LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS

[Guest]

"LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS: by Authors,
KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO.

ABSTRACT. Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdos, we show that

[tex]G(X) \ge f(X) * \frac{\log X \log \log X \log \log \log \log X }{(\log \log \log X)^{2}}[/tex],

where f(X) is a function tending to infinity with X. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes."

Source:

https://arxiv.org/abs/1408.4505

[Guest]

Keywords: Prime Number Theorem (PNT)

Hmm. What are the number of primes less than or equal to X?

What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?

Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),

[tex]G(X) \le \beta(X) * log X[/tex] where [tex]\beta(X)[/tex] tend to infinity as G(X) tend to infinity and where [tex]2 << \beta(X) << X[/tex].

[Guest]

Keywords: Prime Number Theorem (PNT)

Hmm. What are the number of primes less than or equal to X?

What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?

Thus, for the largest prime gap...,\beta G(X), we have simply (via the great PNT),

[tex]G(X) \le \beta(X) * log X[/tex] where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where [tex]2 << \beta(X) << X[/tex].

On the further investigation of [tex]\beta(X)[/tex],
we wish you good luck and the following words of inspiration.

"No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite." -- David Hilbert.

[Guest]

On large and small gaps between consecutive primes, here's a unifying idea:

The repetition and growth of prime gaps are essential for the efficient generation of composites in accordance with the Fundamental Theorem of Arithmetic and in accordance with the Prime Number Theorem.

[Guest]

An Update:

Keywords: Prime Number Theorem (PNT)

Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?

What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?

Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),

[tex]G(X) \le \beta(X) * log X[/tex]

where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where [tex]log X << \beta(X) << \pi(X)[/tex].

[Guest]

An Update:

Keywords: Prime Number Theorem (PNT)

Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?

What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?

Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),

[tex]G(X) \le \beta(X) * log X[/tex]

where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where [tex]log X \le \beta(X) << \pi(X)[/tex].

[Guest]

An Update:

Keywords: Prime Number Theorem (PNT)

Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?

What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?

Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),

[tex]G(X) \le \beta(X) * log X[/tex]

where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where

[tex]log X \le \beta(X) << \pi(X)/log X[/tex].

[Guest]

An Update:

Keywords: Prime Number Theorem (PNT)

Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?

What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?

Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),

[tex]G(X) \le \beta(X) * log X[/tex]

where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where

[tex]log X \le \beta(X) << \pi(X)/log X[/tex].

Hmm. For large X,

is G(X) more or less [tex](log X)^{2}[/tex]?

[Guest]

An Update:

Keywords: Prime Number Theorem (PNT)

Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?

What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?

Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),

[tex]G(X) \le \beta(X) * log X[/tex]

where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where

[tex]log X \le \beta(X) << \pi(X)/log X[/tex].

Hmm. For large X,

is G(X) more or less [tex](log X)^{2}[/tex]?

For large X,

[tex]G(X) < (log X)^{2}[/tex].

[Guest]

FYI: On Prime Gaps,

https://en.m.wikipedia.org/wiki/Prime_gap#Numerical_results

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

Series Expansion of [tex](log X)^{2}[/tex]:

For |-1 + X| > 1, we have

[tex](log X)^{2} = (log (-1 + X))^{2} -2(log (-1 + X)) \sum_{k=1}^{\infty}\frac{(-1)^{k}}{k(-1+X)^{k}} + ( \sum_{k=1}^{\infty}\frac{(-1)^{k}}{k(-1+X)^{k}})^{2}[/tex].

If [tex]G(X) \rightarrow (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?

What is the function, k(X)?

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

Power Series Expansion of [tex](log X)^{2}[/tex]:

For large X, we have

[tex](log X)^{2} = (log X)^{2} -2(logX)\sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}} + ( \sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}})^{2}[/tex].

If [tex]G(X) \rightarrow (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?

Or if [tex]G(X) < (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?

What is the function, k(X)? This is a difficult and deep question!

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

Power Series Expansion of [tex](log X)^{2}[/tex]:

For large X, we have

[tex](log X)^{2} = (log X)^{2} -2(logX)\sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}} + ( \sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}})^{2}[/tex].

If [tex]G(X) \rightarrow (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?

Or if [tex]G(X) < (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?

What is the function, k(X)? This is a difficult and deep question!

Hmm. The problem is deep but not difficult.

FYI: We can approximate log X with the appropriate Harmonic Series...

Relevant Reference Link:

[tex]https://en.m.wikipedia.org/wiki/Euler–Mascheroni_constant[/tex]

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

Power Series Expansion of [tex](log X)^{2}[/tex]:

For large X, we have

[tex](log X)^{2} = (log X)^{2} -2(logX)\sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}} + ( \sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}})^{2}[/tex].

If [tex]G(X) \rightarrow (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?

Or if [tex]G(X) < (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?

What is the function, k(X)? This is a difficult and deep question!

Hmm. The problem is deep but not difficult.

FYI: We can approximate log X with the appropriate Harmonic Series...

Relevant Reference Link:

[url]https://en.m.wikipedia.org/wiki/Euler–Mascheroni_constant[/url]

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

We can do better! For large X we have,

[tex]c_{1 }(log X)^{2} \le G(X) \le c_{2 }(log X)^{2}[/tex]

where [tex]0 < c_{1 } < c_{2} \le 1[/tex].

Moreover, [tex]\pi(X)[/tex]= o[tex](log^{2}(X))[/tex].

[Guest]

Hmm. That's theory! We want to see an example.

Let nth prime, [tex]p_{n }[/tex] = 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227.

According to Wikipedia on prime gaps,
https://en.m.wikipedia.org/wiki/Prime_gap#Simple_observations,

we have,

[tex]p_{n+1 } - p_{n }[/tex] = 8350.

Since [tex]X = p_{n } \approx p_{n+1}[/tex],

we compute,

[tex]\pi(X)/log^{2}(X) \approx[/tex] .00002523,

http://m.wolframalpha.com/input/?i=PrimePi%28293703234068022590158723766104419463425709075574811762098588798217895728858676728143227%29%2F%28log%28293703234068022590158723766104419463425709075574811762098588798217895728858676728143227.%29%29%5E2

Wow! Theory works very well!

[Guest]

Hmm. That's theory! We want to see an example.

Let nth prime, [tex]p_{n }[/tex] = 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227.

According to Wikipedia on prime gaps,
https://en.m.wikipedia.org/wiki/Prime_gap#Simple_observations,

we have,

[tex]p_{n+1 } - p_{n }[/tex] = 8350.

Since [tex]X = p_{n } \approx p_{n+1}[/tex],

we compute,

[tex]\pi(X)/log^{2}(X) \approx[/tex] .00002523 * [tex]\pi(X)[/tex].

http://m.wolframalpha.com/input/?i=PrimePi%28293703234068022590158723766104419463425709075574811762098588798217895728858676728143227%29%2F%28log%28293703234068022590158723766104419463425709075574811762098588798217895728858676728143227.%29%29%5E2

Wow! Theory works very well!

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

We can do better! For large X we have,

[tex]c_{1 }(log X)^{2} \le G(X) \le c_{2 }(log X)^{2}[/tex]

where [tex]0 < c_{1 } < c_{2} \le 1[/tex].

Moreover, [tex]\pi(X)[/tex]= O[tex](log^{2}(X))[/tex].

Note:. We changed o-notation to O-notation.

[Guest]

For large X,

[tex]log X << G(X) < (log X)^{2}[/tex].

This is our best result!

We can do better! For large X we have,

[tex]c_{1 }(log X)^{2} \le G(X) \le c_{2 }(log X)^{2}[/tex]

where [tex]0 < c_{1 } < c_{2} \le 1[/tex].

Moreover, (1) [tex]\pi(X)[/tex]= O[tex](log^{2}(X))[/tex].

Note:. We changed o-notation to O-notation.

Hmm. Result, (1), is still wrong! Instead we should have,

(2) [tex]\pi(X) >> (log^{2}(X))[/tex].