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PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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Cited by 26 (2 self)
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
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Cited by 14 (1 self)
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The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
On the period of the linear congruential and power generators
 Acta Arith
"... We consider two standard pseudorandom number generators from number theory: the linear congruential generator and the power generator. For the former, we are given integers e, b, n (with e, n> 1) and a seed u0, and we compute the sequence ..."
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Cited by 7 (2 self)
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We consider two standard pseudorandom number generators from number theory: the linear congruential generator and the power generator. For the former, we are given integers e, b, n (with e, n> 1) and a seed u0, and we compute the sequence
On pseudosquares and pseudopowers
, 712
"... Introduced by Kraitchik and Lehmer, an xpseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least xpseudosquare that improves on a bound that is exponential in x due to Schinzel. We ..."
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Cited by 2 (2 self)
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Introduced by Kraitchik and Lehmer, an xpseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least xpseudosquare that improves on a bound that is exponential in x due to Schinzel. We also obtain an equidistribution result for pseudosquares. An xpseudopower to base g is a positive integer which is not a power of g yet is so modulo p for all primes p ≤ x. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most exp(agx/log x) for a suitable constant ag. A bound of exp(agxlog log x/log x) is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRHconditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares.
An Epic Drama: The Development of the Prime Number Theorem
, 2010
"... The prime number theorem, describing the aymptotic density of the prime numbers, has often been touted as the most surprising result in mathematics. The statement and development of the theorem by Legendre, Gauss and others and its eventual proof by Hadamard and de al ValléePoussin span the whole ..."
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The prime number theorem, describing the aymptotic density of the prime numbers, has often been touted as the most surprising result in mathematics. The statement and development of the theorem by Legendre, Gauss and others and its eventual proof by Hadamard and de al ValléePoussin span the whole nineteenth century and encompass the growth of a brand new field in analytic number theory. As an outgrowth of the techniques of the proof is the Riemann hypothesis which today is perhaps the outstanding open problem in mathematics. These ideas and occurences certainly constitute an epic drama within the history of mathematics and one that is not as well known among the general mathematical community as it should be. In the present paper we trace out the paper, the development of the proof and a raft of other ideas, results and concepts that come from the prime number theorem.
my lovely sisters:
, 2004
"... I am indebted to Professor Richard Lipton for supervising this thesis. Under his guidance, I was able to complete all the work in this thesis in about two years. I feel fortunate to have him as my advisor. His original mind, his keen insight in approaching hard problems, and his vast knowledge (whic ..."
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I am indebted to Professor Richard Lipton for supervising this thesis. Under his guidance, I was able to complete all the work in this thesis in about two years. I feel fortunate to have him as my advisor. His original mind, his keen insight in approaching hard problems, and his vast knowledge (which extends well beyond computer science) have served as an inspiration for me throughout this thesis. I am particularly thankful to him for sharing with me his prowess in numbertheoretic arguments. The effortless manner in which he can demonstrate some of the hard numbertheoretic results, has inspired me to learn a great deal about this beautiful subject. Indeed, this knowledge has already started to pay its dividends, as is apparent in a substantial part of this thesis. I am also grateful to him for reinforcing his faith in my abilities, for constantly encouraging me to try difficult problems, and for being a great collaborator. I cannot thank him enough for all he has done for me in the three years that I have known him. My fascination with randomness goes back to my undergraduate studies. Under the supervision of Professor Ketan Mulmuley and Professor Sundar Vishwanathan at IIT Mumbai, I studied randomized algorithms for linear programming. During this period, they intro
Totally Goldbach numbers and related conjectures
"... Goldbach’s famous conjecture is that every even integer n greater than 2 is the sum of two primes; to date it has been verified for n up to 1017; see [10, 13]. In order to establish the conjecture for a given even integer n, one optimistic approach is to simply choose a prime p < n, and check to ..."
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Goldbach’s famous conjecture is that every even integer n greater than 2 is the sum of two primes; to date it has been verified for n up to 1017; see [10, 13]. In order to establish the conjecture for a given even integer n, one optimistic approach is to simply choose a prime p < n, and check to see whether n−p is prime. Of course, one has to make a sensible choice of p; if n − 1 is prime, one should not choose p = n − 1, and there is obviously no point choosing a prime p which is a factor of n. In this paper we examine the set of numbers n for which every “sensible choice ” of p works: Definition 1 A positive integer n is totally Goldbach if for all primes p < n − 1 with p not dividing n, we have that n − p is prime. We denote by A the set of all totally Goldbach numbers. It turns out that there are very few totally Goldbach numbers. We find: