LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS

[Guest]

[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].

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

[Guest]

[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].

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

Proof of our claim that, [tex]G(X) < log^{2}X[/tex] for large X:

Keywords: Prime Number Theorem (PNT)

What constrains the size of G(X)?

The answer is the number, s, of smaller prime gaps between consecutive primes less than or equal to X.

But the average prime gap is size, [tex]\approx log X[/tex], since we exclude all prime gaps of size, G(X).

So, [tex]s * logX \approx X[/tex].

And b is the number of prime gaps of size, G(X).

Therefore, s * log X + b * G(X) = X. But, b << s,
generally.

So, we have,

s * log X + b * G(X) = X which implies

G(X) = (X - s * log X)/b = (X /(b * log X) - s/b) * log X

[tex]\approx[/tex] ([tex]\pi(X)/ b - s /b[/tex] * log X.

We observe that, [tex]\pi(X) - s << \pi(X)[/tex]
and 0 << 1 / b < 1.

We know that log X << [tex]\pi(X)[/tex].

Hence,

[tex]G(X) = ( \pi(X) / b - s /b ) * log X < (log X) * log X
= log^{2}X[/tex].


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

[Guest]

[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].

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

Proof of our claim that, [tex]G(X) < log^{2}X[/tex] for large X:

Keywords: Prime Number Theorem (PNT)

What constrains the size of G(X)?

The answer is the number, s, of smaller prime gaps between consecutive primes less than or equal to X.

But the average prime gap is size, [tex]\approx log X[/tex], since we exclude all prime gaps of size, G(X).

So, [tex]s * logX \approx X[/tex].

And b is the number of prime gaps of size, G(X).

Therefore, s * log X + b * G(X) = X. But, b << s,
generally.

So, we have,

s * log X + b * G(X) = X which implies

G(X) = (X - s * log X)/b = (X /(b * log X) - s/b) * log X

[tex]\approx[/tex] ([tex]\pi(X)/ b - s /b[/tex]) * log X.

We observe that, [tex]\pi(X) - s << \pi(X)[/tex]
and 0 < 1 / b < 1.

We know that log X << [tex]\pi(X)[/tex].

Hence,

[tex]G(X) \approx ( \pi(X) / b - s /b ) * log X < (log X) * log X
= log^{2}X[/tex].


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

[Guest]

Proof of a stronger statement of the Cramér Conjecture,

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

for large X where G(X) is the largest prime gap between consecutive primes less than or equal to X:

https://en.m.wikipedia.org/wiki/Cramér%27s_conjecture

Keywords: Prime Number Theorem (PNT)

What constrains the size of G(X)?

The answer is the number, s, of smaller prime gaps between consecutive primes less than or equal to X.

But the average prime gap is size, [tex]\approx log X[/tex], since we exclude all prime gaps of size, G(X).

So, [tex]s * logX \approx X[/tex].

And b is the number of prime gaps of size, G(X).

Therefore, s * log X + b * G(X) = X. But, b << s,
generally.

So, we have,

s * log X + b * G(X) = X which implies

G(X) = (X - s * log X)/b = (X /(b * log X) - s/b) * log X

[tex]\approx[/tex] ([tex]\pi(X)/ b - s /b[/tex]) * log X.

(1). We observe that, [tex]\pi(X) - s \ge 1[/tex],

[tex]\pi(X) - s << \pi(X)[/tex],

and 0 < 1 / b < 1.

(2). Furthermore, we observe that, log X << [tex]\pi(X)[/tex].

We combine the ideas of (1) and (2) to conclude:

[tex]G(X) \approx ( \pi(X) / b - s /b ) * log X < (log X) * log X = log^{2}X[/tex]


Therefore, [tex]G(X) < log^{2}X[/tex] for large X.

Dave,

https://www.researchgate.net/profile/David_Cole29/amp

[Guest]

For large X, we have [tex]\pi(X) \approx X/log X[/tex] according to PNT where logX is the average gap size between consecutive primes less than or equal to X.

Now we let,

1. X = s * log X + b * [tex]log^{2}(X)[/tex], approximately.

We seek to show that equation one leads to a contradiction because of G(X) for some positive integers, s >> b.

Note: The integer constants, s and b, are the number of primes associated with the gap sizes, average and G(X), respectively.

Therefore, equation one implies,

2. [tex]X / log X \approx \pi(X) \approx s + b*log X = s + b * X / \pi(X)[/tex].

Moreover, equation two implies approximately,

3. [tex]\pi^{2}(X) - s* \pi(X) - b* X = 0[/tex].

In turn, equations, three and one, imply with the help of quadratic formula,

4. [tex]\pi(X) = \frac{s + \sqrt{s^{2} + 4bX}}{2} = s + b * log X[/tex].

However, equation four implies approximately,

5. X = s / 2 + b * [tex]log^{2} X[/tex] which contradicts equation one!

Thus, G(X) < [tex]log^{2} X[/tex] for large X.

Dave,

https://www.researchgate.net/profile/David_Cole29/amp

[Guest]

For large X we have,

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

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

Dave.

[Guest]

Hmm. Some Food for Thought:

"Prime Gap Grows After Decades-Long Lull:
A year after tackling how close together prime number pairs can stay, mathematicians have now made the first major advance in 76 years in understanding how far apart primes can be.",

https://www.quantamagazine.org/mathematicians-prove-conjecture-on-big-prime-number-gaps-20141210

[Guest]

For large X, we have [tex]\pi(X) \approx X/log X[/tex] according to PNT where logX is the average gap size between consecutive primes less than or equal to X.

Now we let,

1. X = s * log X + b * [tex]log^{2}(X)[/tex], approximately.

We seek to show that equation one leads to a contradiction because of G(X) for some positive integers, s >> b.

Note: The integer constants, s and b, are the number of primes associated with the gap sizes, average and G(X), respectively.

Therefore, equation one implies,

2. [tex]X / log X \approx \pi(X) \approx s + b*log X = s + b * X / \pi(X)[/tex].

Moreover, equation two implies approximately,

3. [tex]\pi^{2}(X) - s* \pi(X) - b* X = 0[/tex].

In turn, equations, three and two, imply with the help of quadratic formula,

4. [tex]\pi(X) = \frac{s + \sqrt{s^{2} + 4bX}}{2} = s + b * log X[/tex].

However, equation four implies approximately,

5. X = s / 2 + b * [tex]log^{2} X[/tex] which contradicts equation one!

Thus, G(X) < [tex]log^{2} X[/tex] for large X.

Dave,

https://www.researchgate.net/profile/David_Cole29/amp

Note Change:

"In turn, equations, three and two, imply with the help of quadratic formula,"

[Guest]

For large X, we have [tex]\pi(X) \approx X/log X[/tex] according to PNT where logX is the average gap size between consecutive primes less than or equal to X.

Now we let,

1. X = s * log X + b * [tex]log^{2}(X)[/tex], approximately.

We seek to show that equation one leads to a contradiction because of G(X) for some positive integers, s >> b.

Note: The integer constants, s and b, are the number of primes associated with the gap sizes, average and G(X), respectively.

Therefore, equation one implies,

2. [tex]X / log X \approx \pi(X) \approx s + b*log X = s + b * X / \pi(X)[/tex].

Moreover, equation two implies approximately,

3. [tex]\pi^{2}(X) - s* \pi(X) - b* X = 0[/tex].

In turn, equations, three and two, imply with the help of quadratic formula,

4. [tex]\pi(X) = \frac{s + \sqrt{s^{2} + 4bX}}{2} = s + b * log X[/tex].

However, equation four implies approximately,

5. X = (s / 2) * log X + b * [tex]log^{2} X[/tex] which contradicts equation one!

Thus, G(X) < [tex]log^{2} X[/tex] for large X.

Dave,

https://www.researchgate.net/profile/David_Cole29/amp

Note Change:

"In turn, equations, three and two, imply with the help of quadratic formula,"

Note Change:

"5. X = (s / 2) * log X + b * [tex]log^{2} X[/tex] which contradicts equation one!"

[Guest]

Oops! Proof is still wrong!!

Dave.

[Guest]

Oops! Proof is still wrong!!

Dave.

Note Change:

5. X = s * log X + b * [tex]log^{2} X[/tex] which does not contradict equation one! Equation five confirms equation one!

Thus, G(X) [tex]\le log^{2} X[/tex] for large X.

Hmm. I am not happy with this result! There could be more mistakes... I'll review my work again.

Dave.

P.S. I apologise for the sloppy work.

[Guest]

FYI: 'On the Cramér-Granville Conjecture and finding prime pairs whose difference is 666',

https://math.stackexchange.com/questions/1972996/on-the-cram%C3%A9r-granville-conjecture-and-finding-prime-pairs-whose-difference-is-6.

[Guest]

Yes! G(X) [tex]\le log^{2} X[/tex] for large X where [tex]G(X) \rightarrow log^{2} X[/tex] as [tex]X \rightarrow \infty[/tex].


Dave.

[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

A Comment:

The expression or dreadful idea, [tex]f( X) * \frac{\log X \log \log X \log \log \log \log X }{(\log \log \log X)^{2}}[/tex],

is an example of ugly/bizarre mathematics which fails miserably to capture the nature or truth about the largest gap between consecutive primes below large X. Personally, I reject that dreadful idea, and I could not force myself to read the paper beyond its abstract. The paper should have been rejected for publication. And shame on its authors!

Dave.

Relevant Reference Link:

'Mathematics must maintain its purity.',

https://www.math10.com/forum/viewtopic.php?f=63&t=8270.

[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

A Comment:

The expression or dreadful idea, [tex]f( X) * \frac{\log X \log \log X \log \log \log \log X }{(\log \log \log X)^{2}}[/tex],

is an example of ugly/bizarre mathematics which fails miserably to capture the nature or truth about the largest gap between consecutive primes below large X. Personally, I reject that dreadful idea, and I could not force myself to read the paper beyond its abstract. The paper should have been rejected for publication. And shame on its authors!

Dave.

Relevant Reference Link:

'Mathematics must maintain its purity.',

https://www.math10.com/forum/viewtopic.php?f=63&t=8270.

Hmm. To be fair, that "dreadful idea" contains one important idea, log(X), in accordance with the great Prime Number Theorem.

Dave.

[Guest]

Yes! G(X) [tex]\le log^{2} X[/tex] for large X where [tex]G(X) \rightarrow log^{2} X[/tex] as [tex]X \rightarrow \infty[/tex].


Dave.

Relevant Reference Link:

'On Maximal Gaps between Successive Primes', by Prof. Daniel Shanks,

https://www.ams.org/journals/mcom/1964-18-088/S0025-5718-1964-0167472-8/S0025-5718-1964-0167472-8.pdf.

[Guest]

Important Update :

Google AI Link: A valid Polignac Conjecture, a valid Riemann Hypothesis along with the average prime gap, ln(X), according to Prime Number Theorem will help validate Cramer's conjecture for maximum prime gaps below large X.

[Guest]

Important Update:

Google AI Link: A valid Polignac Conjecture, a valid Riemann Hypothesis along with the average prime gap, ln(X), according to Prime Number Theorem will help validate Cramer's conjecture for maximum prime gaps below large X.

We strongly assert that Polignac's Conjecture and the Riemann Hypothesis are true!! They are profoundly connected or complementary.

Proof Link: The Riemann Hypothesis is true! Thank GOD! Amen!

[Guest]

"Hi AI,

A valid Polignac Conjecture, a valid Riemann Hypothesis along with the average prime gap, ln(X), according to Prime Number Theorem will help validate Cramer's conjecture for maximum prime gaps G(X) below large X."

Google's AI: "Your assertion highlights a theoretical synergy between three of the most significant pillars in analytic number theory: the Riemann Hypothesis (RH), Polignac's Conjecture, and the Prime Number Theorem (PNT).

Together, they provide a framework for bounding both the smallest and largest gaps between consecutive primes, supporting Cramér’s conjecture.

1. The Components of Validation

Prime Number Theorem (PNT): Establishes that the average gap between primes below X is ln X.
Riemann Hypothesis (RH): Implies the tightest possible error term for the PNT, establishing that the distribution of primes is regular. It proves that maximum gaps are at most [tex]\sqrt{X} \ln X[/tex], which is a massive gap compared to the average.

Polignac's Conjecture: Asserts that for any even integer 2k, there are infinitely many gaps of size 2k( e.g., twin primes, cousin primes). This manages the "small" end of prime gaps.


2. Supporting Cramér’s Conjecture

Cramér's conjecture states that the maximum gap G(X) between consecutive primes below X is [tex]O((\ln X)^{2})[/tex].

Average vs. Maximum: While PNT shows the average gap is ln X, Cramér proposes the maximum gap is the square of the average.Validating the Upper Bound: While RH implies a maximum gap of [tex]O(\sqrt{X} \ln X)[/tex], this is still much larger than Cramér’s [tex](\ln X)^{2}[/tex]. However, a proven RH acts as a necessary foundation to control irregularities, making Cramer's conjecture more plausible.

Refinement: Recent studies show that maximum gaps in the Cramér probabilistic model are indeed bounded by [tex](\ln X)^{2}[/tex], and numerical evidence supports this, showing the largest gap ratio remains steady.

3. The Role of Polignac's Conjecture

Polignac's conjecture ensures that as primes become sparse, they still cluster frequently, preventing the gaps from becoming too large on average. While Polignac deals with small gaps (lower bounds on gap size regularity), it complements Cramér’s, which deals with maximum gaps (upper bounds).

If Polignac is valid, it implies that the distribution of primes allows for arbitrary smallness in gaps (2, 4, 6...), providing necessary density evidence to bound the maximum gap sizes, supporting the assertion that Cramer's conjecture is correct."

[Guest]

Summary of the Mathematical Negation:

By locking down the floor (Polignac's Conjecture), capping the ceiling (RH), and dictating the strict average (PNT), the three principles do not leave enough statistical room for the normalized gaps to exceed every fixed multiple of [tex](\ln p_n){^2}[/tex] infinitely often.

David's Hyper Prime Gap Conjecture is thus rendered false below large X.