Venn Diagrams are used (yeah) but abused (boo)

In a prior blog entry I speculated about which math phrases will enter the English Language and will they be used correctly.  I thought Prisoner's Dilemma would enter and be used correctly, but Turing Test and Venn Diagram would not enter.

Since then I've seen Turing Test used, but only because of the movie The Imitation Game. I don't think it will enter the human language until a computer actually passes it for real, which might not be for a while. See this excellent post by Scott Aaronson about a recent bogus claim that a machine passed the Turing Test.

Venn Diagrams seem to be used more (Yeah!) but incorrectly (Boo!)

1) In this article (which inspired this post), about who might replace John Boehner as speaker of the house, there is the following passage:

Option 3: An acceptable and respected conservative like Jeb Hensarling or
Tom Price emerges as speaker. Why these two? First, Paul Ryan doesn’t seem
to want the gig, so that leaves us with only a few options for someone who
fits in the Venn diagram of being enough of an outsider, well liked, and
sufficiently conservative.

Is this correct use? They really mean  the intersection of oustider, well-liked,
and suff conservative. One can picture it and it sort of makes sense, but its not
quite correct mathematically.

2)  In this ad for Venn Beer (in celebration of John Venn's 180th birthday!) they really mean union, not intersection.

3) This Venn Diagram about Vladmir Putin's and your Aunt's record collection doesn't really make sense but I know what they mean and its funny.

4) This Venn Diagram about how to woo women is incorrect, not funny, not mathematically meaningful.

5) This Venn Diagram involved Doctors, Prostitutes, and TSA agents. At first it is funny and seems to make sense. But then you realize that the intersection of Doctors and Prostitutes  is NOT People who make more per hours than you make all day, its actually prostitutes with medical degrees. Its still funny and I see what they are getting at.

6)  This Venn Diagram (it's later in the article) of Republican Candidates for the 2016 nomination for Prez is correct for the math and informative, though one may disagree with some of it (Is Trump really Tea-Party or should he be in his own category of Trumpness?)

21st Century Problems

My youngest daughter, Molly, a high school senior talking colleges with a woman about ten years her senior. The woman remembered all her friends watching the clock so they could go home to check their emails to see if they were accepted. Molly said "Sheesh, you had to go home to check email?"

My other daughter Annie, a college junior, went on an overnight last Thursday to a place without cell phone reception. She spent Friday night with her friends in her class catching up on emails, texts and Facebook messages.

Now back in my day (that being the early 80's) we got our college acceptances and rejections by postal mail, where that one crucial bit of information could be determined by the thickness of the envelope. Some of my friends had their mail held by the post office so they could find out a few hours earlier.

In college I did have email access, in fact I wrote an email system for Cornell. But most students didn't use email so we resorted to other means. Student organizations could hire a service that put posters on key locations throughout campus. Chalk on sidewalks worked particularly well. The personals section of the Cornell Daily Sun served as a campus bulletin board. In my freshman dorm we had phones in our rooms but no answering machines. We did put little whiteboards on our doors so people could leave us messages. We had a lounge on our floor where you could find most people and talk to them in person. You young people should try that more often.

We had to coordinate activities and meeting places ahead of time, if someone was late you waited for them. On the other hand I never had to spend my Friday nights catching up on emails and texts.

When did Mathematicians realize that Fermat did not have a proof of FLT?

I recently came across the following passage which is about  Fermat's Last Theorem (FLT).

Pierre de Fermat had found a proof, but he did not bother to write it down. This is perhaps the most frustrating note in the history of mathematics, particularly as Fermat took his secret to the grave.


AH- so at one time people thought Fermat DID have a proof of FLT.  That is, a proof using just the math of his time, likely a very clever proof. I doubt anyone thinks that Fermat had a proof in this day and age. Actually it has been in fiction: in a 2010 episode Dr. Who episode The eleventh hour, the doctor has to prove to some very smart people that they should take his advice. He does this by showing them Fermat's proof of FLT. Good Science Fiction but highly unlikely as Science Fact. In an episode of ST-TNG  (Title: The Royale. Year: 1989) it is claimed that FLT is still open. Whoops. But in an episode of ST-DSN (Title: Facets. Year: 1995) they refer to `Wiles proof of FLT'.

Wikipedia states: It  is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely. I think that understates the case.


So here is a question for all you math historians out there: When did the math community realize that FLT was really hard?

We have one clue- the quote I began with. Its from... 2013. Whoops. The book is  The Simpsons and their mathematical secrets by Simon Singh (Author of Fermat's Enigma  which is about the quest to proof FLT).  I've read the passages about FLT in the Simpsons book over again to make sure he doesn't someplace say that Fermat prob didn't have a proof. No--- he seems to really say the Fermat had a proof. So whats going on here? Possibilities:

1) I'm wrong. There are serious credible people who think Fermat had a proof and he talked to them; perhaps while working Fermat's Enigma. I find this unlikely- I have never, literally never,  heard of  anyone, not even math cranks, who think Fermat had a simple proof.  Some cranks think THEY have a simple proof, though even that seems far less common after FLT was proven.

2) I'm right. He didn't have anyone who was both serious and credible check his book. I find this unlikely. He has written Fermat's 't Enigma so surely he is in contact with people that are both credible and serious.

3) He did have someone check his book but thought the story was better the way he told it. (This was common on the TV show Numb3rs which never let a mathematical truth get in the way of a good story.)
I find this unlikely since a better way to say it is we'll never know if Fermat had a proof!

One problem with such mistakes is that it destroys his credibility on other math things he writes of. That info about Dr. Who I got from the book, but I suspect its correct. And the stuff about the math that appears in the Simpsons is checkable and seems correct. I give one example: in one episode you see in the background

3987¹² + 4365¹²= 4472¹²

which is correct on a calculator with only 10 digits of precision. Cute! But I have stopped reading any of the math history in the book for fear that I will get incorrect notions in my head.

However, back to my original question: Was there a time when people thought Fermat really had a proof?Was  there a time when people thought there was an easy proof? When did that change?

The Theorems Conference

All too often theoretical computer scientists get more obsessed by proofs than the theorems themselves. I suggest a theorems conference. Here's how it would work:

Authors would submit two versions of a paper. One has the statement of the theorem and why that theorem matters, but no proofs. The second version includes the proofs.

The program committee first makes tentative decisions based on the first version of the paper. If tentatively accepted the PC then looks at the second version. The PC can reject the paper if the the proofs have significant flaws, gaps or readability issues. The PC cannot reject the paper for any other aspect of the proof such as length or lack of technical depth or originality.

This way we truly judge papers based on what they prove--what the results add to the knowledge base of the field.

Of course my plan has many flaws. Some papers with their proofs may have already been posted on archive sites which the PC members could have seen. More likely, the PC will guess the difficulty of the proof and judge the paper based on this perceived difficulty, and not on the theorem itself.

We need a culture shift, away from an emphasis on proofs. That's the only way we can judge our results for the results themselves.

An Open(?) Question about Prime Producting Polynomials Part II in 3-D!

(Apologies- No, this post is not in 3-D)

I posted last week about An Open(?) Question about Prime Producing Polynomails

I got several emails about the post with more information, inspiring this post!
All the math in this post can be found in my writeup  Polynomials and Primes,
unless otherwise specified. NONE of the results are mine.

You can read this post without reading the prior one.

1) KNOWN: Let  f(x) ∈ Z[x]  be  a poly of degree d. Then there exists a non prime in f(x)'s image (actually there are an infinite number of  non primes, in fact there are an infinite number of composites).  If f(1) is prime then of  f(1+mf(1)) as m=0,1,2,... at most 2d-2 of them are prime.

2) Algorithm to find an x such that f(x) is not prime: compute f(1), f(1+f(1)),...,f(1+(2d-2)f(1)) until you find one that is not prime. This takes 2d-1 evals. OPEN(?): Is there a better deterministic algorithm where we measure complexity by number of evals? Since this is a simple model of computation lower bounds might be possible.

3) There is the following randomized algorithm: Eval f(1)- if its not prime you're done. If f(1) is prime then pick a random m where 0≤ m ≤ (2d-2)² and eval f(1+mf(1)). This is non prime with prob 1- (1/(2d-1)).

4) What is it about Z that makes this theorem true? In my write up I show that if D is an integral domain with only a finite number of units (numbers  that have mult inverses) then any poly in D[x] has to produce non-primes infinitely often. (A prime in D is a number a such that if a=bc then either b or c is a unit.)

5) What about if D has an infinite number of units? See this  paper for examples of polynomials over integral domains D such that the poly only takes on  only prime or unit values.

6) What about polys over Q? over R? over C? In my write up I prove similar theorems for Q and then use that to get theorems for C.

7)  Looking at polys in Z[x,y] is much harder, see this survey. 

8) If f(x)∈Z[x] is a poly then does there exist a prime in f(x)'s image? An infinite number of primes? Easy but stupid answer is no: f(x)=2x. Better question: assume that f(x)'s coefficients are rel prime.

Dirichlet's theorem: if GCD(a,b)=1 then ax+b is prime infinitely  often.

Open: is x²  + 1 prime infinitely often?

Designer Chips

A computer architecture researcher talked to me about a role theoretical computer science can play for them: creating a new kind of computer processor. Microprocessors stopped getting faster a decade ago due to energy challenges so computer architects look for new ways to improve performance, moving away from the general-purpose CPU towards processors that handle more specific functions. The GPU, Graphics Processor Unit, has long been around to handle the graphics-intensive needs of modern computers and many have used GPUs for other purposes such as machine learning. These days we can program chips using FPGAs (Field-programming gate arrays) and are nearly at the point of cheaply compiling directly to hardware. How does this change the theory we do?

What kind of specialized chips would speed up our algorithms? If we want to find matchings on graphs, for example, is there some routine one could put in a chip that would lead to a much more efficient algorithm?

On the complexity side, how do we model a computer where we can program the hardware as well as the software? What are the right resource bounds and tradeoffs?

In general our notions of computers are changing, now with multi-core, cloud computing and designer chips. Not only should we focus on applying theory to these new models of computing, but we should think about what future changes in computing could yield more efficient algorithms. Theorists should be involved in planning the future of computing and we're not even doing a great job reacting to changes around us.

An Open(?) question about prime-producing-polynomials

Known Theorem: If f(x)∈ Z[x] is  prime for all nat number inputs then  f(x) is a constant.

NOTE- Recall that if p is a prime then so is -p.

Known Proof: Assume f(x) has degree d. f(1) IS prime.  Let f(1)=p. Look at

 f(1+p), f(1+2p),...,f(1+(2d+1)p).

One can easily show that p divides all of these. Hence if they are all primes then they must all be p or -p. Since there are 2d+1 of them, at least d+1 of them are the same, say p. Hence f is the constant p.

END of known proof.

Note that the proof gives the following theorem:

Let f(x)∈ Z[x] of degree d. We assume f(1)≥ 0. Least a st f(a) is NOT prime is  ≤ 1+(2d+1)p.

(This can prob be improved a bit with some cases, but its good enough for now.)

Recall Euler's poly x²-x+41 produces primes for x=0,...,40. But at 41 you get a composite. This is much smaller than the upper bound 1+(2d+1)p = 1+5*41=206.

Wolfram MathWorld has a  page of other polys in Z[x]  that produces lots of primes initially, but NONE come close to the bound.

QUESTIONS:

Proof a better upper bound.

Proof a better lower bound (Fix d and produce and infinite seq of polys of degree d...)

Close the gap!

If this is already known, then let me know please.

Can also ask for polys in Q[x], R[x], C[x]. For Q[x] and R[x] same theorem is true- no poly can produce all primes. I suspect also true for C[x] but I haven't seen it stated anywhere. (ADDED LATER- Proof for C[x] is easy. First proof
for Q[x] and then by Lagrange interpoloation if a poly has inf many times
where f(integer)=integer, poly is in Q[x].)

You can also NOT include negative primes and see how that changes things.

Whiplashed

I recently watched the movie Whiplash, about a college jazz band director, Fletcher played by J.K. Simmons, who torments his musicians to force them to be their best. The movie focuses on a drummer, Andrew, which makes for a great audio/video feast but in its essentials Whiplash is a story of a professor and his student.

I can imagine playing the role, “Do you think your proof is correct? Yes or No? Are you applying Toda’s theorem correctly or are you using the same crazy logic your dad used when he left your mom?” OK, maybe not.

Nevertheless Fletcher has a point. Too often I’m seeing graduate student doing just enough to get a paper into a conference instead of pushing themselves, trying to do great work and still not being satisfied. Fletcher says the two most dangerous words in the English language are “good job”.  While that might be a little cruel, we do need to push our students and ourselves to take risks in research and be okay in failing. To roughly quote John Shedd and Grace Murray Hopper, "the safest place for a ship is in the harbor, but that’s not what ships are for."

Whiplash had a different kind of scene that definitely hit home. Andrew could not impress his family with the fact that he was lead drummer in the top college jazz band in the nation. I’ve been there, trying to get my mother excited by the fact that I had a STOC paper early in my career. "That's nice dear".