On long κ-tuples with few prime factors

On intervals with few prime numbers

We prove that there are long intervals containing fewer prime numbers than the average for intervals of such length.

Intervals with few Prime Numbers

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The Action of a Few Permutations on r - tuples isQuickly

We prove that for every r and d 2 there is a C such that for most choices of d permutations 1; 2; : : : ; d of Sn, the following holds: for any two r-tuples of distinct elements in f1; : : : ; ng, there is a product of less than C log n of the i's which map the rst r-tuple to the second. Although we came across this problem while studying a rather unrelated cryptographic problem, it belongs to ...

Integral Points on Quadrics in Three Variables Whose Coordinates Have Few Prime Factors

Orders of reductions of elliptic curves with many and few prime factors

In this paper, we investigate extreme values of ω(#E(Fp)), where E/Q is an elliptic curve with complex multiplication and ω is the number-of-distinct-prime-divisors function. For fixed γ > 1, we prove that #{p ≤ x : ω(#E(Fp)) > γ log log x} = x (log x)2+γ log γ−γ+o(1) . The same result holds for the quantity #{p ≤ x : ω(#E(Fp)) < γ log log x} when 0 < γ < 1. The argument is worked out in detail...