Did Euler prove theorems by example?

There's some evidence that precisely the opposite can be said: that Euler is aware of the fallacies of proving theorems by example (of course, this does not necessarily mean he has never used it). One memorable instance is his Exemplum Memorabile Inductionis Fallacis, where he described how he was almost led to conjecture a recursive formula for a particular numerical sequence until he found that they disagreed on the 10th term. (There are other reasons for that formula to have been plausible; that and other topics are discussed in this article.)

(Incidentally the "right" formula is now quite well-known.)

"Proof by example" is a technique used by Euclid, who often proved results that hold e.g. for n integers in a typical case, say for 3 integers, as well as by Diophantus, who had to choose values for his parameters due to his lack of algebraic notation. I regard both versions as complete proofs.

This is apparently not what Fraser is referring to; Euler did generalize from examples to theorems in his Algebra, where he transferred correct results from "rings of integers" ${\mathbb Z}[i]$ to general quadratic rings without proof; but Euler wrote his algebra when he was old and completely blind, and perhaps it is fair to say that Euler was collecting evidence for his "method" rather than regarding these examples as proofs. I am not aware of a single example where Euler explicitly said that he regarded the verification of examples as a proof, but I only have read his number theoretical work in detail. The idea that Lagrange proved results by examples is ridiculous.

Not sure if this answer adds anything to the ones already given. I write it because It is an example where Euler explicitly writes about the necessity of giving a proof, and more importantly, calls a proof given by himself "Attempt at a proof". The following is his remarks before his "attempt at a proof" of the sum of two squares in 1758, "On numbers which are the sum of two squares". Two years later, he has another paper with the title " Proof of Fermat’s Theorem That Every Prime Number of the Form 4n + 1 is the Sum of Two Squares". In a way, even the titles of these two papers suggest an answer to your question.

All prime numbers which are sums of two squares, except 2, form this series: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, etc. Not only are these contained in the form 4n + 1, but also, however far the series is continued, we find that every prime number of the form 4n+1 occurs. From this, we can conclude by induction 6 that it is likely enough that there is no prime number of the form 4n+ 1 which is not also a sum of two squares. Nevertheless, induction, however extensive, cannot fulfill the role of proof. Even if no one doubts the truth of the statement that all prime numbers of the form 4n+ 1 are sums of two squares, until now mathematics could not add this to its established truths. Even Fermat declared that he had found a proof, but because he did not publish it anywhere, we properly extend confidence toward the assertion of this most profound man, and we believe that property of the numbers, but this recognition of ours rests on pure faith without knowledge. Although I labored much in vain on a proof to be discarded, nevertheless I have discovered another argument to be given for this truth, which, even it if it is not fully rigorous, still appears to be equivalent to induction connected with nearly rigorous proof.

The following is from the introduction of the second paper where Euler summarizes the first paper.

I next put forth an attempt of the proof from which the validity of this theorem is revealed much more clearly, even if it should be set aside by the standards of rigorous proof.