# Exceptional MathReviews

Mathematical Reviews is a publication of the American Mathematics Society that provides capsule reviews of essentially every mathematics research paper that is published. It has been in operation since 1940, and in recent years the entire database has been on-line, with the name MathSciNet. Some of the reviews are rather entertaining in one way or another, and the availability of this database has led to a new form of diversion among mathematicians: searching on promising words (e.g. “plagiarism”) to see what turns up. My colleague Kimball Martin has put together a new web-page entitled Exceptional MathReviews with links to the best reviews he has heard about.

One of the most famous such reviews turns out to be apocryphal. In the days before the searchable database, I had heard it claimed that one Math Review contained the following devastating evaluation: “This paper fills a much-needed gap in the literature.” It turns out that this isn’t actually true. For the real story and more about Math Reviews, see an article from the 1997 Notices of the AMS.

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### 5 Responses to Exceptional MathReviews

1. Florifulgurator says:

Could be nice, alas I have no subscription account available 🙁

2. Pindare says:

Thanks for the potentially interesting link, too bad I have no subscription. (They use to let people see the Featured Reviews for free, but alas it has been discontinued, to the despair of I guess many students and not-in-academia researchers & amateurs…)

3. Bill Lee says:

A sampler….
Granville on Biaca
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Matches for: MR=MR1418826
84
MR1418826 (97i:11025)
Baica, Malvina14(1-WIW)15
The Euclidean character of the Fermat’s last theorem. (English. English summary)
Notes Number Theory Discrete Math.16 2 (1996), no. 1,17 20–23.
11D41 (11A05)18
Review in linked PDF19 Add citation to clipboard20 Document Delivery Service21 Journal Original Article
References: 0 Reference Citations: 0 Review Citations: 223
Herein the author states “her genuine concern” about Wiles’s purported proof of Fermat’s last theorem \ref[A. Wiles, Ann. of Math. (2) 141 (1995), no. 3, 443–551; MR1333035 (96d:11071)24] which, after all, appeared in an “in-house publication in the Annals of Mathematics at Princeton”. Baica’s concerns seem to stem from a worry that results concerning elliptic curves “may not be equivalent to the result in the Euclidean geometry”. She backs up her concerns by noting that “there is a need to provide Galois’ connection from category theory”. Of course, she has no such worries about the validity of her own, Euclidean-algorithm-inspired, proof of Fermat’s last theorem.

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Matches for: MR=MR0429922
84
MR0429922 (55 #2931)
Hsu, Kuang Yau14
On decomposition of certain formal groups.
Tamkang J. Math.15 6 (1975), 69–78.
14L0516
Review in linked PDF17 Add citation to clipboard18 Document Delivery Service19 Journal Original Article
References: 0 Reference Citations: 0 Review Citations: 0
It is hard to imagine in a single paper such an accumulation of garbled English, unfinished sentences, undefined notions and notations, and mathematical nonsense. The author has apparently read a large number of books and papers on the subject, if one looks at his bibliography; but it is doubtful that he has understood any of them. He speaks blithely of elements of a formal group and claims to prove (!) that two formal groups having the same Lie algebra are isomorphic. One also hears of root systems of a commutative formal group, and of the union of its Borel subgroups, which belongs to the Lie algebra, etc. What is amazing to the reviewer is that such a thing was ever printed.

Reviewed by J. Dieudonne

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MR1786212 (2001f:11085)
D\c abrowski, Andrzej14(PL-SZCZ)15; Wieczorek, Ma\l gorzata16(PL-SZCZ)17
Families of elliptic curves with trivial Mordell-Weil group. (English. English summary)
Bull. Austral. Math. Soc.18 62 (2000), no. 2,19 303–306.
11G0520
Review in linked PDF21 Add citation to clipboard22 Document Delivery Service23 Journal Original Article
References: 0 Reference Citations: 225 Review Citations: 226
This paper contains barely a single correct statement. The first proposition is that an elliptic curve $y^2 = x^3 + A x + B$, with $A,B \in Z$, $A \geq 0$, cannot contain a rational torsion point of order 5 or 7. The proof is erroneous, and indeed the curve $y^2 = x^3 +2160 x + 170\,640$ contains the point $(-24,324)$ of order $5$, and the curve $y^2 = x^3 +206\,037 x + 80\,423\,334$ contains the point $(219, -11\,664)$ of order $7$. (The one-parameter families of curves possessing $5$- and $7$-torsion are misquoted from tables of Kubert, but the mathematical error lies elsewhere.) The second result (which is actually true) is given a proof containing the line “leads to the equation $x^3+ A x + B = 0$ with no real solution (when $A \geq |B| > 0$, of course)”, where the assertion is false, of course. The final example concerns the curve $y^2= x^3 + 5 x+1$ where “one easily checks that $E(Q)_{\rm tors}=Z/3Z \ (=\{(0,1),(0,-1),\infty\})$”. But adding $(0,1)$ to $(0,1)$ results in $(\frac{25}{4}, -\frac{133}{8})$, so the point $(0,1)$ is of infinite order!

Reviewed27 by Andrew Bremner28

4. MathPhys says:

I never knew that MR allows reviewers to be critical.