Average Frobenius distribution for the degree two primes of a number field

James, Kevin; Smith, Ethan

doi:10.1017/s0305004112000631

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Gcd's and Hilbert Class Polynomials1

Average Order Of a Random Subset Of Wieferich Primes1

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2017

2022

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Cited by 2 publications

(2 citation statements)

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“…In the same article, they also prove an averaged version of this conjecture. A similar averaged result is known if both E 1 and E 2 are defined over the rationals, E 1 has complex multiplication, and E 2 does not, see [25,Theorem 10] and [14, pp. 199-200].…”

Section: Gcd's and Hilbert Class Polynomialssupporting

confidence: 52%

Around the support problem for Hilbert class polynomials

Campagna¹,

Dill²

2022

Preprint

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Let HD(T ) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D. We study the rate of growth of the greatest common divisor of HD(a) and HD(b) as |D| → ∞ for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing HD(a) also divides HD(b), what can we say about a and b? If we replace HD(T ) by T n − 1 and the Dedekind domain is a ring of S-integers in some number field, then these are classical questions that have been investigated by Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and Corrales-Rodrigáñez-Schoof.

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Section: Gcd's and Hilbert Class Polynomialssupporting

confidence: 52%

Around the support problem for Hilbert class polynomials

Campagna¹,

Dill²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Quite often, the calculation of average density of an intractable primes distribution problems is the first line approach to solving the individual primes distribution problem, see [53], [4], [30], et cetera.…”

Section: Average Order Of a Random Subset Of Wieferich Primesmentioning

confidence: 99%

Results for Wieferich Primes

Carella

2017

Preprint

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Let v ≥ 2 be a fixed integer, and let x ≥ 1 and z ≥ x be large numbers. The exact asymptotic formula for the number of Wieferich primes p, defined by v p−1 ≡ 1 mod p 2 , in the short interval [x, x + z] is proposed in this note. The search conducted on the last 100 years have produced two primes p < x = 10 15 such that 2 p−1 ≡ 1 mod p 2 . The probabilistic and theoretical information within predicts the existence of another base v = 2 prime on the interval [10 15 , 10 40 ]. Furthermore, a result for the upper bound on the number of Wieferich primes is used to demonstrate that the subset of nonWieferich primes has density 1.

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Average Frobenius distribution for the degree two primes of a number field

Cited by 2 publications

References 17 publications

Around the support problem for Hilbert class polynomials

Around the support problem for Hilbert class polynomials

Results for Wieferich Primes

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