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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.
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 from the area of mathematics if you assume that time is limited?
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?
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?
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?
Would you be so kind as to explain or summarize the connection between hyper-dimensional sphere packing and error-correcting codes?
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!
-- Let us endeavor so to live that when we pass even the undertaker shall be sorry. -- M. Twain
What is your motivation?
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?
"Who are you?" "No one of consequence." "I must know." "Get used to disappointment."
Two is prime you insensitive clod. Which makes it odd.
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?
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?
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."
Confucius say, "Find worm in apple - bad. Find half a worm - worse."