Interviews: Ask Mathematician Neil Sloane a Question 189
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.
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.
what should I learn (Score:5, Interesting)
What should I learn from the area of mathematics if you assume that time is limited?
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Read "What Is Mathematics? An Elementary Approach to Ideas and Methods [amazon.com]" by Richard Courant. It is the single best overview of undergraduate mathematics, IMHO. I really wish I had read this as an undergrad.
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It takes a special type of person to be a mathematician....."working on your favorite sequences"?
Hard to imagine getting excited about number sequences. But, that's just me..takes all types in this world, and thank God for that so there are people that can and WILL do this stuff, but I"d rather sit and watch a car rust than do that stuff for any length of time, unless paid and un-Godly amount of money. And even then......
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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.
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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 e
Math and Politics (Score:2)
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?
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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?
Et 2, Brute? (Score:2)
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Prime factorization for PKI (Score:5, Interesting)
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?
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Why would you ask a combinatorialist this? It's like asking a Linux sysadmin for opinions on C# vs F#.
Really? (Score:2)
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 si
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>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.
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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? (Score:2)
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Anything ending is 2,4,5,6,8 or 0 :)
Except for 2
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And 5.
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Haha, true. ... while bieng an even number ... is still a prime.
For some reason I missed half of my life that 2
So in this example I did not pay attention about the 5, funny.
Anyway, as you might have guessed, I was only nitpicking.
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The sequence in question is:
1 ...
12
123
1234
12345
Neither 2 nor 5 are in the sequence so your original post was fine.
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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.
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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.
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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?
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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}{digi
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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
Is mathematics invented or discovered? (Score:5, Interesting)
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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.
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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
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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?
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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 switc
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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 j
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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 quantitie
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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
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Need Help (Score:2)
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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? (Score:5, Interesting)
hyper-dim sphere packing & error correcting co (Score:2, Interesting)
Would you be so kind as to explain or summarize the connection between hyper-dimensional sphere packing and error-correcting codes?
The Mathmagician (Score:2)
The Mathmagician [twitter.com] 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?
Mathematical theory of life (Score:1)
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.
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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.
Ask a question (Score:3)
Why are you so certain ... (Score:2)
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.
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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.
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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 (Score:2)
why havn't you released the theory of Psychohistory
Do you think pi () is not a normal number? (Score:1)
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You may want to type out whatever symbol you meant in the perens, as Slashdot ate it.
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The source, Kowalski, the source. Where is it?
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I know I always picture this Kowalski when I read APKs posts:
http://madagascar.dreamworks.c... [dreamworks.com]
It makes for some great comedy.
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I'm NOT obligated to give away MY work to be misused as Chrome's was-> http://it.slashdot.org/story/1 [slashdot.org]... [slashdot.org] which you ADMIT I'm not http://it.slashdot.org/comment [slashdot.org]... [slashdot.org]
Isn't your garbage offered up for free? No, you aren't obligated to give it away, but by all appearances you do, you just refuse to submit to a code review to make sure your crap isn't designed to do anything malicious.
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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) 0.1010010001000010000010
Base 10 sequences, other bases of interest? (Score:5, Interesting)
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?
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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...
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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/ [oeis.org]
There! :-p
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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!
This may not be that trivial (Score:2)
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
N
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The sequence is the summary is 123456789, 123456789*10* not 123456789, 123456789*0*
How hard is it to detect user activity??!? (Score:2)
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.
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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...
Question (Score:1)
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?
Mathematics in the future (Score:1)
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 (Score:1)
I hope everybody is familiar with this Wonderful math-computer science pape A personal view of average-case complexity [ldpreload.com] 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 i
Unsolved problems (Score:2, Interesting)
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... (Score:2)
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 t
McEliece cryptosystem (Score:2)
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 (Score:2)
1 + 1, everyone knows that, but what's 2 + 2? Got you there didn't I?
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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 (Score:3)
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 (Score:1)
From A Complete Non-Mathematician (Score:2)
So .. in small words .. what's the point?
What is the use of these things?
Amature/Hobby Math Projects (Score:2)
Best Base (Score:1)
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nine?
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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.