Factoring with Binary Quadratic Forms

Buell, Duncan A.

doi:10.1007/978-1-4612-4542-1_10

Cited by 17 publications

(38 citation statements)

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“…Hua [9, §11.5] gives an algorithm, using continued fractions, for producing the fundamental and base solutions. See [5] for results on the existence and number of base solutions. Now fix (x 0 , y 0 ) ∈ T , and define…”

Section: Theory Of the Pell Equationmentioning

confidence: 99%

Solving constrained Pell equations

Kedlaya¹

1998

Math. Comp.

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Abstract. Consider the system of Diophantine equations x 2 − ay 2 = b, P (x, y) = z 2 , where P is a given integer polynomial. Historically, such systems have been analyzed by using Baker's method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases P (x, y) = cy 2 + d and P (x, y) = cx + d, which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

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Section: Theory Of the Pell Equationmentioning

confidence: 99%

Solving constrained Pell equations

Kedlaya¹

1998

Math. Comp.

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“…We assume that the reader is familiar with the theory of binary quadratic forms ( [2] and [3]). We write f • g for the composition of two forms f and g and f g for the product, which is the form obtained by composition of f and g followed by reduction.…”

Section: Computations In Real Quadratic Fieldsmentioning

confidence: 99%

Computations of class numbers of real quadratic fields

Srinivasan

1998

Math. Comp.

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Abstract. In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field Q( √ d) is presented, which computes the class number in expected time O(d 1/5+ ). The algorithm is a random version of Shanks' algorithm.One of the main steps in algorithms to compute the class number is the approximation of L(1, χ). Previous algorithms with the above running time O(d 1/5+ ), obtain an approximation for L(1, χ) by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for L(1, χ) without assuming the Riemann Hypothesis, by using a new technique that we call the 'Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time O(d 1/5+ ). However, our estimate of O(d 1/5+ ) on the running time of our algorithm to compute the class number is not effective.

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“…[1] has a chart (page 52) that lists which generic characters go with each discriminant. We will use this frequently without further reference.…”

Section: Introductionmentioning

confidence: 99%

Sun's conjectures on fourth powers in the class group of binary quadratic forms

Fitzgerald

2010

Journal of Number Theory

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Sun [5] proved that if p and q are distinct odd primes then (−1) (q−1)/2 q is a quartic residue modulo p iff p is represented by an element of G(−16q 2 ) 4 , where G( ) is the class group of primitive binary quadratic forms of discriminant . In [6] Sun posed a series of conjectures, labeled (8.2) through (8.6), on the order of G( ) 4 , denoted by h 4 ( ). Liu [4] has found counterexamples to conjecture (8.4). Here we prove Sun's conjectures (8.2), (8.3), (8.5) and (8.6) are correct and prove a modified version of (8.4) is also correct. The proofs are elementary.

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Factoring with Binary Quadratic Forms

Cited by 17 publications

References 0 publications

Solving constrained Pell equations

Solving constrained Pell equations

Computations of class numbers of real quadratic fields

Sun's conjectures on fourth powers in the class group of binary quadratic forms

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