ON THE GENERA OF X0(N) 1. Introduction For each positive integer
... We now show a similar result modulo other primes `. We compute the density of positive integers N for which g0 (N ) ≡ 1 (mod `). As above, we may assume N has a prime factor p with 12` | (p + 1); then ` | (g0 (N ) − 1) is equivalent to ` | ν∞ , and this occurs precisely when either `3 | N or some pr ...
... We now show a similar result modulo other primes `. We compute the density of positive integers N for which g0 (N ) ≡ 1 (mod `). As above, we may assume N has a prime factor p with 12` | (p + 1); then ` | (g0 (N ) − 1) is equivalent to ` | ν∞ , and this occurs precisely when either `3 | N or some pr ...
Prime Number Conjecture
... In a letter to the great mathematician Leonard Euler, Goldbach posited that all prime numbers (ℙ′) greater than five (ℙ′ > 5) are the sum of 3 smaller primes, known as Goldbach’s weak conjecture (Bruckman (2006); Bruckman (2008); Chang (2013) & Shu-Ping (2013). The conjecture was recently proven tru ...
... In a letter to the great mathematician Leonard Euler, Goldbach posited that all prime numbers (ℙ′) greater than five (ℙ′ > 5) are the sum of 3 smaller primes, known as Goldbach’s weak conjecture (Bruckman (2006); Bruckman (2008); Chang (2013) & Shu-Ping (2013). The conjecture was recently proven tru ...
-R-ES-O-N-A-N--CE--I-D-e-c-e-m-b-e-T-`-99
... of n eX)! So, Xn-1 = TIdln TI(T,n)=d(X _(T) = TIdln d(X). Here, we have denoted by (r, n) the greatest common divisor of T and n. From the above expression for n(X), it is not at all clear that n(X) has integer coefficients. However, one uses elementary number theory to invert the identi ...
... of n eX)! So, Xn-1 = TIdln TI(T,n)=d(X _(T) = TIdln d(X). Here, we have denoted by (r, n) the greatest common divisor of T and n. From the above expression for n(X), it is not at all clear that n(X) has integer coefficients. However, one uses elementary number theory to invert the identi ...
ABSTRACT On the Goldbach Conjecture Westin King Director: Dr
... function that has period k. An arithmetic function for natural numbers n and m, ψ(n) is one such that ψ(n + m) = ψ(ψ(n) + ψ(m)) and ψ(nm) = ψ(ψ(n)ψ(m)). A period of k means ψ(n) = ψ(n + km), where n, k, and m are natural numbers. The Dirichlet L-function of χk is defined as Lk (s, χ) = ...
... function that has period k. An arithmetic function for natural numbers n and m, ψ(n) is one such that ψ(n + m) = ψ(ψ(n) + ψ(m)) and ψ(nm) = ψ(ψ(n)ψ(m)). A period of k means ψ(n) = ψ(n + km), where n, k, and m are natural numbers. The Dirichlet L-function of χk is defined as Lk (s, χ) = ...
Remarks on number theory I
... which clearly implies that the sum of the reciprocals of the primitive a-abundants diverges . A simple argument shows that the a's for which the sum of the reciprocals of the primitive a-abundants diverges form an everywhere dense G,, in (l, v), i . e.. they are the countable intersection of dense o ...
... which clearly implies that the sum of the reciprocals of the primitive a-abundants diverges . A simple argument shows that the a's for which the sum of the reciprocals of the primitive a-abundants diverges form an everywhere dense G,, in (l, v), i . e.. they are the countable intersection of dense o ...
Construction of Composite Numbers by Recursively
... Then, for r > 0, Pr,1 , Pr,2 , Pr,3 , · · · is a partitioning of all prime numbers. With Birkhoff and Vandiver’s theorem (proposition 2.12), we can show Pr,k is nonempty for r > 1 and k > 0, except the trivial case P2,1 . Hence, we have a non-empty partitioning of primes for each r > 2. Proposition ...
... Then, for r > 0, Pr,1 , Pr,2 , Pr,3 , · · · is a partitioning of all prime numbers. With Birkhoff and Vandiver’s theorem (proposition 2.12), we can show Pr,k is nonempty for r > 1 and k > 0, except the trivial case P2,1 . Hence, we have a non-empty partitioning of primes for each r > 2. Proposition ...
project - William Stein
... If p does not divide a (i.e. the greatest common divisor of p and a, is one), then ap-1=1(mod p). (See [4] for a nice proof of the theorem.) ...
... If p does not divide a (i.e. the greatest common divisor of p and a, is one), then ap-1=1(mod p). (See [4] for a nice proof of the theorem.) ...
Short effective intervals containing primes
... statement has been proved for almost all intervals in a quadratic average sense by Selberg [19] in 1943 assuming the Riemann hypothesis and replacing K by a function KðxÞ tending arbitrarily slowly to infinity. From a numerical point of view, the Riemann hypothesis is known to hold up to a very large ...
... statement has been proved for almost all intervals in a quadratic average sense by Selberg [19] in 1943 assuming the Riemann hypothesis and replacing K by a function KðxÞ tending arbitrarily slowly to infinity. From a numerical point of view, the Riemann hypothesis is known to hold up to a very large ...