.png){kind=icon}
Return
The sharp threshold for making squares
.png){kind=icon}
OA
AI
DOI：10.4007/annals.2018.188.1.2.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
Consider a random sequence of N integers, each chosen uniformly and independently from the set {1,..., x}. Motivated by applications to factorization algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should N be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event and conjectured that it in fact exhibits a sharp threshold in N. More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of 4 / pi + o(1) as x -> infinity and made a conjecture regarding the location of the sharp threshold. In this paper we prove both of these conjectures by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali.
Keywords:
integer factorization
perfect square
random graph process
Journal
IF:
5.3
Papers:
1.4K
Citations:
1.6W
Organization
Cited Papers
Elimination of daptomycin in a patient with acute renal failure undergoing extended daily dialysis
OAAI
OAAIShareSaveShareSaveAntimicrobial resistance and molecular typing of Staphylococcus aureus isolates from raw milk in Hunan Province.png)
PeerJOAAI
PeerJIF0
OAAIShareSaveShareSaveShareSave
View more