How do large prime gaps... affect FTA?

How do large prime gaps... affect FTA?

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

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

Re: How do large prime gaps... affect FTA?

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

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

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

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

...
Guest

Re: How do large prime gaps... affect FTA?

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

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

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

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

...

For large X, we have $$\pi(X) \approx X/log X$$ 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 * $$log^{2}(X)$$.

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

...
Guest

Re: How do large prime gaps... affect FTA?

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

Re: How do large prime gaps... affect FTA?

For large X, we have $$\pi(X) \approx X/log X$$ 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 * $$log^{2}(X)$$, 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. $$X / log X \approx \pi(X) \approx s + b*log X = s + b * X / \pi(X)$$.

Moreover, equation two implies approximately,

3. $$\pi^{2}(X) - s* \pi(X) - b* X = 0$$.

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

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

However, equation four implies approximately,

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

Thus, G(X) < $$log^{2} X$$ for large X.

Dave,

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

Re: How do large prime gaps... affect FTA?

For large X we have,

$$c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X$$

where $$0 < c_{1 } < c_{2} < 1$$.

Dave.
Guest

Re: How do large prime gaps... affect FTA?

Guest wrote:For large X, we have $$\pi(X) \approx X/log X$$ 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 * $$log^{2}(X)$$, 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. $$X / log X \approx \pi(X) \approx s + b*log X = s + b * X / \pi(X)$$.

Moreover, equation two implies approximately,

3. $$\pi^{2}(X) - s* \pi(X) - b* X = 0$$.

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

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

However, equation four implies approximately,

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

Thus, G(X) < $$log^{2} X$$ 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

Re: How do large prime gaps... affect FTA?

Guest wrote:
Guest wrote:For large X, we have $$\pi(X) \approx X/log X$$ 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 * $$log^{2}(X)$$, 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. $$X / log X \approx \pi(X) \approx s + b*log X = s + b * X / \pi(X)$$.

Moreover, equation two implies approximately,

3. $$\pi^{2}(X) - s* \pi(X) - b* X = 0$$.

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

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

However, equation four implies approximately,

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

Thus, G(X) < $$log^{2} X$$ 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 * $$log^{2} X$$ which contradicts equation one!"
Guest

Re: How do large prime gaps... affect FTA?

Oops! Proof is still wrong!!

Dave. Guest

Re: How do large prime gaps... affect FTA?

Guest wrote:Oops! Proof is still wrong!!

Dave. Note Change:

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

Thus, G(X) $$\le log^{2} X$$ 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