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Will someone please help me in understanding the claims of corollary $2$ and $3$?

I understand that corollary $2$ means for large $N$ there are more primes than squares in the interval $[1,N]$ which can also be seen alternatively using Prime Number Theorem. But I want to understand it from the Claim made in this corollary by Euler.

I know the proof given here is wrong and lacks rigor. Actually this proof was given by Euler. Can we anyway reach to corollary by bypassing the wrong proof ?

Any help would be appreciated. Thanks in advance. enter image description here

math is fun
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    @PeterForeman: I think this is an extract of a translation of a work of Euler, and so you shouldn't judge it by 21st century standards. I wish I was able to contribute to mathematics $10^{-6}$ of what Euler did. – Rob Arthan Jun 15 '20 at 21:45
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    No..it is not in terms of bijection, it is rather in terms of density. I am aware of that proof using prime number theorem. I was just curious whether we could also prove that using the corollary. – math is fun Jun 15 '20 at 21:46
  • Exactly @Rob Arthan – math is fun Jun 15 '20 at 21:47
  • @PeterForeman Euler did this all the time, I remember reading about it in an MAA article about the Goldbach-Euler theorem (where he also used this method) – Ryan Shesler Jun 15 '20 at 21:47
  • @PeterForeman - maybe the question could be considered in the context of asymptotic density, in which case they both are 0 – The Chaz 2.0 Jun 15 '20 at 21:47
  • Really the enormous development Euler did with minimum tools available at that time is worth appreciating. – math is fun Jun 15 '20 at 21:48
  • Haha fixed. @Rob Arthan – math is fun Jun 15 '20 at 21:54
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    Thanks! That makes sense now. Apparently, "câlin" is French for cuddle, but I think Euler's theory of cuddles is elsewhere in his collected works $\ddot{\smile}$. – Rob Arthan Jun 15 '20 at 21:56
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    Between $1..N$ there are $\pi(N)$ primes and $\sqrt{N}$ squares and $\pi(N)>\sqrt{N}$, quite a few details are provided here. – rtybase Jun 15 '20 at 22:01
  • Yeah..I was also referring to the same @rytbase – math is fun Jun 15 '20 at 22:09
  • One simple approach is still an open problem: https://en.wikipedia.org/wiki/Legendre%27s_conjecture – lhf Jun 16 '20 at 00:25

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