Results 1 
4 of
4
A Lower Bound for Primality
, 1999
"... Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by s ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC 0 [p] for any prime p. Similar lower bounds are presented for the set of squarefree numbers, and for the problem of computing the greatest common divisor of two numbers. 1 Introduction What is the computational complexity of the set of prime numbers? There is a large body of work presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77]), but  Supported in part by NSF grant CCR9734918. y Supported in part by NSF grant CCR9700239. z Supported in part by ARC grant A69700294. as was pointed out recently in [BDS98a, BDS9...
An invitation to additive prime number theory
, 2004
"... The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sam ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.
A Lower Bound for Primality
, 1999
"... Recent work by Bernasconi, Damm and Shparlinski showed that the set of squarefree numbers is not in AC , and raised as an open question whether similar (or stronger) lower bounds could be proved for the set of prime numbers. In this note, we show that the Boolean majority function is AC  ..."
Abstract
 Add to MetaCart
Recent work by Bernasconi, Damm and Shparlinski showed that the set of squarefree numbers is not in AC , and raised as an open question whether similar (or stronger) lower bounds could be proved for the set of prime numbers. In this note, we show that the Boolean majority function is AC Turing reducible to the set of prime numbers (represented in binary). From known lower bounds on Maj (due to Razborov and Smolensky) we conclude that primality cannot be tested in AC [p] for any prime p. Similar results are obtained for the set of squarefree numbers, and for the problem of computing the greatest common divisor of two numbers.