An approach to Andrica's conjecture

Question

Andrica's conjecture states that $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$.

but solving for $n=1,2,\dotsc$ yields

n=1, $\sqrt{p_{2}}-\sqrt{p_1} < 1$=>$\sqrt{p_{2}}<\sqrt{p_1}+1$

n=2, $\sqrt{p_{3}}-\sqrt{p_2} < 1$=>$\sqrt{p_{3}}<\sqrt{p_2}+1=>\sqrt{p_{2}}>\sqrt{p_3}-1$

implies $\sqrt{p_1}>\sqrt{p_3}-2$

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continue with the same we have,

$\sqrt{p_{n+1}} < (n+\sqrt{2})$ or

$p_{n+1} < (n+\sqrt{2})^2$ which is obviously true

as $p_{n+1}$ according to Pierre Dusart is $p_{n+1} < (n+1)\ln\left((n+1)\ln(n+1)\right)$

Please elaborate how I am wrong !

Answer 1

Your logic feeling is wrong: Assuming $\pi^2>10$ you can easily prove that $\pi>3$, which is true; but in fact $\pi^2< 10$.

Same thing with Andrica's conjecture: If it is false, anything (like $1=0$) can be derived from it, but when it is true it only implies true things. Therefore the true consequence you derived is no corroboration of the conjecture.

Answer 2

You have just shown that we cannot arrive at a contradiction of Andrica's conjecture using this method.
But this doesn't mean that actually Andrica's conjecture is true.
If you are not a mathematician, it was a nice effort.