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I have found this exercise:

Show that there exist infinitely many indices $k$ such that the intervals $[k^2,(k+1)^2]$ contain at least one prime number.

Hint: Use the fact that the set of primes is infinite and the ends of each interval are not prime numbers. This is not the Legendre's Conjecture!

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  • $\begingroup$ What goes wrong when you try to follow the hint? $\endgroup$ – lulu Aug 24 '19 at 10:52
  • $\begingroup$ The hint already obviously answers the question. $\endgroup$ – Peter Aug 24 '19 at 10:52
  • $\begingroup$ @Peter: How this is possible. $\endgroup$ – Safwane Aug 24 '19 at 10:53
  • $\begingroup$ Well, every prime falls in such an interval (is between two consecutive squares), and since infinite many primes exist, there cannot be a largest $k$ with this property. Simply because larger and larger primes fall into larger and larger such intervals. $\endgroup$ – Peter Aug 24 '19 at 10:54
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    $\begingroup$ Your intervals, excluding $(k+1)^2$, form a partition of $\mathbb{R}^{+}$. Obviously, primes will have to "fit" somewhere in that partition. What happens, if by contradiction, you suppose that the number of indexes $k$ is finite? There will be a $K=\max\{k\}<\infty$, right? And primes are not bounded, right? $\endgroup$ – rtybase Aug 24 '19 at 11:10
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Suppose, towards a contradiction, that there are finitely many $k_i$ for which the intervals $[k_i^2, (k_i+1)^2]$ contain at least one prime number. Let $n$ be the number of such indices and we observe that every prime number belongs in the set $$\bigcup_{i=1}^n [k_i^2, (k_i+1)^2].$$ Note that each interval contains finitely many integers, and there are a finite number of such intervals. So there are a finite number of primes. This is absurd of course.

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Hint

Define $I_k=[k^2,(k+1)^2]$. Therefore $$\bigcup_{k=1}^{\infty} I_k=\Bbb N$$Now, what happens if only finitely many $I_k$s contain at least a prime? In other words, can infinitely many primes distributed over $\Bbb N$ be distributed over a finite number of $I_k$s?

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