We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, any results on reverse direction ? If we assume Goldbach Conjecture holds true, can we get any results about GRH ?

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them anymore). The author purports to bound the probability of the Goldbach Conjecture being false as $\approx 10^{–150,000,000,000}$ What is mathematics to make of […]

The List of unsolved problems in mathematics contains varies conjectures of exotic primes like: Mersenne primes (of the form $2^p – 1$ where $p$ is a prime, A000668, $43\%$) Sophie Germain primes ($p$ and also $2p+1$ is prime, A005384, $42\%$) Fermat primes (of the form $2^{2^k} + 1$, A019434, $100\%$) regular primes (A007703, $61\%$) Fibonacci […]

Let us assume there exists some infinite order differential equation whose solution is: $$ y= \sum_{n=1}^\infty A_n \exp(p_n^sx) $$ Where $p_n$ is the $n$’th prime. Substituting $ y=\exp(\lambda x)$ as a trail solution and factorizing. The differential equation must be simplified and factorized to: $$ \prod_{j=1}^\infty (\lambda-p_n^s) = 0$$ $$ \implies \prod_{j=1}^\infty (1-\frac{p_n^s}{\lambda}) = 0$$ […]

I tried to understand Terence Tao’s Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare myself to understand those papers ? Maybe there will be a sequence of books ? I do not have analytic […]

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens function, error terms of Prime Number Theorem, and Farey sequences. Those equivalent statements do not use Riemann Zeta function directly, but provide additional insights about Riemann Hypothesis from […]

My question pertains to two famous groups of related conjectures: Goldbach’s Conjecture (GC); Goldbach’s Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH). These two groups of conjectures both appear to be strong statements about the distribution of the prime numbers. Yet it is unclear to me if each provides independent information […]

Question I think I managed to reformulate a stronger version of Goldbach’s conjecture as an optimization problem: $$ \frac{\partial F_n}{\partial a_n} = \frac{\partial F_n}{\partial \overline a_n} = \frac{\partial F_n }{\partial \lambda_n}=0$$ Where: $$ F_n = |a_n|^2 – \lambda_n \frac{\sin(\pi |b_n|^2)}{|b_n|^2} $$ $$ (\sum_{r=1}^\infty |a_r|^2 x^{2r+1})^2 = \sum_{r=1}^\infty |b_r|^2 x^{2r+4}$$ Is this formulation correct? Is this […]

The result closest to Goldbach conjecture is Chen’s theorem [Sci. Sinica 16 157–176], the proposition “1+2”. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach conjecture is an undecidable proposition.

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ contains at least a pair of primes $p_1$ and $p_2$ whose sum is $n$, so $p_1+p_2=n$. Context: when a number $n$ does not […]

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