How do large prime gaps... affect FTA?

[Guest]

How do large prime gaps, G(X), between consecutive primes less than or equal to X affect the Fundamental Theorem of Arithmetic (FTA)?

If [tex]G(X) \approx log^{2}X[/tex] for large X, are there too few primes being generated which may violate FTA when applied to composites greater than or equal to [tex]X^{2}[/tex]?

[Guest]

How do large prime gaps, G(X), between consecutive primes less than or equal to X affect the Fundamental Theorem of Arithmetic (FTA)?

If [tex]G(X) \approx log^{2}X[/tex] for large X, are there too few primes being generated which may violate FTA when applied to composites greater than or equal to [tex]X^{2}[/tex]?

Hmm. Both FTA and the Prime Number Theorem (PNT) must not be violated.

If [tex]G(X) \approx log^{2}X[/tex] is allowed, then [tex]X^{2}[/tex] = [tex]\prod_{i=1}^{i=j}[/tex] [tex]p_{i}^{n_{i }}[/tex] with some primes, [tex]p_{i} \le X[/tex], must be true even if there is G(X) between some consecutive primes less than or equal to X.

...

[Guest]

How do large prime gaps, G(X), between consecutive primes less than or equal to X affect the Fundamental Theorem of Arithmetic (FTA)?

If [tex]G(X) \approx log^{2}X[/tex] for large X, are there too few primes being generated which may violate FTA when applied to composites greater than or equal to [tex]X^{2}[/tex]?

Hmm. Both FTA and the Prime Number Theorem (PNT) must not be violated.

If [tex]G(X) \approx log^{2}X[/tex] is allowed, then [tex]X^{2}[/tex] = [tex]\prod_{i=1}^{i=j}[/tex] [tex]p_{i}^{n_{i }}[/tex] with some primes, [tex]p_{i} \le X[/tex], must be true even if there is G(X) between some consecutive primes less than or equal to X.

...

For large X, we have [tex]\pi(X) \approx X/log X[/tex] according to PNT where log X 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].

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

...

[Guest]

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

[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]

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

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.