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The second paper is Sur l'usage des séries infinies dans la théorie des nombres Crelle $\mathbf{18}$ (1838), 259--274. The quote in Crelle is near the end, at the top of p. 272. After it he says he …

To extend on Matt's comment about Euler, here is something I wrote up some years ago about Euler's discovery of the functional equation only at integral points. I hope there are no typos. Although Eul …

In the 1st chapter of Eisenbud's book that you had mentioned, he discusses four fundamental theorems of Hilbert that appeared in 1890 and 1893 (see p. 26): the basis theorem, the Nullstellensatz, the …

Euler's attack on the lemma was flawed precisely because of the (implicit) assumption regarding the veracity of a third powers product principle in $\mathbf Z[\sqrt{-3}]$. A close reading of Euler's …

In 1882, Dedekind and Weber developed the theory of Riemann surfaces purely algebraically, by taking as their primary object of study not Riemann surfaces, but instead function fields $K$ over $\mathb …

Cormack and Hounsfield received the 1979 Nobel prize in medicine for their work on CT scans. Cormack, a physicist, published his mathematical work on this in 1963, to essentially no response. Hounsfie …

In algebraic number theory, the existence of a Frobenius element at any prime $p$ in a Galois extension $K/{\mathbf Q}$ is crucial. That is, for any prime ideal $\mathfrak p$ lying over $p$ in $K$ th …

Here is a characterization of entropy functions due to Faddeev in 1956 (see pp. 229-231 of Faddeev's paper here if you read Russian or Chapter 1 of A. Feinstein's 1958 book Foundations of information …

In the first decades of the 20th century, $p$-adic analysis (or valuation theory more generally) was regarded by many as rather exotic. After Hensel's work there was a steady development by Strassmann …

Does acceptance of conjectures before they became theorems count? Example 1. The Artin reciprocity law. When Artin went around to other people describing what he was trying to show, nobody else belie …

Any rational function field over a finite field has genus $0$ and class number $1$, where the class number of a function field over a finite field is the number of degree-zero elements of the divisor …

Mathematicians used to hold plenty of false, but intuitively reasonable, ideas in analysis that were backed up with proofs of one kind or another (understood in the context of those times). Coming to …

In the early 1900s, Dickson introduced what he called generalized quaternion algebras over any field $K$ of characteristic not 2. These are exactly what we'd call quaternion algebras over $K$. His def …

Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p \equiv a \bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p …

I think the Busemann-Petty problem is an example like what you're asking, although changes in opinion would be due to progress (positive and negative) rather than any heuristic analysis. A great de …