Interviews: Ask Mathematician Neil Sloane a Question

Considered by many to be one of the most influential mathematicians alive today, Neil Sloane has made major contributions to the fields of sphere packing, combinatorics, and error-correcting codes. He is probably best known for being the creator and curator of the On-Line Encyclopedia of Integer Sequences (OEIS), known simply as “Sloane” by its many users. The repository is over 50 years old and contains over 260,000 sequences.

Neil recently turned 76 but his passion for mathematics remains as strong as ever. Talking about a recent project, he writes: “Back in September I was looking at an old sequence in the OEIS. The sequence starts 1, 12, 123, 1234, 12345, ..., 123456789, 12345678910, 1234567891011, ... The n-th term: just write all the decimal numbers from 1 to n in a row and think of this as a big number. The entry for the sequence had a comment that it is expected that there are infinitely many terms which are primes, but that no prime was known, even though Dana Jaconsen had checked the first 64,000 terms. So I asked various friends and correspondents about this, and people extended the search somewhat. In fact Ernst Mayer has set up a cloud-source project to look for primes in the sequence, and the sequence has now been checked to nearly n = 270,000 without finding a prime. But I am hopeful that a prime will appear before we get to n = 10^6. When a prime is found, as it surely will be, it probably won't be the largest prime known, but it will be close to the record (which is held by the latest Mersenne prime). We may make it into the top ten. It will certainly be the largest known prime which is easy to write down! (Explicitly, I mean. You may know that 2^32582657-1 is prime, but you won't be able to write down the decimal expansion without using a computer).”

Neil has agreed to take some time away from his favorite sequences and answer any questions you may have. As usual, ask as many as you'd like, but please, one question per post.

Comments

what should I learn

[Anonymous Coward]

What should I learn from the area of mathematics if you assume that time is limited?

Re:what should I learn

[Anonymous Coward]

Read "What Is Mathematics? An Elementary Approach to Ideas and Methods" by Richard Courant. It is the single best overview of undergraduate mathematics, IMHO. I really wish I had read this as an undergrad.

Re:what should I learn

[serviscope_minor]

What should I learn from the area of mathematics if you assume that time is limited?

Depends. Do you want to do maths for some purpose, or do you just want to have fun? For example, I do a lot of linear algebra like stuff for work. I was writing unit tests for some C++ linear algebra code I had and generating the matrices randomly. Then I thought "hey I've heard linear algebra works over finite fields too", so I modified my code slightly and hey presto, it worked.

Turns out the properties of random matrices especially over finite fields are interesting to me (a lot of the nasty things like condition numbers go away and matrices are either perfectly invertible or perfectly singular, so it's much cleaner), and I kind of picked up an interest. So, now I have a few puzzles to solve. I've no idea if they've ever been solved before (I can't find any obvious indication they have) and I don't know if they're even solvable given current mathematics (let alone by me), but it's interesting.

But if you want to learn mathematics that's a mix of interesting and useful for your job, then I have a few suggestions based loosely on my job, though the techniques have allowed me to easily solve problems that others found hard:

1. Linear algebra (over real numbers---though complex ones are a very small extension).
Get comfortable with inversion, decompositions that one use instead of inversion such as LU, pivoted LU, Cholesky. Get to grips with symmetric eigen decomposition and SVD, and have some understanding of the latter algorithms (no need to be able to code up a good one yourself).

Linear algebra doesn't do much on its own, but it serves as the basis for an awful lot of useful things.

1a you'll need some basic calculus, like calculating derivatives.

2. Optimization and curve fitting. Learn basic least squares and understand why it works (requires linear algebra in practice to perform). Lots of statistics packages can do that for you, but it's the first step. Then learn nonlinear least squares. This will allow you to fit more weirdass functions. Then learn a bit about iterative reweighted least squares. At this point, things get really useful in data modelling since you can fit weirdass functions to data with unpleasant kinds of noise which wreck least squares. Basically, the reweighting downweights points far from the curve so they have less influence. When it comes to optimization, learn a few options like Gauss-Newton, Levenberg Marquadt, gradient descent.

Learn RANSAC. It's well known in the computer vision community, but little used outside. It's also very useful and is a fantastic algorithm for getting a good starting point for nonlinear iterative reweighted least squares with very noisy data. Compared to most other techniques it's almost magically good.

Interestingly, none of the above are complex. A basic levbenberg-marquadt optimizer is about 6 or 7 lines of code if you can do maths in your language (e.g. matlab/octave, C++ with a good library, etc), ignoring the problem specific bits about derivatives, though automatic differentiation can solve that for you too. Likewise RANSAC is easy to implement (another few lines of code).

With those, there's an awful lot of problems you can solve. General data analysis is a breeze. Machine learning is just larger scale optimization. Knowing the basics will allow you to implement some algorithms right off the bat and give a good grounding to learn about others.

Lots of problems in 3D, like 3D reconstruction, or fitting models to point clouds from 3D scans are all essentially optimization problems. And all the 3D geometry is just linear algebra too.

And so on. In terms of practical maths for engineering type things those few are very useful.

Re:what should I learn

[david_thornley]

If you're interested in injection molding, here's a tip: nobody else at the picnic will be interested in how many ejector pins were used to make the plastic forks.

Math and Politics

[smittyoneeach]

Rhetorical: is there anybody less interested in political power than a mathematician?
Should we try to get away from so many lawyers and doctors in political office, and try to bring in some (arguably) more thoughtful people, or would this merely succeed in upsetting everyone?

Re:Math and Politics

[smittyoneeach]

Great. Will you please introduce Congress to arithmetic, and explain that all numbers are not, in fact, "morally equivalent"?

Re:Math and Politics

[pla]

explain that all numbers are not, in fact, "morally equivalent"?

Can you deny that the complete exclusion of even numbers from the vast majority of prime number research is just as bad as Hitler's internment of the Polish Jews???

Why do you hate America?

Prime factorization for PKI

[Anonymous Coward]

Is the use of prime factorization as the basis for public key cryptography still considered to be safe against attacks, given advances in number theory and Moore's Law since the '70s?

Are alternative schemes (e.g., Merkle's knapsack packing) under active consideration?

Re:Prime factorization for PKI

[Anonymous Coward]

Why would you ask a combinatorialist this? It's like asking a Linux sysadmin for opinions on C# vs F#.

Really?

[ledow]

It would strike me that a brute-force approach is pretty poor for this.

As the digits of the sequences are well-known and predictable, some ancient mathematical tricks (e.g. if the digits sum to a multiple of three, etc.) and a bit of algebra on the base-10 expression should surely yield more convincing proof one way or another than anything else, certainly if you'd got as far as they have by brute-force.

Anything ending is 2,4,5,6,8 or 0 is gone immediately as non-prime. Three, sixes and nines have rules similar to the above that operate on the digits of base-10 expression. It would seem to rule out vast swathes of such numbers. Past that, there's not much left to check at all.

But because the sequence is highly predictable and can only end in so many things, you're quickly only looking at massively large numbers as factors to see if they "hit".

Re:Really?

[Anonymous Coward]

>Past that, there's not much left to check at all.
There are still plenty to check. The numbers are huge which is why they take so long to test for primality.

Re:Really?

[jpapon]

Anything ending is 2,4,5,6,8 or 0 is gone immediately as non-prime. Three, sixes and nines have rules similar to the above that operate on the digits of base-10 expression. It would seem to rule out vast swathes of such numbers. Past that, there's not much left to check at all.

Yes, because I'm sure a bunch of world-class mathematicians who have spent their lives working with primes aren't aware of those things. Thank goodness they have you around to help them out, or they might have wasted all that time checking even numbers for primality!

Re:Really?

[angel'o'sphere]

Anything ending is 2,4,5,6,8 or 0
Except for 2 :)

Re:Really?

[sexconker]

And 5.

Re:Really?

[wonkey_monkey]

It would strike me that a brute-force approach is pretty poor for this.

Yes, yes it would. What makes you think the people working on this have forgotten to skip the even numbers, and employ all the other tricks at their disposal?

It would seem to rule out vast swathes of such numbers.

Well, yes, in a sense. But since what it rules out is an infinite subset of an infinite sequence...

Past that, there's not much left to check at all.

...you're still left with an infinite set of numbers to check through.

Re: Really?

[johnsnails]

Two is prime you insensitive clod. Which makes it odd.

Re:Really?

[zdavek]

A little thought reveals that any number in this sequence where the number you're adding to the end (n) is has a factor of 3 makes the whole number also divisible by 3. A little more thought reveals that where (n) has a factor of 3 the sequence of (n-1) will also have a factor of 3. This alone knocks out 2/3 of the possible numbers in the sequence that may be prime.

Re:Really?

[wonkey_monkey]

A little thought reveals that any number in this sequence where the number you're adding to the end (n) is has a factor of 3 makes the whole number also divisible by 3.

Ugh. I feel like I should be able to do this, but... why is that the case?

Re:Really?

[angel'o'sphere]

Haha, true.
For some reason I missed half of my life that 2 ... while bieng an even number ... is still a prime.
So in this example I did not pay attention about the 5, funny.
Anyway, as you might have guessed, I was only nitpicking.

Re:Really?

[ChrisTaylor2904]

It took me a few minutes to get this too. It relies on the fact that the sum of the digits of a number is congruent to the number itself mod 3 (which is easy by induction).

This proof covers both parts of the original assertion, i.e. n = 0 mod 3 implies both that nth and (n-1)th terms of sequence are equal to 0 mod 3.

The assertion is clearly true for n=3.

Now for the case where you're adding n on at the end. This number looks like:

{the (n-3)rd number in the sequence}{digits of n-2}{digits of n-1}{digits of n}

The n-3rd number is divisible by 3 by the induction hypothesis. The digits of n-2 and n-1 are respectively -2 and -1 mod 3, so adding them together is 0 mod 3, and n itself is 0 mod 3, so the nth term of the sequence must also be divisible by 3.

Re:Really?

[vux984]

The sequence in question is:

1
12
123
1234
12345 ...

Neither 2 nor 5 are in the sequence so your original post was fine.

Re:Really?

[serviscope_minor]

You're correct of course about those rules. They're actually more general than that. And they particularly help when you're testing all the integers to get primes.

What you're referring to is essentially wheel factorisation, or at least turns into it. The trivial wheel is skipping every number which is a multiple of 2, i.e. by starting at 1 and using n+=2. For larger wheels, you don't add 2 each time, you add a member of a cyclic sequence.

There's a nice paper about the Sieve of Eratosthenese in Haskell which shows the practical implementation of such a wheel here:

https://www.cs.hmc.edu/~oneill...âZ

For numbers which don't follow such a regular sequence, however, applying a test such as "do the decimal digits add up to 9 (apply recursively)" is equivalent to simply testing if the number divides by 9. Doing your checks would save the inevitable conversion to base 2, for moderately large swathes of numbers.

However: that doesn't help all that much. Remember the number of digits goes up as about n log n, and sure you eliminate numbers, but the more you eliminate, the faster you get to really insanely long numbers. And while PRIMES is in P, I believe that to guarantee the correct answer, the order is something horrendous like the 6th power of the number of digits. You can of course limit the number of those tests with a faster randomized test (which will eliminate a large percentage) but you still ultimately need a deterministic test so you're still bounded by O((n log n)^6) running time.

So even if all your nice tricks work (and they do), eventually you'll slam into needing to test the primality of a number with a few hundred thousand digits and that's going to suck.

Re:Really?

[the phantom]

You mean something like the Sieve of Eratosthenes?

Is mathematics invented or discovered?

[urbster1]

In other words, is mathematics a fundamental part of the fabric of reality (i.e. Platonism)? And are concepts like zero, infinity, imaginary numbers, and so on, actually real objects? Or do you think mathematics is mostly a tool created by humans out of convenience (akin to language), and numbers and other concepts are just abstract ideas in our brains?

Re:Is mathematics invented or discovered?

[ShanghaiBill]

In other words, is mathematics a fundamental part of the fabric of reality (i.e. Platonism)?

Another way to ask this question: If we make contact with an advanced alien civilization, would they have "math" similar to ours? They will use different numerical symbols, and likely not use base-10, but would they otherwise have the same basic concepts of zero, rational numbers, transcendental numbers, theorems, proofs, etc?

Re:Is mathematics invented or discovered?

[Anonymous Coward]

Some philosophers disagree. They'd claim that you need to distinguish between reality (what is real) and actuality (what is "Given"), and claim that reality comprises actuality but not vice versa. In this point of view many concepts and abstract objects are real.

Re:Is mathematics invented or discovered?

[khallow]

IMHO completely depends on your definition of reality. But one way is implied by the language you use, such as "concepts" and "abstract ideas" which are only used for certain not real things.

Another thing to consider here is whether it matters. For example, does it matter if ideas are real? Does it matter if only ideas that can be fully described or represented in our universe are real? Does it matter if no ideas are real (though clearly we can still speak of real representations of some of these ideas just due to our ability to describe them)?

Re:Is mathematics invented or discovered?

[khallow]

Said the person who has obviously never read Plato (and probably not a page of any other philosopher, either).

Plato's virtue is that he was the first to write stuff like this down. If he were to do it now, he'd just be another kook with a blog rehashing stale ideas. While there is still some value to his work, a lot of it is valuable just for the dead ends that are illustrated and eventually dismissed so that we may avoid them. The theory of forms is one of those dead ends.

For example, it is completely irrelevant to us whether a concept is a real object or not. We don't use ideal tables or ideal numbers, we use real world objects that happen to have properties in common with these concepts.

Even the concept itself is actually a description or a representation and we can come up with rival descriptions or representations, sometimes quite alien ones, without requiring a concept to be real or seeing even the slightest difference in our lives or arguments for real versus not real.

Re:Is mathematics invented or discovered?

[Prune]

You should be asking this question of a physicist, not a mathematician — mathematical Platonism is just another religion.

Physics is clear on the question: there is a limit of entropy/information density in any finitly-bounded region of space. Initially this was demonstrated for flat spacetime in a result known as the Bekenstein bound, and was later extended to de Sitter spacetimes (and we're in an asymptotically de Sitter spacetime according to accepted cosmology). This means that physical quantities cannot be arbitrary precision (real-valued), because you can encode infinite information in a real number and that contravenes the aforementioned bound. Thus, real numbers are not real, and uncountable infinities do not exist in the physical universe. This severely limits the mathematics that actually applies to reality at a fundamental level.

Combining the above together with the fact that any causally connected system in the universe is finite in size (the limitations being accelerating expansion and the speed of light resulting in a cosmological horizon), any physical entity can be fully described by a non-deterministic linear bound automaton, which is a class of mechanistic information processing entities, even less powerful than Turing machines. That includes the human brain, and also the system comprised of the sum total of all human brains and any intelligent artifacts we ever create interacting together. The class of problems a non-deterministic LBA can solve is pretty limited. So how can mathematicians think and talk about concepts like uncountable infinities and everything in mathematics that depends on them, if their brains are based on physics in which these concepts play no part?

Let's separate the existence of thoughts on such concepts from the concepts themselves having any reality. The former are obviously connected to the physical universe via their neural correlates. As for the latter, they're easily explained as an extension of the sort of heuristics the brain uses in virtually all aspects of its functionality, as is well-known from cognitive psychology. As an example, concepts like the number pi (to a given number of digits) are just shorthands for their generative processes (to a given number of iterations or recursions). Even while mathematicians think about problems that are outside the class of those which are computable, their brains are not applying any magical non-computable processes to solve them. It's a combination of not really solving them (which would be impossible as they're not real) but processing them in other ways based on the assumption they're real, the luck and lack thereof of stochastic search that cognition oft relies on, and, without a doubt in some cases, accepting "solutions" which are wrong but unknowably so.

Re:Is mathematics invented or discovered?

[Anonymous Coward]

In my non-mathematician opinion, I suspect that it would be basically similar, but there'd be some significant differences:

1. Things that we calculate via trigonometric functions might be calculated using different repeating functions. I know that this has been experimented with on Earth too.
2. The might not have the same cartesian bias that we have.
3. They might not use positional notation at all, making the base-10 question moot. Maybe they invented the electric calculator before the common man switched from an accretional number system, or maybe they do something very different.
4. Not sure whether they'd have anything even remotely similar to matrix multiplication.
5. Assuming they haven't become completely reliant on calculators even for simple arithmetic, there's a good chance they use fundamentally different addition, subtraction, multiplication, division, etc. patterns. After all, computers subtract using 2's complement, but we don't teach children to subtract via 10's complement.
6. They might think of imaginary numbers, quaternions, etc. very differently. Either more or less naturally than we do.

Re:Is mathematics invented or discovered?

[jouassou]

Regarding the last point, about thinking very differently about imaginary numbers and quaternions, you might find this paper interesting; it is a readable and easily accessible introduction to the topic of geometric algebra, with an emphasis on its pedagogical applications in physics. This mathematical formalism goes back over a century to Grassmann and Clifford, and has been repopularized in physics by Hestenes. I believe some people are also using the formalism for computer graphics. The short version is that you can unify vectors, quaternions, and complex numbers into a single geometric formalism, if you just treat scalars, vectors, planes, and cubes all as first-class objects in a general geometric space, and that this leads to more intuitive geometric interpretations.

Re:Is mathematics invented or discovered?

[Prune]

The integers extended by some irrational, say sqrt(2), requires use of real numbers but does not involve uncountable infinities or infinite information.

Only if you're limiting the accepted irrational numbers to a countable subset of R. That's not very useful. For example, it was demonstrated in a paper a few years ago that if you could have infinite precision real weights for the connections of an artificial recurrent neural network, that would allow super-Turing processing. However, your restriction would break that and any other such approaches (and, of course, physics also breaks it -- such a thing cannot exist in the universe -- which was my point).

It is just applying logical rules on various abstract structures. There is no requirement that they be "real" in the sense of being the result of some measurement in the real world. That also doesn't require assumptions beyond basic deductive logic working.

You seem to have missed the point. I was saying that humans cannot solve non-computable problems in the general case. Mathematicians have the same theoretical limits as a digital computer. The "just applying logical rules on various abstract structures" is not solving non-computable problems (siome of the simplest examples of these are listed in http://mathoverflow.net/questi... ); it's an action that can be mapped to a computational process (and even in cases where said process cannot be effectively simplified beyond an ab initio molecular dynamics simulation of mathematicians' brains and their environs, it's still computational).

Re:Is mathematics invented or discovered?

[martinfb]

42

Re:Canonical way to speak numbers after decimal

[__aaclcg7560]

How you pronounced the period between two sets of numbers depends on the context. Mathematic users typically use the word point (i.e., ten point one six). Computer users who refer to the dot notation for IP addresses will use the word dot (i.e., 172 dot 0 dot 0 dot 1). I don't think the two words are interchangeable.

Need Help

[lbmouse]

Can you give me a hand with my son's math homework?

Re:Need Help

[__aaclcg7560]

My father loved doing my math homework when I was a kid. He had a sixth grade education from the 1950's that taught him more mathematics than high school graduates today. Having him do my homework didn't help me do well in school. When I got into college, I had re-learn basic math all over again before I could take the introductory math courses.

Re:Need Help

[Hognoxious]

I was helping my eldest boy. He was adding 14 and 17 and getting 21. Then he added 16 and 28 and got 34.

Then Kansas came on the radio and they had the answer: "Carry one my wayward son."

What are the hidden gems?

[khallow]

One of the common problems with any field of science or math is how hard it is for outsiders to understand what's going on inside. What sort of challenging problems, profound conjectures, sublime proofs, or versatile tools and applications do you feel languish in obscurity or are greatly underappreciated by either the layman and/or a knowledgeable mathematician outside your field(s) of interest?

hyper-dim sphere packing & error correcting co

[Anonymous Coward]

Would you be so kind as to explain or summarize the connection between hyper-dimensional sphere packing and error-correcting codes?

The Mathmagician

[gameboyhippo]

The Mathmagician is the most computational local wrestler in sports entertainment today. Unfortunately, he loses a lot. What integer sequence should he study to win his next match?

Re:Ready for death?

[__aaclcg7560]

I had a college instructor who retired from teaching mathematics at 81. Why that age? Because it was nine squared. I think he was 89 (a prime number) when he keeled over.

Mathematical theory of life

[Anonymous Coward]

Do you think that the concept of life can be defined mathematically?

For example, certain states of dynamical systems could be defined as 'alive' if they
reproduce and evolve, where reproduction and evolution would have to be defined as well,
of course.

Then, we could go on and look for criteria for dynamical systems that
would imply that life can or must exist. Or prove that the probability
of a system to be alive is nonzero if parameters are chosen randomly. Etc. etc.

Ask a question

[maestroX]

What is your motivation?

Why are you so certain ...

[angel'o'sphere]

Why are you so certain that that sequence contains any primes at all?

Considering the sequence of infinite numbers: 2, 22, 222, 2222 etc. it contains only one prime and 4, 44, 444, etc. none at all.

Re:Why are you so certain ...

[wonkey_monkey]

Why are you so certain that that sequence contains any primes at all?

Why are you implying that it might not?

Consider the sequence of primes: it consists of nothing but primes! Gasp!

That's about as good as your argument.

Re:Why are you so certain ...

[angel'o'sphere]

Hae? Why do you come to the conclusion that "I'm implying that it does not comtain a prime"?

That was a honest question, so again: why is he so certain that there will be primes somewhere?

You obviously have no answer, so why did you even bother answering to me?

Mathist

[zlives]

why havn't you released the theory of Psychohistory

Do you think pi () is not a normal number?

[Inyu]

If so, why? I'm curious.

Re:Do you think pi () is not a normal number?

[Coren22]

You may want to type out whatever symbol you meant in the perens, as Slashdot ate it.

Re:Do you think pi () is not a normal number?

[Inyu]

There was GREEK SMALL LETTER PI (U+03C0) in the parentheses, please, Slashdot, display it correctly.

Re:Do you think pi () is not a normal number?

[serviscope_minor]

Given I've been messing around in such things recently, I think I'd follow that up with:

Do you think that all algebraic irrationals are normal?

Or, (if I understand correctly, restating exactly the same question to make it relate more to OEIS), do you think that at some point showing that a number has an infinite, non repeating, non normal distribution of digits will be sufficient to prove that it's transcendental?

It would be more than a little bit nice to be able to prove that (say) 0.1010010001000010000010000001... is transcendental just by pointing out it's not normal.

Base 10 sequences, other bases of interest?

[Hussman32]

Some of the sequences being studied (like the example in the summary) use formulations developed from base 10 numbers. Have you explored other bases, in particular prime number bases, or perhaps a rational fraction or even irrational/transcendent number? If so, were there any interesting surprises?

Re:Base 10 sequences, other bases of interest?

[Toshito]

I just tried his sequence but in base 2 (since I'm a programmer!)

1, 10, 101, 1010, 10101, 101010, etc...

The pattern is boring, each binary value in the sequence, when converted in decimal, repeats the following:

previous value x 2
previous value x 2 + 1

The same list in decimal:

1, 2, 5, 10, 21, 42, etc...

Re:Base 10 sequences, other bases of interest?

[caviare]

No, In base 2 his sequence would be 1, 110, 11011, 11011100, 11011100101, 11011100101110, etc...

Re:Base 10 sequences, other bases of interest?

[Toshito]

Yes, I see my error.

I tought his sequence was listing all the numerical symbols sequentially, adding a digit each time, and repeating when all the symbols have been used. Thus having 1234567890123...etc...

Now I see that after 9 it's ten, eleven ,twelve, etc...

So in binary it's indeed

1, 110, 11011, 11011100, etc...

Thanks!

Re:Base 10 sequences, other bases of interest?

[Toshito]

Yes I'm a good programmer, I work in assembly you insensitive clod!

Like I did mention in my reply, I read the summary too fast, and tought that it was another sequence.

Well, it turns out that the sequence I was thinking about exists: https://oeis.org/A057137/

There! :-p

Re:Base 10 sequences, other bases of interest?

[jclaes]

That sequence is in the database as http://oeis.org/A047778 since at least 1999. Funny, just one month ago, Neil Sloan asked "the smallest prime in this sequence is 485398038695407. What is the full subsequence of primes?" For the moment, the first is also the only prime known in this sequence.

This may not be that trivial

[gavron]

1
12 - any number that ends on a multiple of 2 is an even number and hence can't be prime
123 - any number whose sum of digits is divisible by 3 is not a prime
1234 - covered in 12 above
12345 - any number whose last digit is a 5 (or 0) is evenly divisible by and hence can't be a prime
123456 - covered in 12 and 123 above
1234567 - the first possible candidate not immediately eliminatable based on consistuent digits
12345678 - covered in 12
123456789 - covered in 123 above
1234567890 - covered in 12 and 12345 above
Now we start adding the same sequence back
12345678901 - second possible candidate
same rules eliminate all of these up to
12345678901234567

Which means out of each ten digits there are only two candidates for primes - a number ending in 1 or a number ending in 7.

Except that every 3 1-0s they can be eliminated as a multiple of 3
Every 5 1-0s they can be eliminated as a multiple of 5
Every 6 1-0s they can be eliminated as per 3 1-0s above
Every 9 1-0s they can be eliminated as per 3 1-0s above

So while ostensibly there are potentially 20% primes in the 1-0 sequence, 40% of those are eliminated in the up-to 10 such sequences, and so on and so on. In fact as you get larger numbers, the predictive nature of the sum of the digits at any juncture allows eliminating more and more numbers asymptotically reaching zero.

E

Re:This may not be that trivial

[Anonymous Coward]

The sequence is the summary is 123456789, 123456789*10* not 123456789, 123456789*0*

How hard is it to detect user activity??!?

[Thing 1]

C'mon Slashdot. I don't want to disable the auto-load feature, as it's useful. But not while I'm reading! Please detect user scroll and click activity, and put a 5-minute wait after any activity before resuming auto-update.

I was reading this particular summary when it bothered me again, so I'm attaching it here as a public bug report.

Re:How hard is it to detect user activity??!?

[Teckla]

C'mon Slashdot. I don't want to disable the auto-load feature, as it's useful.

God damn I so badly want to disable the auto-load / auto-refresh feature... I hacked it with a blacklist at one point but somehow they worked around that...

Re:Quickly - seven eights?

[Coren22]

nine?

Question

[Anonymous Coward]

What is your view on the validity of computer-generated proofs, specifically those too large to ever be checked by even a concerted group of human beings?

Re:Question

[tehcyder]

My question is, how can we bring more high-wage jobs that use advanced math skills into the business world?

That's like saying "how can we bring more high-wage jobs that use advanced Hittite cuneiform linguistic skills into the business world?"

The business world doesn't care about something unless it helps to make money. You clearly don't need advanced maths to make money as a rule.

Mathematics in the future

[Anonymous Coward]

When asked about the great conjecture of Collatz, Paul Erdos replied with "Mathematics is not ready for such problems".
Do you think we may find a branch of Mathematics that is actually an empirical science, akin to Wolfram's "New Kind of Science"?

Computational Complexity

[Anonymous Coward]

I hope everybody is familiar with this Wonderful math-computer science pape A personal view of average-case complexity by R Impagliazzo

  In this paper he give an excellent outline of the P=NP? problem, and talks about 5 possible words, Algorithmica, Heuristica, Pessiland, Minicrypt, and Cryptomania, where this question is answered differently. Professor Sloane, which land do you think we live in? Do you think that there are more than 5 possibilities?. Do you expect any progress on this question in the near future?

Thanks

Wayne Shanks

Et 2, Brute?

[smittyoneeach]

I blame the oddest prime of them all.

Re:Et 2, Brute?

[the phantom]

You mean 2, right? I mean, it is the only even prime number, which makes it rather odd among primes...

Re:Et 2, Brute?

[smittyoneeach]

Yes, as stated in the Latin pun of the Comment Subject. But hey, a great gags merit running Twice.

Re:Et 2, Brute?

[smittyoneeach]

(and botching the noun/verb agreement)

Re:Canonical way to speak numbers after decimal

[khallow]

How would an enlightened foreigner say "That never gets old?"

Unsolved problems

[Anonymous Coward]

Which of the many unsolved problems (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics) have you tried to solve and for which one do you think you came close?

About that prime sequence...

[gnasher719]

If you check the sum of digits, you find that five out of every six consecutive numbers are divisible by 2 or 3, and only one isn't. Normally two out of six numbers are not divisible by 2 or 3. That means these numbers are only half as likely as your average random number to be primes.

Normally, the probability that a random integer n is a prime number is about 1 / ln n. The probability that a random n digit number is a prime is about 1 / 2.3n. With these numbers, it is about 1 / 4.6n.

We can estimate the number of primes that we should find while adding all (0.9 * 10^k) k-digit numbers, ending with a number of (k - 1/9) * 10^9 digits: That estimate is about 0.5 + (1 / 4.6) / (k - 10/9). That's about 0.5444 for six digit numbers added, about 0.5369 for seven digit numbers, about 0.5315 for eight digit numbers. Hoping for a solution within the first million numbers is optimistic.

McEliece cryptosystem

[slew]

With all the renewed interest in post quantum computer cryptography, why do you think there is minimal research in the error correcting code styles of public-private key encryption? (e.g., the McEliece cryptosystem) Are there ones that you consider to be better candidates?

OK hot shot

[HalAtWork]

1 + 1, everyone knows that, but what's 2 + 2? Got you there didn't I?

Re:OK hot shot

[serviscope_minor]

1 + 1, everyone knows that

1+1=0, because I work in GF(2) today.

but what's 2 + 2?

Still 0 because I love me some finite fields.

combinational neural networks & orthogonal arr

[slew]

One of the current problems with training deep combinational neural networks is that it's often not easy to tell what you are training them to look for. People train NN blindly on vast data sets, but often have no idea how robust this training is before deploying them.

Do you think some of the mathematics surrounding orthogonal arrays can be extended to improve the metrics on how efficient or robust the training is of a neural network might be?

about the sequence idea

[fkodama]

what is incredible for this sequence idea is that instance is base dependent. but definition not. I mean, base-10 : 1,12,123,1234,12345,... base-2:1,110=4 base 10, 11011=27 base 10, 11011100 = 220 base 10, 11011100101 =1765 base 10 base-4:1,12=6 base 10,123=27 base 10,1210=100 base 10, base 8:1,12=10 base 10, 123=83 base 10, 1234=... 123 in base 8 is 83 in base 10 which is prime. So I do believe there is infinite primes on base 10, it's not a solid belief, but anyway it's not the case. the case is that any huge prime generated by this idea can be optimally compressed by its last part.

From A Complete Non-Mathematician

[Toad-san]

So .. in small words .. what's the point?

What is the use of these things?

Amature/Hobby Math Projects

[Spinalcold]

These days, with the internet, there is opportunity to do hobbyist science like Zooniverse and OEIS. Do you know of other math projects like OEIS that the public can contribute too?

Best Base

[ScottAllenMueller]

If we humans could easily change our predominantly decimal number system to a different base, which base would you choose? Hexadecimal as it's easy to translate to and from binary (as well as base 4)? Any other bases or benefits? Is base 12 ideal due to today's frequent usage of dozen counting, time and 12's many useful divisors?

Re:Math question...

[cwsumner]

Why is the square root of -1 so provocative?

It's not.

It just means that your equations have an inherent variable or "dimension" that is orthogonal to the explicit variables.

See the derivation of the calculations of phase shift in AC motors.