Possible title: Can you solve these mathematical challenges?

[Janitor]

Here are three questions. Feel free to answer any or all. Question #1 you can work out with some simple algebra. I will warn you that the other two questions are not ones you are likely to be able to answer just by straining your brain. You will probably only be able to answer #2 and #3 by drawing upon your knowledge of algebra and geometry, respectively. Good luck.

1. Label the end points of a line segment with A and B. Place a point C on the segment such that AC is longer than CB, and more particularly that the ratio of the length of AC to that of CB is the same as the ratio of AB to AC. Calculate the ratio AC/CB. Does the number look familiar to you? Bonus points for identifying which mathematician wrote on this topic: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.”

2. What ninth and final number x completes this set?

{-1, -2, -3, -7, -11, -19, -43, -67, x}

3. What is the pattern in this sequence?

infinity, five, six, three, three, three, three,... (threes continuing forever)

In other words, what sort of mathematical pondering might have led to this sequence, where the ordering of the terms comes out like this in a natural way.

I'll give the answers in a day or two if there are any remaining unanswered.

 

[cookiemonster]

#1 sounds golden...

cookiemonster

 

[Chen]

#1 is golden indeed and #2 is the ninth Heegner number...

 

[Janitor]

I'm impressed.

The answer to #1 is indeed the Golden Ratio, (1/2)[1+sqrt(5)]

The answer to #2 is the remaining Heegner number, -163. The Heegner numbers are the values within the square root for which imaginary quadratic fields are uniquely factorizable.

Hint on #3: it comes up when considering spaces of various dimension.

 

[Chen]

I found the answer for #3 online. Can't imagine anyone could figure it out withuot knowing about it first.

 

[Janitor]

Yes, I ordered the questions by decreasing liklihood that anybody would be able to answer them.

 

[cragwolf]

The answer to #2 is the remaining Heegner number, -163. The Heegner numbers are the values within the square root for which imaginary quadratic fields are uniquely factorizable.

This thread could become really cool if you (or someone else) could explain this in more detail, but without using too much mathemtical machinery, so that a physicist could understand it. :) What's the next Heegner number after -163? How do you actually calculate a Heegner number?

 

[Chen]

There are only nine such numbers, making -163 the last one.

 

[Janitor]

My own understanding is incomplete!

... if you (or someone else) could explain this in more detail... - cragwolf

I suspect a pretty thick book could be written on this topic. As an indicator of how deep it goes, consider that Karl F. Gauss (considered by some the greatest mathematician ever) himself could only conjecture it. A fellow named Heegner finally proved it decades later.

If a one-semester course were going to be taught on Heegner numbers, here are some of the things that the instructor might go over in the first couple of weeks, in order to get the class warmed up for the attack on Heegner numbers.

Groups, the definition of which most Forum members have down cold.

Rings. A ring can be viewed an abelian group under addition, along with an associative multiplication operation which distributes over the addition operation.

Commutative ring: a ring in which the multiplicative operation is commutative.

A field is a commutative ring in which every nonzero element has a multiplicative inverse.

Divisor of zero (others may call it “zero-divisor”): a nonzero element of a ring which, when multiplied by some nonzero element, gives the zero element.

Integral domain: a commutative ring with a unity element, i.e. a 1, and no divisors of zero.

In an integral domain, one says b divides a if a=bc for some c in the integral domain.

An element of an integral domain is called a unit if it divides 1, i.e. if the element has a multiplicative inverse.

An element b is called an associate of a if b=ua for some unit u.

A nonunit c is said to be irreducible if c=ab implies a or b is a unit.

An integral domain is called a unique factorization domain if every nonunit can be written uniquely (up to associates and order) as a product of irreducible elements.

Example of a unique factorization domain: the ring of integers under ordinary addition and multiplication. The units are 1 and -1. The only associates of an element n are n and -n. The irreducible elements are the prime numbers.

Another example, which brings us closer to where the Heegner numbers come in: The set {a+b*sqrt(13)}, where a and b are integers, is an integral domain. It is not a unique factorization domain, since for instance 4=2*2 but also 4=[3-sqrt(13)]*[-3-sort(13)].

The example immediately above is known as a real quadratic field. In general for a real quadratic field, a and b may be any rational numbers. If the square root of a negative number is used, the resulting field is known as an imaginary quadratic field. If in particular your square root is of -1, the field is called the Gaussian rationals.

I feel pretty good about the stuff I wrote above, though if I have bollixed something up, I hope someone will correct me. At this point the instructor will probably start in on Gauss’s class number idea. Maybe one of the math-type people can step in here and post something on that topic. All I can tell you for the moment is that when the square root is of a positive number, there are infinitely many fields that are said to have "class number 1." But when the square root is of a negative number, the class number can be shown to be 1 only for the nine values of the Heegner numbers within the square root. I have never seen the proof; I suspect it is lengthy and difficult.

 

[cragwolf]

Thanks very much Janitor! That sets it up very nicely. You know of any good books that treat this subject?

 

[Janitor]

Some time ago I checked out a library book called The Book of Numbers, co-written by John Horton Conway--a very talented mathematician. I happened to jot down some things from the book, including the Heegner numbers, and I was referring to my jottings when I posted question #2. I didn't jot anything about how they were derived, so there may not be much if anything in that book about the proof. I can't remember.

 

[matt grime]

Anything involving class number ought to be treated with respect and care. Though having said that all the prerequisites you list ought to be taught in the first semester of any reasonable undergraduate degree in maths. Reality and idealism, eh?

 

[Janitor]

Well, I've never been a math major.

So I guess by the time a student is taking a course on class number, it is expected that integral domains and so forth are such old-hat topics that the instructor would not even bother refreshing minds about them.

Answer to bonus question: Good old Euclid himself wrote that, back before anyone was using the term ‘Golden ratio’ or 'Golden section.'

Answer to number 3: the number of symmetrical, convex polytopes in 2, 3, 4, … dimensions of space. In 2 dimensions, there are an infinite number of regular, convex polygons, starting with the equilateral triangle, then the square, then the regular pentagon, etc. In 3 dimensions, they are called polyhedra, and there are the five platonic polyhedra. There is a well-known proof using graph theory that shows there can only be these five. In 4 dimensions, there are six regular polytopes, and then things get simpler in all higher dimensions, with only three regular polytopes in each. My source for this is at John Baez’s site:

http://math.ucr.edu/home/baez/platonic.html

 

[matt grime]

Good third question. It had me pondering.
And as for integral domain etc, they really are very basic definitions, but I'm an idealist like I say.

Here's my thinker then. Given the following sequence 1,2,4,8,16,32, find a none obvious way of continuing it involving circles and points on the perimeter and prove that it does not follow the obvious pattern. (proof can be non-rigorous)

 

[Janitor]

Wolfram’s Mathworld site says very mysteriously in its entry on Heegner numbers: "The Heegner numbers have a number of fascinating connections with amazing results in prime number theory. In particular, the j-function provides stunning connections between e, pi, and the algebraic integers." The j-function is what is called a modular function. Maybe Matt or someone could comment on this.

I once read an article (Scientific American, if I recall) on something called the "Monstrous Moonshine conjecture." John Conway was involved in that work. I just found this at a website on Richard Borcherds: "The moonshine conjecture predicts the existence of an intimate relationship between the monster group, the largest of the sporadic finite simple groups and the theory of modular functions… The monster group F was finally constructed by Griess and a simpler construction was given later by [John Horton] Conway…"

The monster group is the largest of the sporadic groups, which are one of the classes of finite simple groups. A huge amount of effort went into classifying the finite simple groups, and the final piece of the puzzle was put together in the 1970s, or maybe it was the 1980s. So many journal pages went into this project that the overall effort is known as "The Enormous Theorem." I wonder if anybody is actively trying to tighten things up enough to reduce the number of pages it takes to prove this?

 

[matt grime]

I can tell you a little about Moonshine and modular functions. haviing listened to Mackay talk about them, but it reall y feels like a rabbit out of a hat: a few of the calculated characters of the Monster group have (for simple reps obviously) strongly suggestive links with modular forms. In particular the degree of the smallest faithful rep of M is one more than the coefficient of degree 1 is some modular form, but no one quite knows why.

I believe Monster groups are these days best thought of as symmetry groups of certain (Leech?) lattices.

There are still people who do not think the CFSG is correct, just to warn you (class. fin. simple grps).

Modular functions and their galois representatives are something to do with Wiles's proof of FLT.

There's enough top set off a small army of cranks and such in there, and as I don't understand any of it I'll let others ask more questions on it but I won't be able to answer them, only point them in the right direction. (But it might be reasonably good motivation to learn some more number theory if I need to)

 

[Janitor]

Re Matt's challenge-

I take it that 64 may not be the next term! Will you give a hint in 24 hours?

ADDED: And thanks for the additional moonshine info. And I didn't know there was doubt about the conclusions of the Enormous Theorem.

 

[matt grime]

To be honest I don't know that 64 is not the next term, but that the construction I'm thinking of does diverge from the obvious sequence (I think the next term is 63 as it happens)

 

[quddusaliquddus]

I just popped into see what the big-guys are up to. Very impressive guys. My physics A-Level teacher used to tell us that his first love was mathematics. Is this true for more physicists than just my A-Level teacher? I get feeling this is the case.

 

[matt grime]

It's not 24 hours, but I thought I'd elaborate - I don't expect you to find the next terms in the sequence cos that's nasty, just show that it can't continue in the obvious fashion, for one of three reasons (that I know of, I'm sure there are more)

 

[Janitor]

I was goofing around with packing of a circle by circles of 1/2, 1/4, 1/8, ... the original circle diameter today, seeing if I could get a sequence that started out on track. I couldn't, though I tried two methods: discarding all previous generations of packing circles, or keeping previous generations and packing the smaller next generations of circles into the unused spaces. At least I knew I would be dealing with whole numbers, though!

I'm floundering on how to get whole numbers out of points on a circle's circumference. Once I hear the answer, I'll probably say, Now why didn't I think of that?

 

[matt grime]

put n (evenly spaced) points on the cricumference of the circle (n=>2) and join them all up and count the number of segments this divides the interior into. The first few numbers go 2,4,8,16... as it happens, and for no points you have 1 segement. (The fact that there is no number for 1 point is the first of the three reasons we can see that this pattern isn't all that it first appears.) So, I guess the question is can you find any reasons that the pattern can't *obviously* continue like this.

 

[Janitor]

Good one.

I just drew the cases for 2, 3, and 4 points and saw that they give 2, 4, and 8 sectors. I'll take your word on the cases of higher point number.

How much can you relax the conditions of it being a circle, and the point spacing being equal?

 

[matt grime]

Actually i got the statement worng, they should be not equally spaced points and apparently it breaks down when ti should be 31.

 

[Janitor]

I had access to CAD software earlier today. Since I had not yet seen Matt's "they should be not equally spaced points," I made them equally spaced. Here are my results:

Number of points Number of segments
2 ...... 2
3 ...... 4
4 ...... 8
5 ...... 16
6 ...... 30
7 ...... 57
8 ...... 88
9 ..... 163
10 ...... 240

It looks like the pattern for number of segments S is S=mp for even number of points p, and S=mp+1 for odd number of points, where m is an integer. In particular, for p=2, 3, 4,... 10, m is 1, 1, 2, 3, 5, 8, 11, 18, 24. (Looked like it was going to be the Fibonacci series there for a bit.)

Then I tried placing p equally spaced points around an ellipse with a 5:2 ratio of major axis to minor axis. I arbitrarily decided to put the first point at the end of a major axis. My results are:

Number of points Number of segments
2 ...... 2
3 ...... 4
4 ...... 8
5 ...... 16
6 ...... 30
7 ...... 57
8 ...... 92
9 ..... 163

I didn't have time to check the case of 10 points on the ellipse. It is interesting that after getting off track at p=8, the ellipse got back into step with the circle at p=9. I would have expected the number of segments in the ellipse to stay ahead of the number in a circle for all cases of point number 9 and up.

In the circular case, there are what seem to be some perfect meets between three lines at just a single point. If the spacing of the points around the circle were not even, these intersection points would presumably open up into triangles, and so the number of segments would become greater. I should have looked more carefully to check if that was in fact what was going on for the case of 8 points around an ellipse.