The paper claims a proof of Legendre's conjecture, but the final step of the proof does not establish a prime in the required interval, so the conjecture remains open.
Koshy, Elementary Number Theory with application , 2 nd Edition, Aca- demic Press(2007)
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Existence of primes between two consecutive squares
The paper claims a proof of Legendre's conjecture, but the final step of the proof does not establish a prime in the required interval, so the conjecture remains open.