User j. h. s. - MathOverflow

A "couple" of questions on Gauss's mathematical diary

2012-01-25T21:17:10Z

Throughout my upbringing, I encountered the following annotations on Gauss's diary in several so-called accounts of the History of Mathematics:

"... A few of the entries indicate that the diary was a strictly private affair of its author's (sic). Thus for July 10, 1796, there is the entry

ΕΥΡΗΚΑ! num = Δ + Δ + Δ.

Translated , this echoes Archimedes' exultant "Eureka!" and states that every positive integer is the sum of three triangular numbers—such a number is one of the sequence 0, 1, 3, 6, 10, 15, ... where each (after 0) is of the form $\frac{1}{2}n(n+1)$, $n$ being a positive integer. Another way of saying the same thing is that every number of the form $8n+3$ is a sum of three odd squares... It is not easy to prove this from scratch.

Less intelligible is the cryptic entry for October 11, 1796, "Vicimus GEGAN." What dragon had Gauss conquered this time? Or what giant had he overcome on April 8, 1799, when he boxes REV. GALEN up in a neat rectangle? Although the meaning of these is lost forever the remaining 144 are for the most part clear enough." "

The preceding paragraphs have been quoted verbatim from J. Newman's The World of MATHEMATICS (Vol. I, pages 304-305) and the questions that I pose today were motivated from my recent spotting of [2]:

Why is there no mention whatsoever to the REV. GALEN inscription in either Klein's or Gray's work?
What is the reason that E. T. Bell expressed that Gauss had written the Vicimus GEGAN entry on October 11, 1796? According to Klein, Gray, and (even) the Wikipedians it was written on October 21, 1796. As far as I understand, Klein and Gray are just reporting the dates that appear on the original manuscript. Did Bell actually go over it?
Last but not least, is there a compendium out there of all known potential explanations to the Vicimus GEGAN enigma? The only ones whereof I have notice can be found on page 112 of [1]:

"... Following a suggestion of Schlesinger [Gauss, Werke, X.1, part 2, 29], Biermann ... proposed that GA stood for Geometricas, Arithmeticas, so reading GEGAN in reverse as Vicimus N[exum] A[rithmetico] G[eometrici cum] E[xspectationibus] G[eneralibus]. Schumann has since proposed other variants; including, for GA, (La) G(rangianae) A(nalysis)..."

Heartfelt thanks for your comments, reading suggestions, and replies.

References

J. J. Gray. " A commentary on Gauss's mathematical diary, 1796-1814, with an English translation". Expo. Math. 2 (1984), 97-130.
F. Klein. "Gauß' wissenschaftliches Tagebuch 1796–1814". Math. Ann. 57 (1903), 1–34.
M. Perero. Historia e Historias de Matemáticas. Grupo Editorial Iberoamérica, 1994, pág. 40.

Answer by J. H. S. for The four squares theorem from the Gauss-Legendre three squares theorem

2013-03-16T16:35:50Z

Also, the Gauß/Legendre theorem in question implies the following (improved) version of the four-squares theorem:

Every positive integer is the sum of four POSITIVE squares unless it belongs to the set

$A \cup B$

where

$A$ $=$ {$1, 3, 5, 9, 11, 17, 29, 41$}

and

$B$ $=$ {$2\cdot 4^{m}: m \in \mathbb{Z}^{+}$} $\cup$ {$6\cdot 4^{m}: m \in \mathbb{Z}^{+}$} $\cup$ {$14\cdot 4^{m}: m \in \mathbb{Z}^{+}$}.

Proof (d'après Prof. J. H. Conway in "The sensual quadratic form" [page 140]). By Gauß/Legendre and the well-known result on numbers that can be written as a sum of two squares it follows that every natural number of the form $8k+3$ (or $8k+6$) is the sum of three positive squares. Multiplying by $4$, we see that the same conclusion applies to natural numbers of the forms $32k+12$ and $32k+24$. Then, one can show that any integer $>49$ which is not a multiple of $8$ is the sum of four positive integers by subtracting an square so as to obtain a number of one of the aforementioned forms (for instance, from a number of the form $8k+2$ subtract $2^{2}$ in order to obtain a number of the form $8k+6$...). The proof is completed by checking the numbers up to $49$ and verifying that a number $n$ divisible by $8$ is the sum of four positive squares only if $\frac{n}{4}$ is. QED.

Answer by J. H. S. for Awfully sophisticated proof for simple facts

2013-02-05T08:18:33Z

I think that the following proof of the fact that every subgroup of index $2$ of a given group is normal might count too. When I first came up with it (sometime during my sophomore year), I believed that I had just found the entrance to a royal road to mathematics.

Let $H\leq G$ be such that $[G:H]=2$. We'll prove that every right coset of $H$ is equal to a left coset of $H$.

Since $[G:H]=2$, $G$ is both the union of two disjoint right cosets of $H$ and the union of two disjoint left cosets of $H$. Let us suppose that $G=He \cup Hx = eH \cup yH$ where $x,y\in G\setminus H$ and $e$ denotes the identity element of $G$. According to standard lore regarding the symmetric difference of sets,

$He \cup Hx = He \triangle Hx \triangle (He \cap Hx) = He \triangle Hx \triangle \emptyset = H \triangle (Hx\triangle \emptyset) = H\triangle Hx$

and

$eH \cup yH = eH \triangle yH \triangle (eH \cap yH) = eH \triangle yH \triangle \emptyset = H \triangle (yH \triangle \emptyset) = H \triangle yH$.

Therefore, $H\triangle Hx = H\triangle yH$. Canceling $H$ on both sides of the latter equality—which is perfectly valid given that $(2^G, \triangle)$ is a group—we conclude that $Hx=yH$. Done.

If you consider that the prior argument doesn't qualify as awfully sophisticated, there is still another fancy way to derive the result in question. As a consequence of P. Hall's famous marriage theorem, M. Hall proves in Theorem 5.1.7 of his Combinatorial Theory that if $H$ is a finite index subgroup of $G$, there exists a set of elements that are simultaneously representatives for the right cosets of $H$ and the left cosets of $H$ (once he's proven the said theorem, he adds: "Simultaneous right-and-left coset representatives exist for a subgroup in a variety of other circumstances. This problem has been investigated by Ore 1."). In the case $[G:H]=2$, this implies at once that every right coset of $H$ is equal to a left coset of $H$ and we are done...

Last but not least, $[G:H]=2 \Rightarrow H \trianglelefteq G$ in the case when $|G|<\infty$ can also be seen a consequence of the well-known fact according to which any subgroup of a finite group whose index is equal to the smallest prime that divides the order of the group is of necessity a normal subgroup of the group. B. R. Gelbaum showcases in one of his books an action-free proof of this fact. He attributes both the fact and the action-free proof to Ernst G. Straus. Does any of you know on what grounds he did so? I have a Xerox copy of the relevant page in the book here. This is exactly what Gelbaum writes therein:

At some time in the early 1940s Ernst G. Straus, sitting in a group theory class, saw the proof of the ... result [i.e., $[G:H]=2 \Rightarrow H \trianglelefteq G$] ... and immediately conjectured (and proved that night): ... IF G:H [sic] IS THE SMALLEST PRIME DIVISOR P of #(G) THEN H IS A NORMAL SUBGROUP.

P.S. The Galois-theoretic proof given by Matthias Künzer is just fabulous!

Answer by J. H. S. for trigonometric identity needed for sums involving secants

2013-02-19T00:57:23Z

I think that the following article of our very own Roberto Bosch Cabrera might come in handy:

https://www.awesomemath.org/wp-content/uploads/reflections/2008_5/article_2.pdf

Specifically, you should take a look at pages 1 & 2 of that note.

Answer by J. H. S. for "Oldest" bug in computer algebra system?

2010-09-16T01:38:24Z

According to Wolfram Alpha and the tables in [2], $\pi(10^{10}) = 455, 052, 511$. Nevertheless, in Zagier's paper we find that $\pi(10^{10}) = 455, 052, 512$.

Wonder whether someone has already noted this discrepancy between the sources elsewhere. Naturally, the discrepancy implies the existence of a bug in either the routines of Zagier or in WA's implementation of the prime counting function. I don't think that it's only a typo in Zagier' note because, if memory serves me right, there are some other texts in the literature that endorse the computations of Zagier (for instance, see [1, page 7].).

References

[1] A. E. Ingham. The distribution of prime numbers. Cambridge Mathematical Library.

[2] H. Riesel. Prime Numbers and Computer Methods for Factorization. Second Edition, 1994, Birkhäuser.

[3] D. Zagier. "The first 50 million primes". Math. Intelligencer, 0 (1977).

Answer by J. H. S. for When factors may be cancelled in homeomorphic products?

2010-05-30T02:33:12Z

The question's been studied in the category of groups, too. R. Hirshon proved in [1] that finite groups can always be cancelled in direct products.

Hirshon mentions some other sufficient conditions for the cancellation theorem to hold. For instance, he says that in the treatise by L. Fuchs there is a proof of the fact that infinite cyclic abelian groups can also be cancelled (provided that either $B$ or $C$ is a commutative group).

References

[1] R. Hirshon, On Cancellation in Groups, Amer. Math. Monthly. 76 (9) (1969), pp. 1037-1039.

Answer by J. H. S. for The set of orders of elements in a group

2013-01-16T18:37:30Z

For every fixed $n \in \mathbb{N}$, Rolf Brandl and Shi Wujie gave in Finite groups whose elements are consecutive integers (Journal of Algebra, 143, 388-400 (1991).) a complete classification of finite groups whose spectrum is $\{1,2,\ldots,n\}$. A particularly appealing spin-off of their study is the following one:

Let $i$ be a positive integer greater than $8$. There is no finite group $G$ whose spectrum is $\{1,2,\ldots,i\}$.

Answer by J. H. S. for Non-rigorous reasoning in rigorous mathematics

2012-12-01T01:10:30Z

I think that the following "derivation" of the Prime Number Theorem from the well-known identity

$\sum_{d|n}\Lambda(d) = \log n$

is a particularly prominent example of what you are asking. Indeed, it follows from the said identity that

$\sum_{n \leq x} \sum_{d|n} \Lambda(n) = \sum_{d\leq x} \Lambda(d)\sum_{n \leq x, d|n} 1 = \sum_{d\leq x}\Lambda(d)\lfloor \frac{x}{d}\rfloor$

and whence,

$\sum_{n \leq x}\Lambda(n)\lfloor \frac{x}{n} \rfloor = \sum_{n \leq x} \log n \sim x \log x$.

Now if we replaced the $\lfloor x/n \rfloor$ in the previous line by $x/n$, we would get

$\sum_{n\leq x}\frac{\Lambda(n)}{n} \sim \log x \sim \sum_{n \leq x}\frac{1}{n}$.

This might lead us to ascertain that the function $\Lambda$ of von Mangoldt behaves in the average like the arithmetical function that is identically equal to $1$, thus

$\psi(x):=\sum_{n\leq x} \Lambda(n)\sim x.$ (Voilà!)

As to the formal version of the preceding argument you may want to take a look at sections 9.9 through 9.12 of [2]. You are to find there a proof of the Prime Number Theorem (presumably due to Ingham) based on the estimate

$\sum_{n \leq x} \psi(\frac{x}{n}) = x\log x - x+ O(\log x), \quad x \geq 1.$

According to Prof. Balanzario (see [1, page 59]): "This demonstration ... is the correct version of our heuristic reasoning [given above]."

References

[1] E. P. Balanzario. Breviario de Teoría Analítica de los Números. SMM, México, 2003.

[2] W. Rudin. Functional Analysis. Tata McGraw Hill Publishing Company Ltd., 1974.

Answer by J. H. S. for For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.

2012-11-20T00:40:51Z

As a matter of fact, P. L. Chebyshev knew already that for any $\epsilon > \frac{1}{5}$, there exists an $n(\epsilon) \in \mathbb{N}$ such that for all $n\geq n(\epsilon),$

$\pi((1+\epsilon)n)-\pi(n)>0.$

In [2], one can find a short report on the problem of determining the smallest $n(\epsilon)$ explicitly once that $\epsilon$ has been fixed.

References

[1] P. L. Chebyshev. Mémoire sur les nombres premiers. Mémoires de l'Acad. Imp. Sci. de St. Pétersbourg, VII, 1850.

[2] H. Harborth & A. Kemnitz. Calculations for Bertrand's Postulate. Mathematics Magazine, 54 (1), pp. 33-34.

Answer by J. H. S. for FLT from Mochizuki's proof of abc

2012-09-27T19:21:06Z

Let us suppose that $x^{n}+y^{n}= z^{n}$ with $x, y,$ and $z$ relatively prime. By the abc conjecture, $|x^{n}|\ll |xyz|^{1+\epsilon}$, $|y^{n}|\ll |xyz|^{1+\epsilon}$ and $|z^{n}|\ll |xyz|^{1+\epsilon}$. Therefore, $|xyz|^{n}\ll |xyz|^{3+\epsilon}$ which implies that for $|xyz|>1$, $n$ is bounded. So, what we actually have is a proof of an asymptotic version of FLT. Nevertheless, if we have explicit information regarding the constant in the abc conjecture, we could determine explicit bounds for $n$.

Answer by J. H. S. for Chebyshev's approach to the distribution of primes

2010-05-29T08:54:37Z

Erdős and Diamond proved in [1] that Chebyshev could have achieved sharper bounds for the asymptotic behavior of the prime counting function. Nevertheless, their proof does not shed any light on the first question that you posed because they took the PNT for granted throughout their note.

References:

[1] H. G. Diamond; P. Erdős. On sharp elementary prime number estimates, Enseign. Math. (2) 26 (1980) 313-321.

Answer by J. H. S. for Titles composed entirely of math symbols

2011-11-20T08:31:48Z

Professor Luca and his co-authors are surely fond of this kind of titles:

F. Luca & B. de Weger, $\sigma_k(F_m)=F_n$. New Zealand J. Math. 40 (2010), 1–13.
F. Luca & F. Nicolae, $\phi(F_n)=F_m$. Integers 9 (2009), A30, 375–400.
F. Luca & M. Mignotte, $\phi(F_{11})=88$. Divulg. Mat. 14 (2006), no. 2, 101–106.
F. Luca & P. Stănică, $F_1F_2F_3F_4F_5F_6F_8F_{10}F_{12}=11!$. Port. Math. (N.S.) 63 (2006), no. 3, 251–260.

Answer by J. H. S. for Nontrivial question about fibonacci numbers?

2010-01-16T07:28:55Z

Here you have some of the coolest ones I have heard of:

1) Let $a$ be a positive integer. Then $a$ is a Fibonacci number if and only if at least one member of the set {$5a^{2}-4, 5a^{2}+4$} is a perfect square.

I think the result is original with Prof. Ira Gessel.

2) Let $\phi$ denote the Euler totient function. Prove that $\phi(F_{n}) \equiv 0 \pmod{4}$ if $n \geq 5$.

The proof consists of an unexpected application of Lagrange's theorem in Group Theory. Guess there are some other ways to prove it, but that approach will always remain my cup of tea. The problem was posed and solved in the Monthly in the 70's (if my memory serves me right). Look for all entries by Clark Kimberling in that magazine and you'll surely find it.

3) Can you find $(a,b,c) \in \mathbb{N}^{3}$ such that $ 2 < a < b < c$ and $F_{a} \cdot F_{b} = F_{c}$?

This problem would be trivial if instead of the $\cdot$ we had placed a plus sign there. In any case, there is no need to panic with this proposal. All you need to recall is the corresponding primitive divisor theorem.

4) Ben Linowitz mentioned above a beautiful result by Professor Florian Luca, namely:

There aren't any perfect numbers in the Fibonacci sequence.

I read the paper in my junior year and I didn't find it that hard to follow. The easy part of this cute note resides in the proof of the fact that there are no even perfect numbers in the Fibonacci sequence. Guess this result is interesting enough to deserve consideration in those lectures that you intend to give. If this proposal is not exactly your idea of excitement, you can take a look at some of the other papers by Professor Florian. He writes a lot about recurrence sequences. Another theorem of his, closely related with the subject matter of this discussion, ascertains that

There is no non-abelian finite simple group whose order is a Fibonacci number.

5) Last but not least... Prove that the sequence {$F_{n+1}/F_{n}$}$_{n \in \mathbb{N}}$ converges and use this fact to derive the continued fraction development for the golden ratio.

This one should be well-known, yet it would be nice to see what your students come up with...

Added (Nov 20/2010) I've just noticed that the Fibonacci Assn. has made available the articles published in The Fibonacci Quarterly between 1963 and 2003. I'm sure you will find plenty of additional material among those files that they have so generously released for our enjoyment. For instance, the seminal paper by J. H. E. Cohn that K. Buzzard mentions below can be found here.

Answer by J. H. S. for Most interesting mathematics mistake?

2009-12-15T20:46:09Z

In chapter 3 of What Is Mathematics, Really? (pages 43-45), Prof. Hersh writes:

How is it possible that mistakes occur in mathematics?

René Descartes's Method was so clear, he said, a mistake could only happen by inadvertence. Yet, ... his Géométrie contains conceptual mistakes about three-dimensional space.

Henri Poincaré said it was strange that mistakes happen in mathematics, since mathematics is just sound reasoning, such as anyone in his right mind follows. His explanation was memory lapse—there are only so many things we can keep in mind at once.

Wittgenstein said that mathematics could be characterized as the subject where it's possible to make mistakes. (Actually, it's not just possible, it's inevitable.) The very notion of a mistake presupposes that there is right and wrong independent of what we think, which is what makes mathematics mathematics. We mathematicians make mistakes, even important ones, even in famous papers that have been around for years.

Philip J. Davis displays an imposing collection of errors, with some famous names. His article shows that mistakes aren't uncommon. It shows that mathematical knowledge is fallible, like other knowledge.

...

Some mistakes come from keeping old assumptions in a new context.

Infinite dimensionl space is just like finite dimensional space—except for one or two properties, which are entirely different.

...

Riemann stated and used what he called "Dirichlet's principle" incorrectly [when trying to prove his mapping theorem].

Julius König and David Hilbert each thought he had proven the continuum hypothesis. (Decades later, it was proved undecidable by Kurt Gödel and Paul Cohen.)

Sometimes mathematicians try to give a complete classification of an object of interest. It's a mistake to claim a complete classification while leaving out several cases. That's what happened, first to Descartes, then to Newton, in their attempts to classify cubic curves (Boyer). [cf. this annotation by Peter Shor.]

Is a gap in a proof a mistake? Newton found the speed of a falling stone by dividing 0/0. Berkeley called him to account for bad algebra, but admitted Newton had the right answer... Mistake or not?

...

"The mistakes of a great mathematician are worth more than the correctness of a mediocrity." I've heard those words more than once. Explicating this thought would tell something about the nature of mathematics. For most academic philosopher of mathematics, this remark has nothing to do with mathematics or the philosophy of mathematics. Mathematics for them is indubitable—rigorous deductions from premises. If you made a mistake, your deduction wasn't rigorous, By definition, then, it wasn't mathematics!

So the brilliant, fruitful mistakes of Newton, Euler, and Riemann, weren't mathematics, and needn't be considered by the philosopher of mathematics.

Riemann's incorrect statement of Dirichlet's principle was corrected, implemented, and flowered into the calculus of variations. On the other hand, thousands of correct theorems are published every week. Most lead nowhere.

A famous oversight of Euclid and his students (don't call it a mistake) was neglecting the relation of "between-ness" of points on a line. This relation was used implicitly by Euclid in 300 B.C. It was recognized explicitly by Moritz Pasch over 2,000 years later, in 1882. For two millennia, mathematicians and philosophers accepted reasoning that they later rejected.

Can we be sure that we, unlike our predecessors, are not overlooking big gaps? We can't. Our mathematics can't be certain.

The reference to the said article by Philip J. Davis is:

Fidelity in mathematical discourse: Is one and one really two? Amer. Math. Monthly 79 (1972), 252–263.

From that article, I find particularly amusing the following couple of paragraphs from page 262:

There is a book entitled Erreurs de Mathématiciens, published by Maurice Lecat in 1935 in Brussels. This book contains more than 130 pages of errors committed by mathematicians of the first and second rank from antiquity to about 1900.There are parallel columns listing the mathematician, the place where his error occurs, the man who discovers the error, and the place where the error is discussed. For example, J. J. Sylvester committed an error in "On the Relation between the Minor Determinant of Linearly Equivalent Quadratic Factors", Philos. Mag., (1851) pp. 295-305. This error was corrected by H. E. Baker in the Collected Papers of Sylvester, Vol. I, pp. 647-650.

...

A mathematical error of international significance may occur every twenty years or so. By this I mean the conjunction a mathematician of great reputation and a problem of great notoriety. Such a conjunction occurred around 1945 when H. Rademacher thought he had solved the Riemann Hypothesis. There was a report in Time magazine.

On order of subgroups in abelian groups

2010-03-23T06:49:14Z

I wonder whether any of you guys has already read the homonymous note by R. Beals in the December 2009 issue of the Monthly.

If so, would you be so kind as to let me know about the main ideas in Beal's approach? As you know, the whole point of his note is to present a solution to the following exercise in Herstein's Topics in Algebra:

Let G be an abelian group having subgroups of order m and n. Prove that G also possesses a subgroup of order lcm(m, n).

The funny thing about this proposal is that in subsequent editions of his book, Prof. Yitz would proclaim that he himself didn't have a solution using the authorized tools. Besides, he even went on to saying: "I've had more correspondence about this problem than about any other point in the whole book.".

Being aware of some of the history behind this little pearl, I'd really like to know what it is that Beals came up with. Is his approach crystal-clear? Is it somehow related to the standard attack of proving it first for the case gcd(m, n)=1?

Thanks in advance for you insightful replies.

P.S. The local library is the only access that I have to the literature. Unfortunately, they don't subscribe to any of the MAA periodicals.

On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...

2009-11-08T04:48:03Z

It is well-known that

A: The series of the reciprocals of the primes diverges

My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.

Property A tells us that the primes are a rather fat subset of $\mathbb{N}$. Is there a way to define a topology on $\mathbb{N}$ such that every dense subset of $\mathbb{N}$ (under this topology) corresponds to a fat subset of the natural numbers?

What do you think about this?

Gauss sums over multiplicative subgroups

2012-05-17T17:40:06Z

Hello,

Is anyone here aware of a well-motivated exposition of the Bourgain-Glibichuk-Konyagin estimate for exponential sums (or Gauss sums) over multiplicative subgroups? If any of you has a write-up on the subject, I would be more than glad to have an opportunity to take a look at it.

Thank you very much!

Character sums: reference request

2012-04-20T16:42:18Z

This one will be quick...

Wonder if anybody knows or remembers the title of the paper in which Karatsuba introduced his approach at Burgess's bound on character sums.

Thanks for your support.

EDIT. It might have been in "Sums of characters and primitive roots in finite fields". Since this appeared in Russian in Dokl. Akad. Nauk. SSSR, would any of you guys be so kind as to provide us with a review of the translation that, according to MathSciNet, appeared in Soviet Math. Dokl. 9 (1968), 755–757? Thanks again.

ADDED. I've been trying to get a copy of the corresponding issue of the Soviet Mathematics Doklady in the libraries to which I have access (real or virtual). It all has been to no avail. Does any of you know if there is a possibility to get a copy of it through the AMS? I recently took at look at ams.org to find out, but I didn't find info regarding the acquisition of back issues of the said journal.

An anecdote by R. Schmidt

2010-11-06T00:53:01Z

Did any of you guys ever read those lines by Roland Schmidt where he talked about the terseness of articles in Group Theory in the days prior to the conclusion of the classification of the finite simple groups?

The complete story might have appeared in the Notices of the AMS, a similar publication, or an online recollection of group-theoretical anecdotes, but I'm not sure about it. All I can recall is that, at some point of his write-up, R. Schmidt brought up that famed Group Theory Year in UC (1960-1961) and a hilarious exploit by J. H. Conway and David Wales from the epoch.

I really hope that somebody out there has read it recently and can tell me where I can find it again.

Thanks.

Added (Nov 8/2010) The said Conway incident went something like this: "one day he (Professor Freese, as you can read in the first reply to the thread) found Conway and Wales working with some relations and generators. Conway and Wales said they wanted to see if it was a group what they had there. Professor Freese could not help but ask

How could it not be?,

to which, they immediately replied

It could be infinite!"

Added (Nov 19/2010) I'm sure that in that elusive note one could also find the following story:

"A typical journal page(?) in those days would look something like this

Theorem. All groups are finite.

Proof. Deny."

I also remember that I didn't really get the joke at the time, but that's another story...

Does anybody here remember reading something like this some time ago? C'mon fellas! I know I could not have been the only one in here to catch it back in the day.

Answer by J. H. S. for Gauss's views on pure mathematics

2012-04-23T00:56:08Z

I consider that the following lines of Hardy might help the OP to form a definite idea as to the accuracy of such an asseveration:

"... I must deal with a misconception. It is sometimes suggested that pure mathematicians glory in the uselessness of their work*, and make it a boast that it has no practical applications. The imputation is usually based on an incautious saying attributed to Gauss, to the effect that, if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme uselessness, the queen of mathematics—I have never been able to find an exact quotation. I am sure that Gauss’s saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.

*I have been accused of taking this view myself. I once said that ‘a science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life’, and this sentence, written in 1915, has been quoted (for or against me) several times. It was of course a conscious rhetorical flourish, though one perhaps excusable at the time when it was written. "

P.S. a) The emphasis is mine. b) The excerpt comes from the last paragraph of section 21 of Hardy's Apology.

A question regarding a claim of V. I. Arnold

2010-04-08T06:31:39Z

In his Huygens and Barrow, Newton and Hooke, Arnold mentions a notorious teaser that, in his opinion, modern mathematicians are not capable of solving quickly. Then, he adds that the exception that proved the rule in this case of his was the German mathematician G. Faltings.

My question is whether any of you knows the complete story behind those lines in Arnold's book. I mean, did Arnold pose the problem somewhere (Квант Magazine?) and G. Faltings was the only one that submitted a solution after his own heart? Is the previous conjecture totally unrelated to the actual development of things?

I thank you in advance for you insightful replies.

P.S. Guess it'd be just great for MO if we somehow managed to get first-hand information about this query of mine.

Answer by J. H. S. for German mathematical terms like "Nullstellensatz"

2012-03-30T22:41:02Z

I would like to mention a handful of examples that may be considered passé nowadays, but were prominent at some point in time.

schlicht: I dare to address this one again because I consider that the feedback in the comments below Gottfried's entry is kind of misleading. About this one, Boas says that (see [1, page 97]):

«... When I was an undergraduate, there was no regular colloquium Harvard, but there was a Mathematical Club, whose meeting were regularly attended by faculty. Once somebody gave a talk on schlicht functions. After the talk, Julian Lowell Coolidge asked plaintively whether there was an English word for 'schlicht'. Osgood replied, "Well, you could call them univalent functions, and everybody would know that you meant 'schlicht'". You need to know that Osgood had been trained in Germany, wrote his treatise on complex analysis in German, and was apt to tell German jokes to his classes. »

limes: That's right... It was not a typo in Ahlfors's text on Complex Analysis. I recently came across this one in another book, but I just can't recall which one it was.
eine Drehstreckung: Tristan Needham recalls this one when he apologizes for the coinage of the term 'amplitwist'. More specifically, he writes

«... To the expert reader I would like to apologize for having invented the word 'amplitwist' ... as a synonym (more or less) for 'derivative', as well the component terms 'amplification' and 'twist'. I can only say that the need for some such terminology was forced on me in the classroom: if you try teaching the ideas in this book without using such language, I think you will quickly discover what I mean! Incidentally, a precedence argument in defence (sic) of 'amplitwist' might be that a similar term was coined by the older German school of Klein, Bieberbach, et al. They spoke of 'eine Dhrestreckung', from 'drehen' (to twist) and 'strecken' (to stretch). »

Last but not least, in several works of old (z.B., Perron's Die Lehre von den Kettenbrüchen, Knopp's Theory and Application of Infinite series, Khinchin's Continued Fractions), there appears the following notation for general continued fractions:

$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$

Guess what the $\mathrm{K}$ stands for...

References

[1] Lion Hunting & Other Mathematical Pursuits: A Collection of Mathematics, Verse and Stories by Ralph P. Boas Jr.

Totient function inequality

2010-10-23T17:59:06Z

Does any of you know if the inequality

$\displaystyle \frac{\phi(\sigma(n))}{n} < (\log \log \log n)^{-1/2}$

is true for all $n$ sufficiently large?

I remember reading something to that effect sometime ago, but a detailed statement of the result eludes me now and that's the reason that you find me asking it here.

I thank you all for your replies.

Answer by J. H. S. for Nontrivial question about fibonacci numbers?

2010-01-17T00:03:18Z

The relation used by Matiyasevich (Matijasevich, Матиясевич), the one to which N. Takenov alluded above/below, is the following:

If $F_{n}^{2}|F_{m}$ then $F_{n}|m$ ...... (20)

In the 1992 Fall issue of the Intelligencer there was a note by Matiyasevich where he explained, among other things, the importance of that relation on his work concerning Hilbert's tenth problem. Here you have an excerpt from that note:

"It is not difficult to prove this remarkable property of Fibonacci numbers after it has been stated, but it seems that this beautiful fact was not discovered until 1969. My original proof of (20) was based on a theorem proved by the Soviet mathematician N. Vorob'ev in 1942 but published only in the third argumented (sic) edition of his popular book [on the Fibonacci sequence]... I studied the new edition of Vorob'ev book in the summer of 1969 and that theorem attracted my attention at once. I did not deduce (20) at that time, but after I read Julia Robinson's paper I immediately saw that Vorob'ev's theorem could be very useful. Julia Robinson did not see the third edition of Vorob'ev's book until she received a copy from me in 1970. Who can tell what would have happened if Vorob'ev had included his theorem in the first edition of his book? Perhaps, Hilbert's tenth problem would have been "unsolved" a decade earlier!"

Answer by J. H. S. for Interesting applications of the Pigeon-hole Principle

2009-12-15T20:39:10Z

M. A. Lukomskaya's proof of van der Waerden's theorem on arithmetic progressions is a remarkable application of the Schubfachprinzip and induction.

Fermat numbers and the infinitude of primes

2010-04-23T08:24:57Z

Wonder whether any of you guys know why it is that the proof of the infinitude of primes that is based on the coprimality of any pair of (distinct) Fermat numbers is commonly attributed to Pólya.

In the first paragraph of this letter from Golbach to Euler there is already an argument along those lines, but since documents crediting it to Professor Pólya are not rare out there, it seems like it's passed unnoticed by a nonzero number of persons.

So, what do you think about this? It's not like Fermat numbers are essential to the proof or that there are no other demonstrations of the result... It's just that I'd really like to know about the origins of this discrepancy between the sources.

On figurate numbers

2011-06-07T04:53:42Z

Do you know a text where I can find a definition of polygonal number that is both geometrically and operationally sound?

I've basically seen two ways in which this topic is approached in the literature. For instance, in the celebrated historical opus by Dickson (volume II, chapter 1), the ancient treatment by a Hypsicles is mentioned:

"If there are as many numbers as we please beginning with one and increasing by the same common difference, then when the common difference is 1, the sum of all the terms is a triangular number; when 2, a square..."

Clearly enough, the above definition suits just right the needs of those persons interested in determining explicitly the $k$-element of the sequence of $n$-gonal numbers, but it gives no clue about the possibility of representing geometrically all those sequences.

That it is actually an issue becomes apparent when one notices that just about every presentation of this topic begins by providing diagrams that illustrate, to some extent, the process by which the first $n$-gonal numbers are built (they will typically focus their attention in the cases $n=3$, $n=4$, and $n=5$) and then, rush to introduce the general formulas without mentioning the way in which the corresponding patterns are supposed to be preserved by them (e.g., M. B. Nathanson. A short proof of Cauchy' polygonal number theorem. Proc. Amer. Math. Soc. 99 (1987) no. 1, 22-24.)... As it turns out, what it's at stake here is the possibility of a definition that reconciles the geometry inherent to these sequences and the ease of manipulation offered by an approach akin to that of Hypsicles.

Prof. Murty's B. Sc. Thesis

2010-09-27T05:57:11Z

Can any of you guys help me to find out if there is a retrodigitized copy of M. Ram Murty's 1976 thesis available on the online database of Carleton University Library?

I really hope this question is appropriate for MO. Thanks in advance for your replies.

References

Carleton University Library
M. Ram Murty. On the existence of euclidean proofs of Dirichlet's theorem on primes in arithmetic progressions. B. Sc. Thesis, 1976, Carleton University Library.

UPDATE (approx. one year later): I contacted the person in charge of the theses at the university library, she said there wasn't a copy of the said work in the central library and contacted the Math Dept. to see if they had a copy of it, but they answered in the negative. Is there a rationale for this situation?

The Wiener-Ikehara approach to the PNT

2010-08-21T17:03:31Z

Was providing an alternative proof of the PNT one of the main impulses that led to the discovery of the tauberian theorem of Wiener and Ikehara or the other way around?

In any case, does any of you people know who was the first individual to realize that, in order to prove the PNT, all one needs to have is the non-vanishing of the Riemann zeta function on the line $\sigma =1$ plus a tauberian theorem of the Wiener-Ikehara persuasion?

I thank you in advance for your insightful replies.

References

[1] P. T. Bateman; H. G. Diamond. A hundred years of prime numbers. 103 9 (1996), pp. 729-741.

Answer by J. H. S. for Famous mathematical quotes

2009-12-08T19:55:49Z

Here you have one of my all-time favorites:

" The ultimate goal of Mathematics is to eliminate any need for intelligent thought."

R. L. Graham (?)

Can any of you guys tell me where that quote first appeared? Same thing for the quote of Atiyah entered by Petrunin.

Comment by J. H. S.

2013-02-10T18:34:27Z

The link is broken... Does anybody know if the file was in fact an article of Alon?

Comment by J. H. S.

2012-12-04T10:33:36Z

While we are it, why should we even call them "groups"? C'mon! As Juliet would say: "What's in a name? That which we call a rose by any other name would smell as sweet."

Comment by J. H. S.

2012-11-22T22:39:26Z

@Igor Belegradek: Why don't you take a look at this thread? mathoverflow.net/questions/114190/schlicht-domain

Comment by J. H. S.

2012-11-22T22:33:42Z

More on the "schlicht property": mathoverflow.net/questions/62218/…

Comment by J. H. S.

2012-11-06T06:01:49Z

That pair doesn't give a solution of the equation, dear Lwins.

Comment by J. H. S.

2012-10-05T05:46:19Z

@quid: That was obviously a typo... Thanks.

Comment by J. H. S.

2012-05-02T05:53:13Z

The n-th term does not even go to 0.

Comment by J. H. S.

2012-04-21T02:32:46Z

Guess this might be a nice example of a borderline case on the copyright theme. Anyways, I've edited the paragraph that seemed to antagonize the audience. Voting to re-open.

Comment by J. H. S.

2012-04-14T06:16:11Z

Unfortunately, I don't. Anyway, thanks a lot for taking a look at my entry.

Comment by J. H. S.

2012-03-30T21:29:36Z

retro.seals.ch/digbib/…

Comment by J. H. S.

2012-03-26T18:10:13Z

Take a look at this thread: mathoverflow.net/questions/36299/…

Comment by J. H. S.

2012-03-23T09:35:53Z

That's not at all clear from the way he stated his question. Anyway, it seems to me that the thread will be closed in no time.

Comment by J. H. S.

2012-03-18T00:09:10Z

I really wonder what this answer has to do with my original question. Sounds to me like: A. What's your name? B. My best friend is C.

Comment by J. H. S.

2012-03-10T19:43:58Z

I was thinking of this "example", too. Nevertheless, I decided not to enter it because I arrived at the conclusion that Hilbert's problem was in some sense a number-theoretical one (a tough one at that). Anyway, I suppose that I'm misunderstanding something here. P.S. The second link doesn't work.

Comment by J. H. S.

2012-02-21T04:30:29Z

Are you sure about the trace-formula tag?