Average twin prime conjecture for elliptic curves
نویسندگان
چکیده
منابع مشابه
Average Twin Prime Conjecture for Elliptic Curves
Let E be an elliptic curve over Q. In 1988, N. Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy–Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s Conjecture is still widely open. In this paper we prove that Koblitz’s Conjecture is true on average...
متن کاملOn Silverman's conjecture for a family of elliptic curves
Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
متن کاملAn Introduction to The Twin Prime Conjecture
Twin primes are primes of the form (p, p + 2). There are many proofs for the infinitude of prime numbers, but it is very difficult to prove whether there are an infinite number of pairs of twin primes. Most mathemati cians agree that the evidence points toward this conclusion, but numerous attempts at a proof have been falsified by subsequent review. The prob lem itself, one of the most famou...
متن کاملThe Uniform Primality Conjecture for Elliptic Curves
An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang’s conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techni...
متن کاملPairing-Friendly Elliptic Curves of Prime Order
Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree k 6 6. More general methods produce curves over Fp where the bit length of p is often twice as large as that of the order r of the subgroup with embedding degree k; the best published results achieve ρ ≡ log(p)/ log(r) ∼ 5/4. In this paper we make the first step towar...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: American Journal of Mathematics
سال: 2011
ISSN: 1080-6377
DOI: 10.1353/ajm.2011.0033