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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 16 (2 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
Dedekind Zeta Functions and the Complexity of Hilbert’s Nullstellensatz
, 2008
"... Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the ..."
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Cited by 5 (4 self)
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Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication HN ̸∈P = ⇒ P ̸=NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.
A counting function for the sequence of perfect powers
 Austral. Math. Soc. Gaz
"... A natural number of the form mn where m is a positive integer and n ≥ 2 is called a perfect power. Unsolved problems concerning the set of perfect powers abound throughout much of number theory. The most famous of these is known as the Catalan conjecture, which states that the only perfect powers wh ..."
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Cited by 3 (0 self)
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A natural number of the form mn where m is a positive integer and n ≥ 2 is called a perfect power. Unsolved problems concerning the set of perfect powers abound throughout much of number theory. The most famous of these is known as the Catalan conjecture, which states that the only perfect powers which differ by unity are the integers 8 and 9. It is of
Interpolating Between Quantum and Classical Complexity Classes
, 2008
"... We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized ..."
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We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m = 2 (and no classical randomized polynomial time algorithm is known), (⋆) is nearly NPhard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (⋆) would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to new. 1 Introduction and Main Results Thanks to quantum computation, we now have exponential speedups for important practical problems such as Integer Factoring and Discrete Logarithm [Sho97]. However, a fundamental
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"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholary research and ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholary research and I understand that my thesis may be made electonically available to the public. iii This thesis is dedicated to the loving memory of George and Alice Wolczuk. v Acknowledgements First, I would like to thank Dr. Cameron Stewart for all his assistance and understanding with my thesis and degree. I would also like to thank Shonn Martin, Lis D’Alessio and Kim Gingerich for all the support, help and kindness they gave me. I would further like to thank my readers YuRu Liu and Michael Rubinstein for their corrections and Agnieszka Zygmunt for all her encouragement. Finally, I will give a very special thank you to Carol Clemson whose moral support, encouragement and friendship made this thesis possible. vii
A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes
, 2008
"... We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized ..."
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We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m = 2 (and no classical randomized polynomial time algorithm is known), (⋆) is nearly NPhard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (⋆) would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to new. 1 Introduction and Main Results Thanks to quantum computation, we now have exponential speedups for important practical problems such as Integer Factoring and Discrete Logarithm [Sho97]. However, a fundamental
A Heuristic for the Prime Number Theorem This article appeared in The Mathematical Intelligencer 28:3 (2006) 6–9, and is copyright by Springer
"... Why does ‰ play such a central role in the distribution of prime numbers? Simply citing the Prime Number Theorem (PNT), which asserts that pHxL ~ x ê ln x, is not very illuminating. Here "~ " means "is asymptotic to " and pHxL is the number of primes less than or equal to x. So w ..."
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Why does ‰ play such a central role in the distribution of prime numbers? Simply citing the Prime Number Theorem (PNT), which asserts that pHxL ~ x ê ln x, is not very illuminating. Here "~ " means "is asymptotic to " and pHxL is the number of primes less than or equal to x. So why do natural logs appear, as opposed to another flavor of logarithm? The problem with an attempt at a heuristic explanation is that the sieve of Eratosthenes does not behave as one might guess from pure probabilistic considerations. One might think that sieving out the composites under x using primes up to è!!! x would lead to x P è!!! J1p< x 1 ÅÅÅÅ N as an asymptotic estimate of the count of p numbers remaining (the primes up to x; p always represents a prime). But this quantity turns out to be not asymptotic to x ê ln x. For F. Mertens proved in 1874 that the product is actually asymptotic to 2 ‰g ê ln x, or about 1.12 ê ln x. Thus the sieve is 11 % (from 1 ê 1.12) more efficient at eliminating composites than one might expect. Commenting on this phenomenon, which one might call the Mertens Paradox, Hardy and