n 1 1 (mod n) for every positive integer a (log n) 463/5+ε then n is squarefree. The proof of Theorem 1 is based on the original idea of Lenstra [16], which relates l p to the distribution of smooth numbers, which we also supplement by some recent results on the distribution of elements of multiplicative subgroups of residue rings of Bourgain, Konyagin, and Shparlinski [3] combined with a bound of Heath-Brown and Konyagin [10] for Heilbronn exponential sums. Also, using these results we can prove the following. Theorem 3. For every ε>0, there is δ>0such that for all but one prime Q 1 δ <p Q, we have l p (log p) 59/35+ε. The proof of the next result is based on a large sieve inequality with square moduli that is due to Baier and Zhao [1]. Theorem 4. For every ε>0, there is δ>0such that for all but O(Q 1 δ ) primes p Q, we have l p (log p) 5/3+ε. We note that = , = , 5 = Throughout the paper, the implied constants in the symbols O and may occasionally depend on the positive parameters ε and δ, and are absolute otherwise. We recall that the notations U = O(V ) and U V are both equivalent to the assertion that the inequality U cv holds for some constant c>0.. Smooth Numbers For any integer n we write P(n) for the largest prime factor of an integer n with the convention that P(0) = P(±1) = 1.

3 On the Divisibility of Fermat Quotients 315 For x y we define S(x, y) as the set y-smooth numbers up to x, that is S(x, y) ={n x : P(n) y} and put (x, y) = #S(x, y). We make use of the following explicit estimate, which is due to Konyagin and Pomerance [15, Thm..1] (see also [11] for a variety of other results). Lemma 5. If x 4 and x y, then (x, y)>x 1 log log x/log y. 3. Heilbronn Sums For an integer m 1 and a complex z,weput e m (z) = exp(πiz/m). Let Z n be the ring of integers modulo an n 1 and let Z n be the group of units of Z n. Now, for a prime p and an integer λ, we define the Heilbronn sum p H p (λ) = e p (λb p ). For x Z p denote f(x) = x + x + + xp 1 p 1 Z p. (5) Also, define for u Z p F(u) ={x Z p : f(x) = u}. (6) We now recall the following two results due to Heath-Brown and Konyagin that are [10, Thm. ] and [10, Lemma 7], respectively. Lemma 6. Lemma 7. Uniformly over all s 0 mod p, we have p H p (s + rp) 4 p 7/. r=1 Let U be a subset of Z p and T = #U. Then #F(u) (pt ) /3. u U Since H p (rp) = 0ifr 0 mod p and H p (rp) = p if r 0 mod p, we immediately derive from Lemma 6 that H p (u) 4 p 9/. (7) p u=1

4 316 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski 4. Distribution of Elements of Multiplicative Subgroups in Residue Rings Given a multiplicative subgroup G of Z n, we consider its coset in Z n (or multiplicative translate) A = λg, where λ Z n. For an integer K and a positive integer k, we denote J(n, A, k, K) = #({K + 1,..., K + k} A). We need the following estimate from [3]. Lemma 8. Let A be a coset of a multiplicative subgroup G of Z n of order t. Then, for any fixed ε>0, we have J(n, A, k, K) kt n + k M n (w; Z, G) tn e n (uw), w Z n u A where Z = min{n 1+ε k 1, n/} and M n (w; Z, G) is the number of solutions to the congruence w zu (mod n), 1 z Z, u G. Let N(n, G, Z) be the number of solutions of the congruence ux y (mod n), where 0 < x, y Z, and u G. We use Lemma 8 in a combination with yet another result from [3], which gives an upper bound on N(n, G, Z). We note that the proof given in [3] works only for Z n 1/, which is always satisfied in this paper; however it is shown in [4] that the result holds without this condition too, exactly as it is formulated in [3]. Lemma 9. Let ν 1 be a fixed integer and let n. Assume #G = t n. Then for any positive number Z we have N(n, G, Z) Zt (ν+1)/(ν(ν+1)) n 1/((ν+1))+o(1) + Z t 1/ν n 1/ν+o(1). 5. Large Sieve for Square Moduli We make use of the following result of Baier and Zhao [1, Thm. 1]. Lemma 10. Let α 1,..., α N be an arbitrary sequence of complex numbers and let N N Y = α n and S(u) = α n exp(πiun). n=1 Then, for any fixed ε>0and arbitrary Q 1, we have q ( ) a S (QN ) ε (Q 3 + N + min{nq 1/, N 1/ Q })Y. q 1 q Q a=1 gcd(a,q)=1 n=1

5 On the Divisibility of Fermat Quotients Proof of Theorem 1 For a positive integer k<p, let N p (k) denote the number of elements v [1, k] of the subgroup G Zp of order p 1, consisting of nonzero pth powers in Z p. We fix some ε>0. To get an upper bound on N p (k) we use Lemma 8, which we apply with n = p, A = G, t = p 1, and K = 0. For every integer a with a p 1 1 (mod p ) there is a unique integer b with 1 b p 1 such that a b p (mod p ). Thus the corresponding exponential sums of G are Heilbronn sums, defined in Section 3. We derive N p (k) = J(p, G, k, K) k p + k M p 3 p (w; Z, G)( H p (w) +1). (8) By the Hölder inequality, we obtain ( ) 4 M p (w; Z, G) H p (w) w Z p w Z p ( = M p (w; Z, G) 1/ (M p (w; Z, G) ) 1/4 ( H p (w) 4 ) 1/4 ) 4 w Z p ( ) M p (w; Z, G) M p (w; Z, G) H p (w) 4. (9) w Z p Trivially, we have w Z p w Z p w Z p M p (w; Z, G) = Z (p 1) p 3+ε k 1. (10) We also see that w Z p M p (w; Z, G) = (p 1)N(p, G, Z). We now choose k = p 463/5+3ε. Lemma 9 applies with ν = 6 and leads to the estimate N(p, G, Z) Zp 13/84 (p ) 1/14+o(1) + Z p 1/6 (p ) 1/6+o(1) Zp 13/84 (p ) 1/14+o(1) (since for Z p 41/5 the first term dominates). Hence, N(p, G, Z) p +1/84+3ε k 1. Therefore M p (w; Z, G) p 3+1/84+3ε k 1. (11) w Z p

6 318 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski Substituting (7), (10), and (11) in (9) and then using (8), we deduce that N p (k) k p + k p 3 (p3+ε k 1 ) 1/ (p 3+1/84+3ε k 1 ) 1/4 (p 9/ ) 1/4 + p ε k p + k1/4 p 17/336+ε, provided p is large enough. Recalling our choice of k, we see that N p (k) k p (1) for the preceding choice of k and sufficiently large p. Since a p 1 1 (mod p ) for all positive integers a l p, this also holds for any a that is composed of primes l l p. In particular it holds for any a S(k, l p ). Thus (k, l p ) N p (k). (13) Now, using Lemma 5 and the bound (1), we derive from (13) that which implies that log log k log l p Therefore log p log k + O k 1 log log k/log l p k p, ( ) 1 = log k ( ) ( ) ε + O. log p ( ) ( ) 463 log log p log l p 5 + 3ε log log k + O log p ( ) ( ) = 5 + 3ε log log p + O(1) 5 + 4ε log log p, provided that p is large enough. Taking into account that ε is arbitrary, we conclude the proof. 7. Proof of Theorem Preliminaries We need several statements about the groups of pth powers modulo p, which may be of independent interest. Fix a prime p. Let again G be the group of order p 1, consisting of nonzero pth powers modulo p. Lemma 11. If n 1, n G are such that n 1 n (mod p) then we also have n 1 n (mod p ).

7 On the Divisibility of Fermat Quotients 319 Proof. Since n 1, n G we can write n 1 m p 1 (mod p ) and n m p (mod p ) (14) for some integers m 1 and m. Therefore m 1 m m p 1 mp n 1 n 0 (mod p). Then m 1 = m + pk for some integer k, which, after substitution in (14), yields the desired congruence. For v Z p, let D p (v) ={(m 1, m ) :0 m 1, m p 1, m p 1 mp v (mod p )}. (15) We can rewrite Lemma 7 in the following form. Lemma 1. Let V be a subset of Zp, T = #V, and v 1/v / G for any distinct v 1, v V. Then #D p (v) (pt ) /3. v V Proof. We follow the arguments of the proof of Lemma from [10]. For v Zp denote λ(v) = v 1 p Zp. Since the cardinality #D p (v) is invariant under multiplication by elements of the group G we have #D p (λ(v)) = #D p (v). Next, we always have λ(v) 1 (mod p). Therefore, the congruence λ(v) m p 1 mp (mod p ) implies m 1 m λ(v) 1 (mod p). Hence λ(v) m p 1 (m 1 1) p (mod p ). But m p 1 (m 1 1) p 1 pf (m 1 ) (mod p ) where the function f(x) is defined by (5). Hence, #D p (v) = #F(U(v)) (16) where U(v) = (1 λ(v))/p Z p and the set F(u) is defined by (6). The assumption that v 1 /v / G for any distinct v 1, v V implies λ(v 1 )/λ(v )/ G and U(v 1 ) = U(v ). Applying Lemma 7 to the set U ={U(v) : v V } and using (16) we get as required. v V #D p (v) = u U #F(u) (pt ) /3

8 30 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski Now we consider two primes p 1 = p and the corresponding subgroups G ν Zp ν consisting of nonzero p ν th powers modulo pν, ν = 1,. Also, we denote by Ḡ ν the subsets of Z formed by the integers belonging to G ν modulo pν. That is, while G ν is represented by some elements from the set {1,..., pν 1}, the set Ḡ ν is infinite, ν = 1,. Lemma 13. Let x, K, and L be positive integers with x<p1 p. Suppose that a set A [1, x] Ḡ 1 Ḡ satisfies the following conditions: (i) there are at least L pairs (n 1, n ) A with n 1 >n and such that n 1 n (mod p ); (ii) there are at most K elements of A in any residue class modulo p 1. Then L K p/3 1 Z 1/3 N(p1, G 1, Z) 1/3 where Z = x/p. Proof. Denote M i = #{n A : n ip A}, i = 1,..., Z. By Lemma 11 and the condition (i) we have Z M i L. i=1 Next, let m i = #{n G 1 : n ip G 1}, i = 1,..., Z. Then by condition (ii) we have Z m i 1 Z M i L K K. (17) i=1 i=1 We observe also that for i = 1,..., Z m i #D p1 (ip ). (18) Moreover, we have Z<p1. In particular, if a positive integer i Z is divisible by p 1 then, by Lemma 11, m i = #D p1 (ip ) = 0. Assume that the residues of ip modulo p 1, i = 1,..., Z, are contained in J distinct cosets C 1,..., C J of the group G 1. For j = 1,..., J, we denote s j = #{i :1 i Z, ip C j} and also t j = #D p1 (v) for some element v C j (clearly, this quantity depends only on the coset C j and does not depend on the choice of v).

9 On the Divisibility of Fermat Quotients 31 Therefore, using (18) we can rewrite (17) as J s j t j L K. (19) To estimate the left-hand side of (19) from above we consider that the cosets C 1,..., C J are ordered so that the sequence {t 1,..., t J } is nonincreasing. By Lemma 1 we have for j = 1,..., J Hence, Clearly, t 1 + +t j (p 1 j) /3. t j p /3 1 j 1/3. (0) J s j = Z. (1) By the definition of N(p 1, G 1, Z),wehave J s j N(p 1, G 1, Z). () We notice that Z 1; otherwise there are no (n 1, n ) A with n 1 >n and such that n 1 n (mod p ). Define J 0 = Z /N(p 1, G 1, Z) and J 1 = min{j 0, J }. It is easy to see that J 0 1. Therefore, J 1 1. To estimate the left-hand side of (19) we consider separately the cases j J 1 and j>j 1 (the second case can occur only if J 0 = J 1 ). By (0), (), and the Cauchy Schwarz inequality, we have ( J1 ) J 1 s j t j Therefore, s j J 1 J 1 t j J s j J 0 tj N(p 1, G 1, Z)p 4/3 1 J 1/3 0. s j t j p /3 1 Z 1/3 N(p 1, G 1, Z) 1/3. (3) If J 0 = J 1 then we also have to estimate the sum over j > J 0. To do so we use (0) and (1): J j=j 0 +1 s j t j t J0 Z p /3 1 Z 1/3 N(p 1, G 1, Z) 1/3. (4) Combining (19), (3), and (4), we complete the proof.

10 3 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski Now we prove a combinatorial statement demonstrating that if a set [1, x] Ḡ 1 Ḡ is large then we can choose a set A [1, x] Ḡ 1 Ḡ satisfying the conditions of Lemma 13 with K and L such that L/K p. Let I 1 and I be nonempty finite sets. For a set A I 1 I we denote the following horizontal and vertical lines : A(x, ) ={y I : (x, y) A}; A(, y) ={x I 1 : (x, y) A}. Lemma 14. For any set A I 1 I there exist a subset B A and positive integers k 1 and k such that: (i) #B 1 #A; (ii) #B(x, ) k 1 for any x I 1 ; (iii) #B(, y) k for any y I ; (iv) #B(x, ) (v) x I 1 #B(x, )>k 1 / y I #B(,y)>k / #B(, y) 1 log(#i 1 + #I ) #A; 1 log(#i 1 + #I ) #A. Proof. The case A = is trivial, so we now consider that #A > 0. Let U be the smallest integer such that U #I 1 + #I,so1 U log(#i 1 + #I ). We construct the following sequence of sets {A ν }, ν = 0,1,... Set A 0 = A. Assume that A ν has been constructed. We now define u ν as the smallest integer u such that #A ν (x, ) x I 1 #A ν (x, )> u 1 #A. (5) 8U Similarly, let v ν be the smallest integer v such that 1 #A ν (, y) #A. (6) 8U y I #A ν (,y)> v Define A ν+1 = A ν {(x, y) : y A ν (x, )} x I 1 #A ν (x, )> uν Clearly, for any ν = 0,1,... we have {(x, y) : x A ν (, y)}. (7) y I #A ν (,y)> vν A ν+1 A ν, 0 u ν+1 u ν <U, 0 v ν+1 v ν <U. There exists a number N<Usuch that

11 On the Divisibility of Fermat Quotients 33 u N+1 = u N and v N+1 = v N. Set B = A N+1, k 1 = u N, k = v N. Now, from (5), (6), and (7), we derive N N #(A \ B) #A ν (x, ) + #A ν (, y) ν=0 x I 1 ν=0 y I #A ν (x, )> uν #A(,y)> vν (N + 1) #A 1 8U #A. So, condition (i) is satisfied. By the definition of B, k 1, and k we see that conditions (ii) and (iii) are satisfied too. Next, if k 1 = 1 then #B(x, ) = #B. x I 1 #B(x, )>k 1 / If k 1 > 1 then we deduce from the equality u N+1 = u N that #B(, y) > 1 8U #A. x I 1 #B(,y)>k 1 / In either case the condition (iv) holds. Analogously, we also have condition (v) satisfied. 7.. Conclusion of the Proof We suppose that Q is large enough while ε and δ are small enough and define x = Q 59/4 3δ and y = ((1 δ) log Q) 59/35+ε. Assume that there are two primes p 1 = p with Q 1 δ <p 1, p Q and such that a p1 1 1 (mod p1 ), ap 1 1 (mod p ) for all positive integers a y. As before, for ν = 1,, we use G ν to denote the subgroup of Zp ν consisting of nonzero p ν th powers modulo pν and use Ḡ ν for the subset of Z formed by the integers belonging to G ν modulo pν. Then S(x, y) Ḡ 1 Ḡ (here we take into account that y < min{p 1, p }). Since ( ) 1 (59/4 3δ) 1 > 1 + δ 59/35 + ε provided δ is small enough compared to ε, we derive from Lemma 5 that (x, y)>q 1+δ (8) (provided ε and δ are small enough).

12 34 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski We now associate with any integer n S(x, y) the pair of residues (n (mod p 1 ), n (mod p )) Z p1 Z p. Using Lemma 14 we conclude the existence of a set A S(x, y) [1, x] Ḡ 1 Ḡ and positive integers k 1, k and an absolute constant c 0 satisfying the following conditions: (a) #A (x, y)/; (b) there are at most k 1 elements of A in any residue class modulo p 1 ; (c) there are at most k elements of A in any residue class modulo p ; (d) there are at least c 0 (x, y)/(k 1 log Q) residue classes modulo p 1 containing at least k 1 / elements from A; (e) there are at least c 0 (x, y)/(k log Q) residue classes modulo p containing at least k / elements from A. Without loss of generality we can assume that k k 1. In particular, we see from property (a) and (8) that #A Q 1+δ. Therefore, by properties (a) and (e) we have (x, y) Q p c 0 k log Q Q1+δ k log Q. Hence, k Qδ log Q, provided that Q is large enough. If a residue class modulo p contains at least k / elements from A, then there are at least k /10 pairs (n 1, n ) A such that n 1 >n and n 1 n (mod p ). Therefore, the conditions of Lemma 13 are fulfilled with K = k 1 and k L = 10 c0 (x, y) (x, y)k k log Q log Q Q1+δ k log Q. Considering again that Q is large enough we obtain that L K k Q Q. k 1 Applying Lemma 13, we obtain p /3 1 Z 1/3 N(p1, G 1, Z) 1/3 Q (9) where x Z = Q 11/4 δ p 11/4 δ/ 1. p On the other hand, Lemma 9 applies with ν = and yields

13 On the Divisibility of Fermat Quotients 35 N(p1, G 1, Z) Zp 5/1 1 (p1 ) 1/6+o(1) + Z p 1/ 1 (p1 ) 1/+o(1) p 13/4 δ/+o(1) 1. Consequently, p /3 1 Z 1/3 N(p1, G 1, Z) 1/3 p 1 δ/3+o(1) 1 Q 1 δ/3+o(1), which disagrees with (9) for Q large enough. This contradiction completes the proof. 8. Proof of Theorem 4 Let P y be the set of all primes p for which a p 1 1 (mod p ) (30) for all primes a y. We need the following estimate, from which Theorem 4 follows quickly. Lemma 15. Suppose Q y. Then for all δ>0and any x, we have #{p P y : Q/ <p Q} (xq)δ (Q + xq 1 + min(xq 1/, x 1/ Q)). (x, y) Proof. For real u, let T (u) = exp(πiun) n S(x,y) and put Y = T(0) = (x, y). Let p P y. By the Parseval identity, we have for each prime p p a=1 (a,p)=1 ( ) a p T ( ) = a p T p a=1 p ( ) b T p p = p N(p, a) p a=1 p N(p, b), (31) where N(q, a) is the number of elements of n S(x, y) in the progression n a (mod q). For p P y we see that n p 1 1 (mod p ) for every n S(x, y). By Lemma 11, for each b {1,..., p 1} there is a unique residue a b modulo p with a b b (mod p) and a p 1 b 1 (mod p). Consequently, N(p, a b ) = N(p, b). Therefore p p p N(p, a) = N(p, a b ) = N(p, b), a=1 which, after substitution in (31), implies that ( ) a p T = p(p 1) N(p, b). p Since 1 a p (a,p)=1

14 36 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski p N(p, b) = Y and clearly N(p,0) = 0 for p>q/ y, by the Cauchy Schwarz inequality, we obtain ( ) a p 1 T = p(p 1) N(p, b) py. p Therefore 1 a p (a,p)=1 p P y 1 a p Q/<p Q (a,p)=1 ( ) a T QY #{p p P y : Q/ <p Q}. (3) By Lemma 10, ( ) a T (xq) δ (Q 3 + x + min{xq 1/, x 1/ Q })Y. (33) q q Q 1 a q (a,q)=1 Comparing (3) and (33), we obtain the desired estimate. To finish the proof of Theorem 4, we take x = Q 5/ and y = (log Q) 5/3+ε in Lemma 15. Inserting the bound from Lemma 5, we have (x, y)>x 1 1/(5/3+ε) Q 1+5δ for a suitable δ>0. Therefore, for the previous choice of y we obtain #{p P y : Q/ <p Q} Q 1 δ, which implies the desired estimate. 9. Comments Lemmas 6, 8, and 9 can easily be obtained in fully explicit forms with concrete constants. Thus, the bound of Theorem 1 can also be obtained in a fully explicit form, which can be important for algorithmic applications. For example, it would be interesting to get an explicit formula for n 0 (ε) such that for n n 0 (ε) the conclusion of Corollary holds. It is interesting to establish the limits of our approach. For example, the bound N p (k) kp 1+o(1) for values of k = p 1+o(1) (or larger), which is the best possible result about N p (k), leads only to the estimate l p (log p) 1+o(1), which is still much higher than the expected size of l p. Furthermore, if instead of Lemma 10 we have the best possible bound

15 On the Divisibility of Fermat Quotients 37 q 1 q Q a=1 gcd(a,q)=1 ( ) a S Q δ (Q 3 + N)Y, q the exponent 5/3 of Theorem 4 can be replaced with 3/. Certainly, improving and obtaining unconditional variants of the estimate (3) and, more generally, investigating other properties of set W(p) is of great interest owing to important applications outlined in [1]. It is quite possible that Lemma 6 can be used for this purpose as well. Congruences with Fermat quotients q p (a) modulo higher powers of p have also been considered in the literature; see [6; 13]. Using our approach with bounds of generalized Heilbronn sums p H p,m (λ) = e p m(λb pm 1 ) due to Bourgain and Chang [] or Malykhin [17] (which is fully explicit), one can estimate the smallest a with q p (a) 1 (mod p m ) for fixed m. References [1] S. Baier and L. Zhao, An improvement for the large sieve for square moduli, J. Number Theory 18 (008), [] J. Bourgain and M.-C. Chang, Exponential sum estimates over subgroups and almost subgroups of ZQ, where Q is composite with few prime factors, Geom. Funct. Anal. 16 (006), [3] J. Bourgain, S. V. Konyagin, and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm, Internat. Math. Res. Notices (008), 1 9. [4], Corrigenda to: Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm, Internat. Math. Res. Notices (009), [5] R. Crandall, K. Dilcher, and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), [6] R. Ernvall and T. Metsänkylä, On the p-divisibility of Fermat quotients, Math. Comp. 66 (1997), [7] W. L. Fouché, On the Kummer Mirimanoff congruences, Quart. J. Math. Oxford Ser. () 37 (1986), [8] A. Granville, Some conjectures related to Fermat s last theorem, Number theory (Banff, 1988), pp , de Gruyter, New York, [9], On pairs of coprime integers with no large prime factors, Exposition. Math. 9 (1991), [10] D. R. Heath-Brown and S. V. Konyagin, New bounds for Gauss sums derived from kth powers, and for Heilbronn s exponential sum, Quart. J. Math. Oxford Ser. () 51 (000), 1 35.

16 38 J. Bourgain, K. Ford, S. V. Konyagin, & I. E. Shparlinski [11] A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), [1] Y. Ihara, On the Euler Kronecker constants of global fields and primes with small norms, Algebraic geometry and number theory, Progr. Math., 850, pp , Birkhäuser, Boston, 006. [13] W. Keller and J. Richstein, Solutions of the congruence a p 1 1 (mod p r ), Math. Comp. 74 (005), [14] J. Knauerand and J. Richstein, The continuing search for Wieferich primes, Math. Comp. 74 (005), [15] S.V. Konyagin and C. Pomerance, On primes recognizable in deterministic polynomial time, The mathematics of Paul Erdős, vol. I, pp , Springer-Verlag, Berlin. [16] H. W. Lenstra, Miller s primality test, Inform. Process. Lett. 8 (1979), [17] Y.V. Malykhin, Estimates of trigonometric sums modulo p r, Mat. Zametki 80 (006), (Russian); English translation in Math. Notes 80 (006), J. Bourgain Institute for Advanced Study Princeton, NJ bourgain@ias.edu S. V. Konyagin Department of Mechanics and Mathematics Moscow State University Moscow Russia konyagin@ok.ru K. Ford Department of Mathematics University of Illinois Urbana, IL ford@math.uiuc.edu I. E. Shparlinski Department of Computing Macquarie University Sydney, NSW 109 Australia igor.shparlinski@mq.edu.au