Is this a valid reduction of Ingham's upper bound for prime gaps?

Question

Let $p_n$ denote the $n^{th}$ prime number. Ingham showed that:

$$p_{n+1} - p_n \lt K p_n^{\frac{5}{8}}$$

where $K$ is a fixed positive integer, is an upper bound for the prime gaps.

(A.E.Ingham, On the difference between consecutive primes, Quart. J. Math. Oxford Ser. vol. 8 (1937) pp. 255-266)

I would like to know if the following manipulations would be valid to reduce the value $\frac{5}{8}$ up to $\frac{1}{8}$. Initially I cannot see it, but surely there is an error and I would like to learn where the mistake is:

1. Both sides to the eighth:

$$(p_{n+1} - p_n)^8 \lt K^8 p_n^{5}$$

2. Def. $K_2 = K^8$ a new fixed constant $K_2 \gt K$:

$$(p_{n+1} - p_n)^8 \lt K_2 p_n^{5}$$

3. By Fermat's Little Theorem, we know that $p_n^{5}$ can be replaced as follows:

$$p_n^{5}= 5K^{'} + p_n$$

For a given positive constant $K^{'}$. Thus:

$$(p_{n+1} - p_n)^8 \lt K_2 (5K^{'} + p_n)$$

And the right side of the inequality can be replaced as follows:

$$(p_{n+1} - p_n)^8 \lt K_2 (5K^{'} + p_n) = (K_2 \cdot 5K^{'}) + K_2 p_n$$

4. Def. $K_3 = K_2 \cdot 5K^{'}$ an even bigger new fixed constant $K_3 \gt K_2$:

$$(p_{n+1} - p_n)^8 \lt K_3 + K_2 p_n$$

5. As $K_3 \gt K_2$, it is also true that:

$$(p_{n+1} - p_n)^8 \lt K_3 + K_2 p_n \lt K_3 + K_3 p_n = K_3 (1+p_n) \lt K_3 (2p_n) = (2K_3) \cdot p_n$$

6. Def. $K_4 = 2K_3$ as an even bigger new fixed constant $K_4 \gt K_3$:

$$(p_{n+1} - p_n)^8 \lt K_4 \cdot p_n$$

7. Now we make again the $8^{th}$ root in both sides:

$$p_{n+1} - p_n \lt K_4^{\frac{1}{8}} \cdot p_n^{\frac{1}{8}}$$

8. Def. $K_5 = K_4^{\frac{1}{8}}$ as a new fixed constant. In this case $K_5 \lt K_4$ but still fixed and positive:

$$p_{n+1} - p_n \lt K_5 \cdot p_n^{\frac{1}{8}}$$

Thus, if the manipulations and redefinition of constants are correct, Ingham's result might be reduced from a $\frac{5}{8}$ exponent to a $\frac{1}{8}$ for a different fixed constant $K_5$.

This manipulations are easy, so surely I am making a forbidden step, I think that the use of Fermat's Little Theorem is valid, but not sure if redefining the fixed constants is correct, or moving the power of eight is valid or not. Indeed the last constant $K_5$ is smaller than the previous one, that also could be an error. Any explanations and insights are very appreciated.

Answer 1

The mistake is in the use of Fermat's little theorem. Indeed, the map

$$ K' : \mathbb{P} \to \mathbb{N} : p \mapsto \frac{p^5 - p}{5}$$

is not constant (contrary to what was claimed in the argument) and is in fact unbounded from above. So in the end, the obtained estimate is true if one keeps in mind that $K_5$ is not a constant, but a 'fast' increasing function of $p_n$.