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15
An Investigation of Bounds for the Regulator of Quadratic Fields
 Experimental Mathematics
, 1995
"... This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large ..."
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Cited by 12 (6 self)
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This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large
An Upper Bound on the Least Inert Prime in a Real Quadratic Field
"... It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D ? 3705, there is always at least one prime p ! p D=2 such that the Kronecker symbol (D=p) = \Gamma1. 1991 Mathematics Subject Classification 11R11, 11Y40 The first author is a Pre ..."
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Cited by 7 (2 self)
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It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D ? 3705, there is always at least one prime p ! p D=2 such that the Kronecker symbol (D=p) = \Gamma1. 1991 Mathematics Subject Classification 11R11, 11Y40 The first author is a Presidential Faculty Fellow. His research is partiallly supported by the NSF. The research of the second two authors is partially supported by NSERC of Canada 1 1 Introduction Let D be the fundamental discriminant of a real quadratic field and let S = f5; 8; 12; 13; 17; 24; 28; 33; 40; 57; 60; 73; 76; 88; 97; 105; 124; 129; 136; 145; 156; 184; 204; 249; 280; 316; 345; 364; 385; 424; 456; 520; 609; 616; 924; 940; 984; 1065; 1596; 2044; 2244; 3705g: At the end of Chapter 6 of [5], the second author made the following conjecture. Conjecture. The values of D for which the least prime p such that the Kronecker symbol (D=p) = \Gamma1 satisfies p ? p D=2 are precisely those in S. He also veri...
Results and estimates on pseudopowers
 Math. Comp
, 1996
"... Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a power ..."
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Cited by 3 (0 self)
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Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a powerof2. Next, we define an xpseudopower of the base 2tobeapositiveintegern that is not a power of 2, but looks like a power of 2 modulo all primes p ≤ x. Let P2(x) denote the least such n. We give an unconditional upper bound on P2(x), a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that P2(x)isaboutexp(c2x/log x) for a certain constant c2. We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than 2 are also given. 1.
Doublyfocused enumeration of pseudosquares and pseudocubes
 In Proceedings of the 7th International Algorithmic Number Theory Symposium (ANTS VII
, 2006
"... Abstract. This paper offers numerical evidence for a conjecture that primality proving may be done in (log N) 3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences— doublyfocused enumer ..."
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Abstract. This paper offers numerical evidence for a conjecture that primality proving may be done in (log N) 3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences— doublyfocused enumeration — is examined. This technique, first described by D. J. Bernstein, allowed us to obtain recordsetting sieve computations in software on general purpose computers. 1
Doubly Focused Enumeration Of Locally Square Polynomial Values
"... Let f be an irreducible polynomial. Which of the values f(c + 1); f(c + 2); : : : ; f(c + H) are locally square at all small primes? This paper presents an algorithm that answers this question in time H=M 2+o(1) for an average small c as H ! 1, where M = H 1=log 2 log H . In contrast, the usu ..."
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Let f be an irreducible polynomial. Which of the values f(c + 1); f(c + 2); : : : ; f(c + H) are locally square at all small primes? This paper presents an algorithm that answers this question in time H=M 2+o(1) for an average small c as H ! 1, where M = H 1=log 2 log H . In contrast, the usual method takes time H=M 1+o(1) . This paper also presents the results of two computations: an enumeration of locally square integers up to 24 2 64 , and an enumeration of locally square values of x 3 + y 7 for small x and y.
The Pseudosquares Prime Sieve
"... Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses t ..."
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Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p ≤ 2(log n) 2, but it is conjectured that p ∼ 1 log nlog log n. Thus, log2 the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n) 2) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, spaceefficient implementation of the wheel datastructure. 1
A Problem Concerning a Character Sum (Extended Abstract)
"... ? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p ..."
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? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p the Legendre symbol and S(k) = P p 1 n=1 n k n p . The problem of nding a prime p such that S(3) > 0 was one of the motivating forces behind the development of several of Shanks' ideas for computing in algebraic number elds, although neither he nor D. H. and Emma Lehmer were ever successful in nding such a p. In this extended abstract we summarize some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1 Introduction Let d denote a fundamental discriminant of an imaginary quadratic eld IK = Q( p d ) and let h(d) denote the class number of IK. Let p be a prime ( 3(mod 4)), n p the Legendre symbol and S...
Faster Algorithms To Find NonSquares Modulo WorstCase Integers
"... This paper presents two algorithms that, given an nbit positive integer m 2 1 + 8Z that is not a square, nd an element of Z=m that is a nonsquare or a nonzero nonunit. Under a standard conjecture, the rst algorithm takes time O(n(lg n) 3 lg lg n). Under a new but plausible conjecture, the sec ..."
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This paper presents two algorithms that, given an nbit positive integer m 2 1 + 8Z that is not a square, nd an element of Z=m that is a nonsquare or a nonzero nonunit. Under a standard conjecture, the rst algorithm takes time O(n(lg n) 3 lg lg n). Under a new but plausible conjecture, the second algorithm takes expected time O(n).
Class Number and Class Group Problems for Some NonNormal Totally Real Cubic Number Fields
, 2000
"... Let fKmgm4 be the family of nonnormal totally real cubic number elds associated with the Qirreducible cubic polynomials Pm (x) = (m + 1)x 1, m 4. We determine all these Km 's with class numbers hm 3: there are 14 such Km 's. Assuming the Generalized Riemann hypothesis for all the real qu ..."
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Let fKmgm4 be the family of nonnormal totally real cubic number elds associated with the Qirreducible cubic polynomials Pm (x) = (m + 1)x 1, m 4. We determine all these Km 's with class numbers hm 3: there are 14 such Km 's. Assuming the Generalized Riemann hypothesis for all the real quadratic number elds, we also prove that the exponent e m of the ideal class groups of these Km goes to in nity with m and we determine all these Km 's with ideal class groups of exponents e m 3: there are 5 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.