What is your favorite math conjecture in number theory (nt)?

[Guest]

My favorite math conjecture (nt) is hands down the great Goldbach Conjecture (GC) (https://en.m.wikipedia.org/wiki/Goldbach%27s_conjecture). This wonderful conjecture had inspired me to formulate a mathematical model of prime numbers... And I have tried on numerous occasions to prove it. And along the way, the Riemann Hypothesis (RH) became a major obstacle as well. But fortunately, I believe I helped to prove both conjectures.

And I also believe the the great work of the great Math Professor Hugh L. Montgomery of University of Michigan was most influential... I am very indebted to him and to the U--M Math Department. Moreover, I greatly appreciate the help I received from great Math Professor Boris Mordukhovich of Wayne State University.

So, I would take some credit for the proof of my beloved GC and less so for the proof of RH. I believe Professor Montgomery and the late great G. H. Hardy are most responsible for the proof of RH.

Sincerely,

David Cole.

Go Blue!

[Guest]

My favorite math conjecture (nt) is hands down the great Goldbach Conjecture (GC) (https://en.m.wikipedia.org/wiki/Goldbach%27s_conjecture). This wonderful conjecture had inspired me to formulate a mathematical model of prime numbers... And I have tried on numerous occasions to prove it. And along the way, the Riemann Hypothesis (RH) became a major obstacle as well. But fortunately, I believe I helped to prove both conjectures.

And I also believe the great work of the great Math Professor Hugh L. Montgomery of the University of Michigan was most influential... I am very indebted to him and to the U-M Math Department. Moreover, I greatly appreciate the help I received from the great Math Professor Boris Mordukhovich of Wayne State University.

So, I would take some credit for the proof of my beloved GC and less so for the proof of RH. I believe Professor Montgomery and the late great G. H. Hardy are most responsible for the proof of RH.

Sincerely,

David Cole.

Go Blue!

Goldbach conjecture states that all positive even integers, e, greater than four is the sum of two odd primes.

Now, please consider the following proposition:

Probability( [tex]e \ne p + q[/tex] ) [tex]= \prod_{j=1}^{k \to \infty }\frac{\pi (\sqrt{e - p_{j}})}{\pi (\sqrt{e - p_{j}}) + 1} \rightarrow 0[/tex]

where p and q are any odd primes less than e.

Moreover, [tex]p_{j}[/tex] is any distinct odd prime less than any even integer, e, greater than 100.

We let k be the number of distinct odd primes less than e.

And we let π() be the odd prime-counting function.

Note: One counts (one is important), and it is counted.

Relevant Reference Link:

'Proof of Goldbach Conjecture (GC)',

https://www.researchgate.net/publication/310845846_Proof_of_Goldbach_Conjecture.

Remarks: Our proof of GC depends on the distribution of odd primes along the natural number line or the prime-counting function. And we assume RH. And therefore, the error associated with the prime-counting function is known, and it does not violate our proof.

Dave.

[Guest]

"Note: One counts (one is important), and it is counted."

We had assumed e = p + n *q for some odd primes, p and q. And therefore, n > 1. That explains why one is counted...

[Guest]

"Note: One counts (one is important), and it is counted."

We had assumed e = p + n * q for some odd primes, p and q. And therefore, n > 1. That explains why one is counted...

Furthermore, we had also assumed n [tex]\ge[/tex] q where n * q = e - p. And there are [tex]\pi (\sqrt{e - p})[/tex] possible prime divisors (q) for the term, n * q.

The square root symbol which represents the exponent, [tex]\frac{1}{2}[/tex], in the expression, [tex]\pi (\sqrt{e - p})[/tex], is a big indicator of the truth of the Riemann Hypothesis. Go figure! Go Blue!

Dave.

[Guest]

What is the probability that 19,106 is not a sum of two prime numbers? Our approximate answer is [tex]10^{-43}[/tex].

And therefore, we expect that 19,106 is a sum of two primes. Example: 19,106 = 19 + 19,087.

[Guest]

What is the probability that 81,234,238 is not a sum of two prime numbers? Our approximate answer is [tex]10^{-324}[/tex].

And therefore, we expect that 81,234,238 is a sum of two primes. One of many examples: 81,234,238 = 11 + 81,234,227.

[Guest]

What is the probability that 81,234,238 is not a sum of two prime numbers? Our approximate answer is [tex]10^{-324}[/tex].

And therefore, we expect that 81,234,238 is a sum of two primes. One of many examples: 81,234,238 = 11 + 81,234,227.

Wolfram Mathematica Code:

f[evn_]:=(pn = 3;
prob = 1.;
j =0;

While[pn <evn,
prob = prob *PrimePi[(evn -pn)^.5] /( PrimePi[(evn -pn)^.5] +1);
pn = NextPrime[pn];
If[PrimeQ[evn -pn] && j==0,j=1];
If[ j==1,eureka=List[evn,pn,evn-pn]];
If[ j==1,j=2]];

Return[Flatten[{prob,eureka}]]);
--------end of code--------

Input: f[81234238]
Output: {5.*10^-324,81234238,11,81234227}.

Remarks: The prime, 2, is counted in our computations. Overall, its impact is irrelevant...

Relevant Reference Link:

https://www.wolframcloud.com/.

Dave.

[LilaSmith]

Great question. After reading all the answers, I learned many interesting theories.

[Guest]

Great question. After reading all the answers, I learned many interesting theories.

Thanks!

Dave