A natural number n is called a k-almost prime if n has precisely k prime factors, counted with multiplicity. For any fixed k ! 2, let Fk(X) be the number of k-th powers mk " X such that Φ(n) = mk for some squarefree k-almost prime n, where Φ(·) is the Euler function. We show that the lower bound Fk(X) ! X1/k/(log X)2k holds, where the implied constant is absolute and the lower bound is uniform over a certain range of k relative to X. In particular, our results imply that there are infinitely many pairs (p, q) of distinct primes such that (p ? 1)(q ? 1) is a perfect square.
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机译：如果n具有恰当的k个主要因素，则自然数n被称为k - 几乎素质，这是k的主要因素，计算多样性。任何固定的k！如图2所示，让fk（x）是k-th力量mk“x，使得一些平方的k-几乎prime n的φ（n）= mk，其中φ（·）是欧拉函数。我们表明较低的绑定的fk（x）！x1 / k /（log x）2k保持，其中隐含的常数是绝对的，下限在相对于x的一定范围内均匀。特别地，我们的结果意味着无限的不同的原始的对（p，q），使得（p？1）（q？1）是一个完美的正方形。
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