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Extra info for Equidistribution in Number Theory, An Introduction
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3 is motivated by the above construction. We use a lower bound sieve and the Bombieri–Vinogradov theorem to construct many primes p such that (p − 1)/2 has no small prime factors and then consider pairs p, of such primes. Using elementary arguments and some applications of the upper bound sieve we show that most of the products p we have constructed have suﬃciently large Carmichael function. Although we do not match the constant 18 that follows from the conjecture, we actually obtain a large number of integers of the required type using this unconditional argument.