The danger of confusing cosets and numbers Friday, Aug 24 2012 

As I mentioned yesterday, I’d like to consider a proposed proof of the Goldbach Conjecture that has garnered some attention, at least some attention from people who ask me about things like the validity of proofs of the Goldbach Conjecture. I like this in particular because it illustrates how I look through some papers (those towards which I’m a bit skeptical) and it illustrates a problem I’ve seen before: switching between interpreting a number as an element of the integers and an element of \mathbb{Z}/n\mathbb{Z}. (There is a certain problem with this, in that although I ‘do number theory,’ were the conjecture proved it is almost certain that I would be not at all familiar with the methods of proof).
In particular, I’ll be looking at the 19 August 2012 preprint “The Goldbach’s conjecture proved” by Agostino Prastaro (the pdf is here).  The rest after the fold –

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Three number theory bits: One elementary, the 3-Goldbach, and the ABC conjecture Friday, Jun 15 2012 

I’ve come to realize that I’m always tempted to start my posts with “Recently, I’ve…” or “So and so gave me such and such a problem…” or “I happened across this on…” It is as if my middle school English teachers (all of whom were excellent) succeeded so well in forcing me to transition from one idea to the next that I can’t help it even today. But, my respect for my middle school teachers aside, I think I’m going to try to avoid that here, and just sort of jump in.

Firstly, as announced at Terry Tao’s Blog, two new polymath items are on the horizon.  There is a new polymath proposal at the polymath blog on the “Hot Spots Conjecture”, proposed by Chris Evans, and that has already expanded beyond the proposal post into its first research discussion post. (To prevent clutter and to maintain a certain level or organization, the discussion gets cut up into 100-comment size chunks or so, and someone summarizes some of the key points in the header each time. I think it’s a brilliant model). And the mini-polymath organized around the IMO will happen at the wiki starting on July 12.

Now, onto some number theory – (more…)

A new toy problem Sunday, Oct 23 2011 

First, a short math question from Peter:

Question: What is the coefficient of x^{12} in the simplified expression of (a-x)(b-x) \dots (z-x)?

I often hate these questions, but this one gave me a laugh. Perhaps it was just at the right time.

A police car passed me the other day with sirens wailing and I became reminded full on about the Doppler Effect. But the siren happened to agree with a song I was whistling to myself at the time, and this made me wonder – suppose we had a piece of music (or to start, a scale) that we wanted to hear, and we stood in the middle of a perfectly straight train track. Now suppose the train had on it a very loud (so that we could hear it no matter how far away it was) siren that always held the same pitch. If the train moved so that via the Doppler effect, we heard the song (or scale), what would its possible movements look like? How far away would it have to be to not run us over?

Some annoying things come up like the continuity of the velocity and pitch, so we might further specify that we have some sort of time interval. So we have a scale, and the note changes every second. And perhaps we want the train to have the exact right pitch at the start of every second (so that it would have constant acceleration, I believe – not so exciting). Or perhaps we are a bit looser, and demand only that the train hit the correct pitch each second. Or perhaps we let it have instantaneous acceleration – I haven’t played with the problem yet, so I don’t really know. I’m just throwing out the idea because I liked it, and perhaps I’ll play with it sometime soon.

Now, the reason I like it is because we can go up a level. Suppose we have a car instead, and we’re in an infinitely large, empty, parking lot (or perhaps not empty – that’d be interesting). Suppose the car had a siren that wailed a constant pitch, too. What do the possible paths of the car look like? How does one minimize the distance the car travels, or how far from us it gets, or how fast it must go (ok – this one isn’t as hard as the previous 2)? It’s more interesting, because there’s this whole other dimension thing going on.

And even better: what about a plane? I sit on the ground and a plane flies overhead. What do its paths look like?

All together, this sounds like there could be a reasonable approach to some aspect of this problem. Under the name, “Planes, trains, and automobiles” – or perhaps in order – “Trains, automobiles, and planes,” this could be a humorous and fun article for something like Mathematics Magazine or AMMonthly. Or it might be really hard. I don’t know – I haven’t played with it yet. I can only play with so many things at a time, after all.

The Collatz Conjecture – recent development? Monday, Jun 6 2011 

On his site, John D. Cook recently proliferated a paper by Gerhard Opfer that claimed to solve the Collatz Conjecture. The Collatz Conjecture is simple to state:

Collatz (or the 3n + 1 conjecture):
Starting at any number do the following: if n is even, divide by 2; if n is odd, multiply by 3 and add 1.
The conjecture states that no matter what positive integer you start at, you will end up at 1 (the so-called 1-4-2 loop).

At first, I had high hopes for the paper. It seems relatively well-written and was submitted to the Mathematics of Computation, a very respectable journal. I even sent out a brief email about the paper. But the paper is flawed. The problem, I think, can be succinctly summarized by the following: he relies on the assumption that starting with any number n_0, one will eventually hit a number that is less than n_0. When stated like this, it seems obvious that there is a problem, but he only relied on that one number (rather than the apparent infinite descent that could follow). The exact problem occurs with his ‘annihilation argument’ on page 11 of the pdf above. He more or less states that one can start at 1 and reach every number by doing a sort of reverse Collatz function (he’s actually a bit wittier than that), but does not prove it.

More commentary can be found on reddit, reddit again, and on math.SE (a question protected by Qiaocho Yuan – go him).

I use this as an intro to a sort of joke that goes around mathematician’s circles. A while back, Sean Carroll wrote up ‘The Alternative-Science Respectability Checklist,’ and it’s awesome. Find it here. It turns out that Scott Aaronson wrote up a similar article, inspired by Sean Carroll, that is titled “Ten Signs a Claimed Mathematical Breakthrough is Wrong.”

His inspiration was the time-old problem that simply stated problems encourage generations up people to attack them, and frequently to think that they have made progress. So he asks :

Suppose someone sends you a complicated solution to a famous decades-old math problem, like P vs. NP. How can you decide, in ten minutes or less, whether the solution is worth reading?

And thus his 10 signs were created. I happen to have heard a few people say that this most recent paper on the Collatz Conjecture only failed three: #6 (The paper jumps into technicalities without presenting a new idea), #8 (The paper wastes lots of space on standard material), and #10 (The techniques just seem too wimpy for the problem at hand). {though perhaps #8 is debateable – some say it’s related to a different convention of writing papers, but I don’t know about any of that}

In my experience, I rely mostly on #1 (it’s not written in \TeX), #4 (it conflicts with some impossibility result), and #7 (it doesn’t build on any previous work). But both of these articles are very funny, though not exactly precise nor entirely true.

A Question on Stoplights Sunday, Mar 27 2011 

Consider stopping at a traffic light: the normal way of things is that all the cars are packed tightly together. Decorum decrees that one waits for the car ahead to accelerate at a green light, lest accidents occur. But is that really necessary? What if instead, the cars left somewhere between a half and a full car length between them so that they could all accelerate at a green light (and still have enough time to slow down/stop if they accelerate too quickly)? Is this efficient or not? Is there any change in traffic flow?

My thinking is that in long lines, to have every car accelerate at the (almost) same time would yield faster traffic flow, but perhaps that’s misguided. But I’m thinking of doing a simple study of the idea, just to see what happens.

Happy weekend!

Containers of Water II Thursday, Mar 24 2011 

In a previous post, I considered the following two questions:

Questions
What sets S maximize |{\bf F}(S;p)| for various p?
What sets S maximize \lfloor {\bf F}(S; p) \rfloor for various p?

I then changed the first question, which I think is perhaps not so interesting, to the following:

What sets S maximize |{\bf F}(S;p)|_c, where |\cdot|_c denotes the largest connected interval of results?

Let’s explore a few cases to see what these answers might look like.

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Containers of Water: Maybe an interesting question. Tuesday, Mar 22 2011 

Consider the old middle-school type puzzle question: Can you measure 6 quarts of water using only a 4 quart bottle and a 9 quart bottle? Yes, you can, if you’re witty. Fill the 9 quart bottle. Then fill the 4 quart bottle from the 9 quart bottle, leaving 5 quarts in the 9-bottle. Empty the 4-bottle, and fill it again from the 9-bottle, leaving only 1 quart in the 9-bottle.  Again empty the 4-bottle, and pour the 1 quart from the 9-bottle into it. Now fill the 9-bottle one last time and pour into the 4-bottle until it’s full – at this time, the 4-bottle has 4 quarts and the 9 bottle has 6 quarts. Aha!

But consider the slightly broader question: how many values can you get? We see we can get 1, 4, 5, 6, and 9 already. We can clearly get 8 by filling the 4-bottle twice and pouring this bottle into the 9. With this, we could get 4 by filling the 4-bottle again and then pouring as much as possible into the 9-bottle when its filled with 8 – as this leaves 3 quarts in the 4-bottle. Finally, we can get 2 by taking 6 quarts and trying to pour it into an empty 4-bottle. (With 2, we get 7 by putting the 2 into the 4-bottle and then trying to pour a full 9-bottle into the 4-bottle). So we can get all numbers from 1 to 9. If we were really picky, we could note that we can then easily extend this to getting all numbers from 0 to 13 quarts, albeit not all in one container.

If you go through these, it turns out it is possible to get any number between 0 and 13 quarts with a 4-bottle and a 9-bottle in at most 10 pours, where a pour includes filling a bottle from the water source (presumably infinite), pouring from one bottle into another, or emptying the water into the water source. Now I propose the following question:

Given only 2 bottles (of an integer size) and up to 10 pours, what is the largest N s.t. we can measure out 0, … , N quarts (inclusive)?

As is natural, let’s extend this a bit further. For any subset of the natural numbers S and for a number of pours p, define

{\bf F}(S; p) := the set R of possible results after using at most p pours from containers of sizes in S

For example, we saw above that {\bf F}(4,9;10) = (0,1, ... , 13). But is this maximal for two containers and 10 pours? If we only allow 5 pours, the set R reduces to size 7; and if we consider the smallest result not attainable, we get 2 quarts. That is very small! I’m tempted to use \lfloor {\bf F} \rfloor to denote the smallest unattainable positive integer amount, but that’s only a whim.

Ultimately, this leads to two broad, open (as far as I know) questions:

Questions
What sets S maximize |{\bf F}(S;p)| for various p?
What sets S maximize \lfloor {\bf F}(S; p) \rfloor for various p?

Perhaps these have already been explored? I don’t even know what they might be called.