The Experts below are selected from a list of 6912 Experts worldwide ranked by ideXlab platform
Florian Luca - One of the best experts on this subject based on the ideXlab platform.
-
Primitive Root bias for twin primes ii schinzel type theorems for totient quotients and the sum of divisors function
Journal of Number Theory, 2020Co-Authors: Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe UdellAbstract:Abstract Garcia, Kahoro, and Luca showed that the Bateman–Horn conjecture implies φ ( p − 1 ) ⩾ φ ( p + 1 ) for a majority of twin-primes pairs p , p + 2 and that the reverse inequality holds for a small positive proportion of the twin primes. That is, p tends to have more Primitive Roots than does p + 2 . We prove that Dickson's conjecture, which is much weaker than Bateman–Horn, implies that the quotients φ ( p + 1 ) φ ( p − 1 ) , as p , p + 2 range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of φ ( p + 1 ) φ ( p ) and σ ( p + 1 ) σ ( p ) , in which σ denotes the sum-of-divisors function.
-
Primitive Root bias for twin primes ii schinzel type theorems for totient quotients and the sum of divisors function
arXiv: Number Theory, 2019Co-Authors: Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe UdellAbstract:Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $\phi(p-1) \geq \phi(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more Primitive Roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{\phi(p+1)}{\phi(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{\phi(p+1)}{\phi(p)}$ and $\frac{\sigma(p+1)}{\sigma(p)}$, in which $\sigma$ denotes the sum-of-divisors function.
-
Primitive Root bias for twin primes
Experimental Mathematics, 2019Co-Authors: Stephan Ramon Garcia, Elvis Kahoro, Florian LucaAbstract:ABSTRACTNumerical evidence suggests that for only about 2% of pairs p, p + 2 of twin primes, p + 2 has more Primitive Roots than does p. If this occurs, we say that p is exceptional (there are only...
-
Primitive Root biases for prime pairs i existence and non totality of biases
Journal of Number Theory, 2018Co-Authors: Stephan Ramon Garcia, Florian Luca, Timothy SchaaffAbstract:Abstract We study the difference between the number of Primitive Roots modulo p and modulo p + k for prime pairs p , p + k . Assuming the Bateman–Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs p , p + k , the dominant inequality is reversed.
-
Primitive Root bias for twin primes
arXiv: Number Theory, 2017Co-Authors: Stephan Ramon Garcia, Elvis Kahoro, Florian LucaAbstract:Numerical evidence suggests that for only about $2\%$ of pairs $p,p+2$ of twin primes, $p+2$ has more Primitive Roots than does $p$. If this occurs, we say that $p$ is exceptional (there are only two exceptional pairs with $5 \leq p \leq 10{,}000$). Assuming the Bateman-Horn conjecture, we prove that at least $0.47\%$ of twin prime pairs are exceptional and at least $65.13\%$ are not exceptional. We also conjecture a precise formula for the proportion of exceptional twin primes.
Tim Trudgian - One of the best experts on this subject based on the ideXlab platform.
-
the least Primitive Root modulo p2
Journal of Number Theory, 2020Co-Authors: Bryce Kerr, Kevin J Mcgown, Tim TrudgianAbstract:Abstract We provide an explicit estimate on the least Primitive Root mod p 2 . We show, in particular, that every prime p has a Primitive Root mod p 2 that is less than p 0.99 .
-
explicit upper bounds on the least Primitive Root
Proceedings of the American Mathematical Society, 2019Co-Authors: Kevin J Mcgown, Tim TrudgianAbstract:We give a method for producing explicit bounds on $g(p)$, the least Primitive Root modulo $p$. Using our method we show that $g(p) 10^{56}$ where $r\geq 2$ is an integer parameter. This result beats existing bounds that rely on explicit versions of the Burgess inequality. Our main result allows one to derive bounds of differing shapes for various ranges of $p$. For example, our method also allows us to show that $g(p)
-
the least Primitive Root modulo p 2
arXiv: Number Theory, 2019Co-Authors: Bryce Kerr, Kevin J Mcgown, Tim TrudgianAbstract:We provide an explicit estimate on the least Primitive Root mod $p^{2}$. We show, in particular, that every prime $p$ has a Primitive Root mod $p^{2}$ that is less than $p^{0.99}$.
-
on the least square free Primitive Root modulo p
Journal of Number Theory, 2017Co-Authors: Stephen D Cohen, Tim TrudgianAbstract:Abstract Let g □ ( p ) denote the least square-free Primitive Root modulo p. We show that g □ ( p ) p 0.96 for all p.
-
on the least square free Primitive Root modulo p
arXiv: Number Theory, 2016Co-Authors: Stephen D Cohen, Tim TrudgianAbstract:Let $g^{\square}(p)$ denote the least square-free Primitive Root modulo $p$. We show that $g^{\square}(p)< p^{0.96}$ for all $p$.
Pieter Moree - One of the best experts on this subject based on the ideXlab platform.
-
irregular primes with respect to genocchi numbers and artin s Primitive Root conjecture
Journal of Number Theory, 2019Co-Authors: Minsoo Kim, Pieter Moree, Min ShaAbstract:Abstract We introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-(ir)regularity and is based on Genocchi rather than Bernoulli numbers. We say that an odd prime p is G-irregular if it divides at least one of the Genocchi numbers G 2 , G 4 , … , G p − 3 , and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each Primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound x as x tends to infinity. As a byproduct, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-Primitive Root.
-
irregular primes with respect to genocchi numbers and artin s Primitive Root conjecture
arXiv: Number Theory, 2018Co-Authors: Minsoo Kim, Pieter Moree, Min ShaAbstract:In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each Primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound $x$ as $x$ tends to infinity. As a by-product, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-Primitive Root.
-
character sums for Primitive Root densities
Jean-Morlet Chair - Doctoral school : Frobenius distribution on curves;Chaire Jean-Morlet - Ecole doctorale : distribution de Frobenius sur des courbe, 2014Co-Authors: H W Lenstra, Peter Stevenhagen, Pieter MoreeAbstract:It follows from the work of Artin and Hooley that, under assumption of the generalised Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is a Primitive Root modulo q can be written as an infinite product ∏p δp of local factors δp reflecting the degree of the splitting field of Xp - r at the primes p, multiplied by a somewhat complicated factor that corrects for the ‘entanglement’ of these splitting fields.We show how the correction factors arising in Artin's original Primitive Root problem and several of its generalisations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors.The method not only applies in the setting of Galois representations of the multiplicative group underlying Artin's conjecture, but also in the GL2-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.
-
computing higher rank Primitive Root densities
Acta Arithmetica, 2014Co-Authors: Pieter Moree, Peter StevenhagenAbstract:We extend the character sum method for the computation of densities in Artin Primitive Root problems developed by H. W. Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the key parameters of the situation at hand, and obviates the lengthy analytic multiplicative number theory arguments that used to go into the computation of actual densities. It yields a conceptual interpretation of the formulas obtained, and enables us to extend their range of application in a systematic way.
-
artin s Primitive Root conjecture a survey
Integers, 2012Co-Authors: Pieter MoreeAbstract:One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin Primitive Root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Wojciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean
Stephan Ramon Garcia - One of the best experts on this subject based on the ideXlab platform.
-
Primitive Root bias for twin primes ii schinzel type theorems for totient quotients and the sum of divisors function
Journal of Number Theory, 2020Co-Authors: Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe UdellAbstract:Abstract Garcia, Kahoro, and Luca showed that the Bateman–Horn conjecture implies φ ( p − 1 ) ⩾ φ ( p + 1 ) for a majority of twin-primes pairs p , p + 2 and that the reverse inequality holds for a small positive proportion of the twin primes. That is, p tends to have more Primitive Roots than does p + 2 . We prove that Dickson's conjecture, which is much weaker than Bateman–Horn, implies that the quotients φ ( p + 1 ) φ ( p − 1 ) , as p , p + 2 range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of φ ( p + 1 ) φ ( p ) and σ ( p + 1 ) σ ( p ) , in which σ denotes the sum-of-divisors function.
-
Primitive Root bias for twin primes ii schinzel type theorems for totient quotients and the sum of divisors function
arXiv: Number Theory, 2019Co-Authors: Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe UdellAbstract:Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $\phi(p-1) \geq \phi(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more Primitive Roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{\phi(p+1)}{\phi(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{\phi(p+1)}{\phi(p)}$ and $\frac{\sigma(p+1)}{\sigma(p)}$, in which $\sigma$ denotes the sum-of-divisors function.
-
Primitive Root bias for twin primes
Experimental Mathematics, 2019Co-Authors: Stephan Ramon Garcia, Elvis Kahoro, Florian LucaAbstract:ABSTRACTNumerical evidence suggests that for only about 2% of pairs p, p + 2 of twin primes, p + 2 has more Primitive Roots than does p. If this occurs, we say that p is exceptional (there are only...
-
Primitive Root biases for prime pairs i existence and non totality of biases
Journal of Number Theory, 2018Co-Authors: Stephan Ramon Garcia, Florian Luca, Timothy SchaaffAbstract:Abstract We study the difference between the number of Primitive Roots modulo p and modulo p + k for prime pairs p , p + k . Assuming the Bateman–Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs p , p + k , the dominant inequality is reversed.
-
Primitive Root bias for twin primes
arXiv: Number Theory, 2017Co-Authors: Stephan Ramon Garcia, Elvis Kahoro, Florian LucaAbstract:Numerical evidence suggests that for only about $2\%$ of pairs $p,p+2$ of twin primes, $p+2$ has more Primitive Roots than does $p$. If this occurs, we say that $p$ is exceptional (there are only two exceptional pairs with $5 \leq p \leq 10{,}000$). Assuming the Bateman-Horn conjecture, we prove that at least $0.47\%$ of twin prime pairs are exceptional and at least $65.13\%$ are not exceptional. We also conjecture a precise formula for the proportion of exceptional twin primes.
Bo Chen - One of the best experts on this subject based on the ideXlab platform.
-
explicit upper bound on the least Primitive Root modulo p2
International Journal of Number Theory, 2021Co-Authors: Bo ChenAbstract:In this paper, we give an explicit upper bound on h(p), the least Primitive Root modulo p2. Since a Primitive Root modulo p is not Primitive modulo p2 if and only if it belongs to the set of intege...
