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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.
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Interviews: Ask Mathematician Neil Sloane a Question

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  • what should I learn (Score:5, Interesting)

    by Anonymous Coward on Monday November 09, 2015 @02:55PM (#50894813)

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

    • Re: (Score:2, Interesting)

      by 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.

      • Wow.....

        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......

        • 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.

    • 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

  • 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?
  • by Anonymous Coward on Monday November 09, 2015 @03:00PM (#50894863)

    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?

    • by Anonymous Coward

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

  • 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

    • by 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.

    • by jpapon ( 1877296 )

      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!

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

      • And 5.

        • 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.

          • by vux984 ( 928602 )

            The sequence in question is:

            12345 ...

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

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

    • by zdavek ( 75457 )

      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.

      • 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?

        • 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}{digi

    • 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

  • by urbster1 ( 871298 ) on Monday November 09, 2015 @03:02PM (#50894881)
    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?
    • 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?

      • by 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 switc

        • 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
    • by khallow ( 566160 )
      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 j
    • by Prune ( 557140 )
      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 quantitie
    • 42
  • Can you give me a hand with my son's math homework?
    • by creimer ( 824291 )
      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.
      • 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."

  • by khallow ( 566160 ) on Monday November 09, 2015 @03:26PM (#50895145)
    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?
  • by 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 [] 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?

  • by 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.

  • by maestroX ( 1061960 ) on Monday November 09, 2015 @04:20PM (#50895763)
    What is your motivation?
  • 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.

    • 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.

      • 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?

  • why havn't you released the theory of Psychohistory

  • If so, why? I'm curious.
    • You may want to type out whatever symbol you meant in the perens, as Slashdot ate it.

    • 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

  • by Hussman32 ( 751772 ) on Monday November 09, 2015 @04:46PM (#50896041)

    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?

    • by Toshito ( 452851 )

      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...

      • by caviare ( 830421 )
        No, In base 2 his sequence would be 1, 110, 11011, 11011100, 11011100101, 11011100101110, etc...
        • by jclaes ( 980059 )
          That sequence is in the database as [] 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.
  • 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

    • by Anonymous Coward

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

  • 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.

    • by Teckla ( 630646 )

      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...

  • by 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?

  • by 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"?

  • by 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 i

  • Unsolved problems (Score:2, Interesting)

    by Anonymous Coward

    Which of the many unsolved problems ( have you tried to solve and for which one do you think you came close?

  • 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 t
  • 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?

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

    • 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.

  • by slew ( 2918 ) on Monday November 09, 2015 @08:32PM (#50897743)

    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?

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

    What is the use of these things?

  • 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?
  • 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?

"The algorithm to do that is extremely nasty. You might want to mug someone with it." -- M. Devine, Computer Science 340