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K−2s M ≤ x s! ,α s ≥2 αi =k 1 α1 ! · · · α s ! k! 2 s s equals the number of ways of writing k = using that k−s s αi ≥ 2. The proposition follows. αi with each One way to apply Proposition 4 is to take P to be the set of all primes below z with |g(p)| small. If there are not too many values of p with |g(p)| large, then we would expect that g(a) is roughly the same as gP (a) for most a. In such situations, Proposition 4 which furnishes the distribution of gP (a) would also furnish the distribution of g(a).

Pνν )} f f for m = p11 · · · pνν . The Carmichael function has the following relation to the power generator, which, as we recall, was defined as un ≡ uen−1 (mod m), u0 = α. If e, λ(m) = 1 then this is purely periodic and t = ord s e where s = ordm α. Thus, the largest possible value for t as α, e vary is λ λ(m) . We would like to know that this is large. , 1991). They showed that λ(n) is usually large, in particular that λ(n) = n exp − log2 n log3 n − c log2 n + O(log2 n) for all but o(N) integers n ≤ N.

T are uniformly distributed in the unit cube. 2 by means of H¨older’s inequality. Indeed, t |S abc (p, t)| = e p (aθ x + bθy + cθ xy ) x=1 y=1 t ≤ t t e p (aθ x + cθ xy ) y=1 x=1 ≤ t3/4 Vac (p, t) 1/4 ≪ t5/3 p1/4 , so that the Weyl criterion implies the result. 2 has been improved by Garaev who replaced 34 + ε by 21 + ε and by Bourgain who reduced it to ε. EXPONENTIAL SUMS, AND CRYPTOGRAPHY 35 One learns a little more by showing uniformity of distribution results for various subsets of the triples.

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