Arithmetic quiz

Discussion in 'Physics & Math' started by arfa brane, Feb 19, 2017.

  1. DaveC426913 Valued Senior Member

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    Right.

    Please Register or Log in to view the hidden image!

    It's not in the menu, but it didn't occur to me to just type it in.
     
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  3. arfa brane call me arf Valued Senior Member

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    What the sequence is, is part of the codomain of a function defined over \( \mathbb N \cup \{0\} \), (the domain).

    If we look at a sequence that does start with 0, it will have to be written as a sum of at least two powers of 0, in fact of any powers of 0. The same is true of the smallest sum of 0s and 1s, and of the smallest sum of 1s, which is 2 which is also the first even prime number.

    So you can say 1 is the first nonzero smallest sum (a sum with two summands) and 2 is the first smallest sum, of finite powers of numbers from {0, 1}. Oh joy.
     
    Last edited: Feb 23, 2017
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  5. DaveC426913 Valued Senior Member

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    So, do I win?
     
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  7. arfa brane call me arf Valued Senior Member

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    Yes, congratulations. Did you use sums of squares?

    Ed. damn, I meant to have "2 is the first smallest sum of powers of numbers which is prime".
    Right off, I can think critically about why the first two numbers in a series given by sums of powers of two numbers, where the domain includes 0, are and aren't prime.

    The Mobius function treats 1 as a special point, whereas the function that treats 1 as the sum of squares of 1 and 0 doesn't output 0 anywhere, 0 isn't in its domain or codomain.
    Now, you get more numbers in the sequence if you increase the number of summands to three, then four etc. Which numbers won't appear in each case and does the Mobius function say anything about it?
     
    Last edited: Feb 24, 2017
  8. Randwolf Ignorance killed the cat Valued Senior Member

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  9. DaveC426913 Valued Senior Member

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  10. arfa brane call me arf Valued Senior Member

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    It's a bit like detective work, this critical thinking about numbers. Numbers have exact values (if there are 12 sheep in a paddock, there are not 24 half-sheep), some natural numbers are prime.

    The natural numbers, it seems, may or may not include 0. Hmm, I can think about a zeroth sheep in a paddock with n sheep in it? Maybe. Sometimes 0 is just an important part of the arithmetic you're doing. Including division in arithmetic means 0 is excluded, or needs to explicitly be excluded from the set of divisors of a number.

    So you have a|b meaning "a divides b with a remainder of 0". Every number, including 0, is divisible by 1, implying every number divided by one of itself is equal to 1. So we have that every number greater than 0 has a set of at least two divisors, and a proof that every prime number has at most two divisors.

    One thing the Mobius function does is pick out the multiples of 4, which is the first non-squarefree number. Is there a connection between multiples of 4 and sums of squares?
    With sums of two squares there are missing numbers, as with sums of three squares; is a sum of four squares sufficient to cover the whole of \( \mathbb N \)?

    What does arithmetic modulo 4 have to do with it?
     
    Last edited: Feb 24, 2017
  11. rpenner Fully Wired Staff Member

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    No, it isn't critical thinking in a maths sense.

    Mathematically the question "what number comes next in this sequence" has no best answer. It's really a question of aesthetics and culture which is disguised as a math question.

    OEIS, for example, has at least three alternatives of non-trivial importance.
     
  12. arfa brane call me arf Valued Senior Member

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    In the case of the partial sequence in the OP, wouldn't it change the "best" answer if the sequence was longer?
     
    Last edited: Feb 25, 2017
  13. rpenner Fully Wired Staff Member

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    Making the sequence longer makes it more likely for someone who shares your culture and aesthetics to guess what motivated you to think the sequence was special, but does not mathematically compel any solution.

    2, 4, 11, 40, 484, 2185, 21495, 101344, 934648, 21286199, 120709720, 2046908123, 21457834229, 120069770663, 976324029182, 16729312538380, 313147820134761, 2029786539491584, 28645763753804896, 292577131561254543, ...

    This sequence is generated by a short mathematical instruction, but OEIS has never heard of it. Though written in numbers, it's not naturally interesting to mathematics.
     
    Last edited: Feb 25, 2017
  14. arfa brane call me arf Valued Senior Member

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    So I guess all the culture and aesthetics, such as they may be, appear more on the RHS, so to say.

    If I write out a sequence like: \( 1^2 + 0^2,\;1^2 + 1^2,\;2^2 + 0^2,\; \) and ask what the next sum of squares is, is it more compelling?
     
  15. rpenner Fully Wired Staff Member

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    Why wouldn't the next term be \(2^2 + 2^2\) ?

    Going forward, assuming we guess you want all unique ways of pairing sums of squares, how are we to guess if you want them ordered by their sum or by the values of the first and second squares. Which one, or both, of 7^2 + 1^2 and 5^2 + 5^2 is to be included? What about when gcd of the two squares is more than 1.
     
  16. arfa brane call me arf Valued Senior Member

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    Ah cha. Here a computationist might step in and say something about how the sequence can be/is generated, and how you need to describe it a certain way--you need explicit instructions.

    As in: write down the ordered-by-value sequence of numbers which are the sums of squares of numbers from {0,1,2,3,...}. (Take pairs from \( \{0^2,1^2,2^2,3^2, ...\} \) under addition.)
     
    Last edited: Feb 28, 2017

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