Euler's errors?

Question

What mathematical errors is Leonhard Euler known to have made?

PS: As I wrote in a comment below: "However, I would not consider proof to be an error merely because it's not a proof by present-day standards." Everybody knows Euler wrote about infinitely large integers and about infinitesimals in ways differing from what today is considered logically rigorous. I had in mind actually erroneous conclusions or arguments that we cannot today replace with any we consider rigorous.

Answer 1

Euler conjectured that for $n=2\pmod 4$ there are no mutually orthogonal Latin squares of size $n\times n$. Bose and Shrikande disproved it by construction and earned the name Euler's Spoilers. See http://en.wikipedia.org/wiki/Graeco-Latin_square

Answer 2

Euler liked to play fast and loose with divergent series. Mathematicians of that era did not seem to be concerned with convergence issues.

For a more concrete example, Euler made a large mistake in trying to prove Fermat's Last Theorem for $n=3$. For details, check out http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_last_theorem.html

Answer 3

Euler apparently had some trouble deriving the Jacobian used in change of variables for double integrals.

He began by considering congruent transformations consisting of (affine) linear functions, and got something like $$\mathrm{d}x\,\mathrm{d}y=m\sqrt{1-m^2}\,\mathrm{d}t^2+(1-2m^2)\,\mathrm{d}t\,\mathrm{d}v-m\sqrt{1-m^2}\,\mathrm{d}v^2$$ which he described as "obviously wrong and even meaningless." He then got

$$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}\right)\,\mathrm{d}t\,\mathrm{d}v$$ which was not symmetric in the variables, and therefore would not do. Finally, he derived the correct

$$\mathrm{d}x\,\mathrm{d}y=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}-\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and lamented that simply multiplying out $$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial x}{\partial t}\,\mathrm{d}t+\frac{\partial x}{\partial v}\,\mathrm{d}v\right)\left(\frac{\partial y}{\partial t}\,\mathrm{d}t+\frac{\partial y}{\partial v}\,\mathrm{d}v\right)=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}+\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and shredding the squared differentials yielded an incorrect but annoyingly close answer.

But let us remember, if Euler committed errors it was only because of the unrivaled breadth of his work. If I could finish with a quote from the article cited below: "As a developer of algorithms to solve problems of various sorts, Euler has never been surpassed."

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Source: For an excellent review of the history of the Jacobian, and to learn more about the details of what I have written, I highly recommend reading this article by Prof. Victor J. Katz (Internet Archive, jstor.

Answer 4

This isn't a bona fide mistake but it's certainly a pitfall. Hopefully someone can verify the following. In Euler's original proof of the Basel Problem $(\zeta(2)=\pi^2/6$), he used the fact that

$$\sin(z)=z\prod_{n\geq 0}\left(1-\frac{z^2}{n^2\pi^2}\right).$$

This was well before Weierstrass's factorization theorem, which allows for a prefactor of $e^{g(z)}$ and in the case of sine, this prefactor is just 1. Rigorously showing that the above factorization holds and that the prefactor is 1 is nontrivial and as far as I know Euler had no solid proof of this fact.

Answer 5

It can be read on Peter Schumer's book "introduction to number theory" page 80, that Euler gave a defective proof that all primes have primitive roots

Answer 6

In the Introduction of Rational Points on Elliptic Curves by Silverman and Tate, it is claimed that Euler, in the 1730s, provided an incorrect solution to a question posed by Fermat in the 1650s, which was to show that the equation $$ y^2 - x^3 = -2 $$ has only two solutions in integers, namely $(3,±5).$ There's no citation given, though.