Let’s define the quantity , and assume Cramer’s conjecture. Then . Let’s denote by the event is -central, and by the event is a couple of twin primes. Then from de Bayes’ theorem, one get...
https://ideasfornumbertheory.com/2016/04/05/cramers-conjecture-implies-twin-prime-conjecture/
See the attached document in http://www.les-mathematiques.net/phorum/read.php?43,1210447 Just click on Ouvrir (open) or télécharger (download) to read the pdf. The assumption of Langlands func...
https://ideasfornumbertheory.com/2016/02/03/langlands-functoriality-conjecture-implies-grh/
We defined to be a primality radius of iff both and are primes, which requires . If we slightly soften this assumption, writing , we can say that a classical primality radius is a natural primali...
https://ideasfornumbertheory.com/2015/12/21/relative-primality-radius/
The Prime Number Theorem (PNT for short) says that the average gap between two consecutive primes of size is . Defining the quantity as , where is the -th typical primality radius of , one can ex...
https://ideasfornumbertheory.com/2015/11/18/a-pnt-based-variance-inequality-for-cramers-conjecture/
Assume that there are only a finite number of twin primes greater than sorted in increasing order and let’s denote them . Let’s now formulate the following Prim conjecture: There exists a pri...
https://ideasfornumbertheory.com/2015/09/17/a-possible-way-to-tackle-the-twin-prime-conjecture/
Suppose is a multiple of . One can assume without loss of generality that . Hence is a Goldbach gap greater than and thus there are at least three different Goldbach gaps, namely , and , hence . ...
https://ideasfornumbertheory.com/2015/08/20/proof-that-etan2-case-n-multiple-of-3/
Suppose is coprime with . Then every potential typical primality radius of is a multiple of . But as is a multiple of less than , and is a multiple of less than , it follows that and can’t be p...
https://ideasfornumbertheory.com/2015/08/19/proof-that-etan2-case-n-coprime-with-3/
Numerical computations seem to show that is always less than whenever . Solving the equation , one gets as a threshold the value , hence every even integer greater than is the sum of two primes, ...
https://ideasfornumbertheory.com/2015/07/03/explicit-upper-bound-for-r0n/
As mentioned in the previous article, it appears that the upper bound would follow from the following reasonable assumption: is « rather close » to This follows from the very definition of w...
https://ideasfornumbertheory.com/2015/04/16/sharp-upper-bound-for-alpha_n/
The definition of a « typical » primality radius of leads to the following assumption: the following ratio: is asymptotically equal to the ratio: Hence, for large enough, the quantity should ...
https://ideasfornumbertheory.com/2015/03/07/a-conjectural-expression-for-n2n/