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- Ernest Croot, Iii
- 2003

We prove an old conjecture of Erdős and Graham on sums of unit fractions: There exists a constant b > 0 such that if we r-color the integers in [2, br ], then there exists a monochromatic set S such that ∑ n∈S 1/n = 1.

- Ernest Croot
- 1999

In this paper we pove that for any given rational r > 0 and all N > 1, there exist integers N < x 1 < x 2 < · · · < x k < e r+o(1) N such that r = 1 x 1 + 1 x 2 + · · · + 1 x k .

- Terence Tao, Ernest Croot, Harald Helfgott
- Math. Comput.
- 2012

Given a large positive integer N , how quickly can one construct a prime number larger than N (or between N and 2N)? Using probabilistic methods, one can obtain a prime number in time at most log N with high probability by selecting numbers between N and 2N at random and testing each one in turn for primality until a prime is discovered. However, if one… (More)

The large sieve has its origins in the work of Linnik and Rényi. It was developed to deal with sequences that avoid a positive proportion of residue classes. It was later simplified by Roth, Bombieri, Davenport, Halberstam, Montgomery, Gallagher and many others. For a survey see Montgomery [12] and Bombieri [1]. It is known that Montgomery’s large sieve… (More)

- Ernest Croot
- 2002

Let L(c, x) = e √ log x log log . We prove that if a1 (mod q1), ..., ak (mod qk) are a maximal collection of non-intersecting arithmetic progressions, with 2 ≤ q1 < q2 < · · · < qk ≤ x, then x L( √ 2 + o(1), x) < k < x L(1/6− o(1), x) . In the case for when the qi’s are square-free, we obtain the improved upper bound k < x L(1/2− o(1), x) .

- Ernest Croot
- 2008

In this paper we will use the following notation. Given a set of positive integers S, we let S(x) denote the number of elements in S that are ≤ x, and we let |S| denote the total number of elements of S. Given two sets of positive integers A and B, we denote the sumset {a + b : a ∈ A, b ∈ B} by A + B; and so, the number of elements in A + B that are ≤ x… (More)

- Ernest Croot, Kai Huang
- 2008

The inventory cycle offsetting problem (ICP) is a strongly NPcomplete problem. We study this problem from the view of probability theory, and rigorously analyze the performance of a specific random algorithm for this problem; furthermore, we present a “local search” algorithm, and a modified local search, which give much better results (the modified local… (More)

- Ernest Croot
- 2007

We further examine Kummer's incorrect conjectured asymptotic estimate for the size of the rst factor of the class number of a cyclotomic eld, h 1 (p). Whereas Kummer had conjectured that h 1 (p) G(p) := 2p(p=4 2) p?1 4 we show, under certain plausible assumptions, that there exist constants a ; b such that h 1 (p) G(p) for a x= log b x primes p x whenever… (More)

- Ernest Croot
- 1990

For multiplicative functions f(n), which take on the values ±1, we show that under certain conditions on f(n), for all x sufficiently large, there are at least x exp(−7(log log x) √ log x) values of n ≤ x for which f(n(n + 1)) = −1.

- Ernest Croot, Kai Huang
- IJMOR
- 2013

In a multi-item inventory system, given the order cycle lengths and unit volumes of the items, the determination of the replenishment times (i.e. “cycle offsets”) of items, so as to minimize the resources needed to store the items, is known as the inventory cycle offsetting problem. In this paper we show that so long as the cycle times and inventory unit… (More)