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When and where did Legendre first publish or write about his conjecture that there is a prime between consecutive square numbers?
$$n^2 < p < (n+1)^2$$
I have looked through edition 1 and 2 of his Essai sur la Théorie des Nombres but I can't find any reference.
Actually Legendre proved, starting from a false theorem, that between $L$ and $L+2\sqrt{L}$ there is always a prime number. See the second edition of Essai sur la Théorie des Nombres at page 406 (paragraph 409). From the same theorem Desboves proved in 1855 as a corollary (p. 290, Corollary II) that there is always a prime number between two consecutive squares, and actually that there are at least two primes between two consecutive squares (Theorem II). Desboves was well aware that Legendre's proof of the main proposition was wrong, and in the article assumes Legendre's result as a postulate. Also notice that Desboves explicitly states that Legendre did not enunciate Corollary II, but only "Corollary I", i.e., that there is a prime number between $n$ and $n+2\sqrt{n}+1$ (this is not exact as Legendre discarded the "$+1$"), but it is clear that the result is the same. Summarizing: Legendre stated the conjecture (as a theorem) in a slightly different form in his Essai, while Desboves was the first to explicitly consider the problem of finding primes between two consecutive squares.
