# Proof of a stronger statement of the Cramér Conjecture

### Proof of a stronger statement of the Cramér Conjecture

Proof of a stronger statement of the Cramér Conjecture,

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

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, $$\approx log X$$, since we exclude all prime gaps of size, G(X).

So, $$s * logX \approx X$$.

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

$$\approx$$ ($$\pi(X)/ b - s /b$$) * log X.

(1). We observe that, $$\pi(X) - s \ge 1$$,

$$\pi(X) - s << \pi(X)$$,

and 0 < 1 / b < 1.

(2). Furthermore, we observe that, log X << $$\pi(X)$$.

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

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

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

Dave.
Guest

### Re: Proof of a stronger statement of the Cramér Conjecture

Furthermore, for large X, we have:

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

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

Dave,

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

### Re: Proof of a stronger statement of the Cramér Conjecture

Hmm. Can we improve our 'proof'?
Guest

### Re: Proof of a stronger statement of the Cramér Conjecture

Guest wrote:Hmm. Can we improve our 'proof'?

Hmm. Here's a minor improvement:

$$0 < 1 / b \le 1$$.
Guest

### Re: Proof of a stronger statement of the Cramér Conjecture

Guest wrote:
Guest wrote:Hmm. Can we improve our 'proof'?

Hmm. Here's a minor improvement:

$$0 < 1 / b \le 1$$.

Here's a significant development in support of our proof:

If $$b * G(X) \approx b* log^{2}X$$, then

b * G(X) is not << X and therefore,

$$s * logX \approx X$$ is false

for 1 << b << $$\pi(X)$$ and for large X.
Guest

### Re: Proof of a stronger statement of the Cramér Conjecture

How does $$G(X) \approx log^{2}X$$ for large X affect the Fundamental Theorem of Arithmetic (FTA)?

Are there too few primes being generated which may violate FTA when apply to composites greater than or equal to $$X^{2}$$?
Guest

### Re: Proof of a stronger statement of the Cramér Conjecture

Guest wrote:How does $$G(X) \approx log^{2}X$$ for large X affect the Fundamental Theorem of Arithmetic (FTA)?

Are there too few primes being generated which may violate FTA when applied to composites greater than or equal to $$X^{2}$$?
Guest

### Re: Proof of a stronger statement of the Cramér Conjecture

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: Proof of a stronger statement of the Cramér Conjecture

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: Proof of a stronger statement of the Cramér Conjecture

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: Proof of a stronger statement of the Cramér Conjecture

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: Proof of a stronger statement of the Cramér Conjecture

Oops! Proof is still wrong!!

Dave.
Guest

### Re: Proof of a stronger statement of the Cramér Conjecture

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