Prime Oracle @Prime_Oracle
Are you interested in prime numbers? ... follow me! Joined May 2013-
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Can you evaluate the Riemann zeta function at an even number using the residue theorem and the Laurent expansion of \pi* cot(\pi*z)?
Complex Analysis... fundamental tool for Analytic Number Theory. Here 1/(e^(2*i*3.141592*cosh(0.8813*(z+1)/2)/z) -1)
the largest konwn pair of twin number is 3,756,801,695,685 x 2^666,669 – 1 and 3,756,801,695,685 x 2^666,669 + 1
"abc conjecture": mathworld.wolfram.com/abcConjecture.… ;)
1898129658527122994504117104765989622686413200853629 is a #primenumber
x_1 := 6, x_n:= x_{n-1}^2 - 2. x_2 = 34... and so on. Observe that: F_2= 17 | x_2 F_3 = 257 | x_5 F_4 = 65537 | x_11 ...amazing...
Prime Oracle retweeted
If you're into #primenumbers this app works beautifully: Lookup & Inquiry For All Primes Up To 1,000,000,000,000,000 |markknowsnothing.com/cgi-bin/primes…
17 follower, 19 tweet, 233 following... all #primenumbers ;)
Using continued fraction of sqr(2), you get a series of rational numbers P_n/Q_n converging to it. What's the matter with #primenumbers? ;)
You can test primality using continued fractions. There are some examples: Lucas series and Lehmer test for Mersenne numbers.
Murty: Let f ∈ Z[x], (a,b)=1. If (with finitely many exceptions) prime divisors of f are either ≡ 1 or b (mod a). Then b^2 ≡ 1 (mod a).
Given coprimes a,b with b^2 = 1(mod a), there are infinitely many #primenumbers of type ax + b, you can show that WITHOUT Dirichlet L-series
One can compute the number of solutions of {x^2 + y^2 = 1} in F_p with the famous Burnside Formula applied on characters of SL(2,p), p prime
An elliptic curve is non singular and of genus 1. Then it satisfies Riemann Hypothesis. For large #primenumber p, it has many point in F_p.
{x^4+y^4=1} in the affine space A^2(F_p) contains p+1+2A points, where p=5 (mod 8) and A =1 (mod 4), p = A^2 + B^2
{x^4+y^4=1} in the affine space A^2(F_p) contains p-3-6A points, where p=1 (mod 8) and A =1 (mod 4), p = A^2 + B^2
If C is a non-singular algebraic curve in P^n(F) of genus g, where F is a finite field, then C satisfies the Riemann Hypothesis.
Points of {x^3+y^3=1} in the affine space A^2 over the fields of p elements is p-2+A where 4p=A^2+27B^2, 3 divides A-1.
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