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Negative prime numbers?

Discussion in 'Mathematics' started by pipipi, Aug 29, 2011.

  1. During a slow swim this morning I was running through the first few lessons I'm planning to teach, and one of them is about primes. I was trying to come up with good questions for primes.
    Is 158 a prime? No need to check cos it's even.
    Is 9 a prime? No although it's an odd number.
    Or is Pi prime (cos there's only two numbers that go into it, but it's not an integer).
    How about -7?
    And here is where I forgot about swimming. Initially I'd thought that -7 would not be prime, because its factors could be -1 and 7 OR 1 and -7 which means there are two many factors (although someone will argue that they are the same number). So far so good for me. But then I thought about normal 7. That could be 1 and 7 OR -1 and -7. And that's too many factors as well!
    Have I forgotten something about the definition of primes that you are only allowed to consider positive integers as factors?
     
  2. pencho

    pencho New commenter

  3. It's more the definition of a prime rather than that of a factor - primes must be natural numbers.
    [Someone beat me to it!]
     
  4. I thought I'd forgotten something about the definition. Will go and look it up.

    Thanks for the link Pencho, but it doesn't really answer it. 3 different explanations? That's when the internet can be more confusing!

    Woosterite- primes must be natural numbers. Ok, so how does that rule in/out 7 being made of 1&7 or -1&-7 so it's got 4 factors.


    I think I've been a bit lazy in my definitions previously saying that it's only 2 factors. and then suggest they go and look up the special case of 1. Maybe that's just me forgetting stuff over the summer!
     
  5. 'Prime' is a definition, a manmade construct, and so there is no straight answer, it just depends what definition you accept. The definition most wideley accepted (including, I believe, by exam boards) is that only natural number factors and primes fall under the definition of being prime.
    With a bright group it could definitely be worth discussing the differences between the definitions given in pencho's link.
     
  6. The definition implicitly suggests the divisors also need to be positive.
    But these kind of questions are not really in the spirit of number theory anyway, which predominantly concerns positive numbers, and I see little point in asking say whether pi is prime when (as has been pointed out), it's ultimately just about the definition of a prime, rather than looking at more interesting properties of primes.
    If you wanted to do something interesting with your class, how about look at Euler's Theorem? It's a way you can generate new primes given an existing list of primes, and shows that an infinite number of primes exist.
     
  7. If there's an alternative definition of a prime number, it's news to me! [​IMG]

     
  8. I don't believe there is; I suppose I was referring to the way the definition is interpreted, as in pencho's link.
    However actually I am aware that 1 used to be considered prime, and so that it no longer is could either be viewed as a different definition or a different way of interpreting the same definition.
     
  9. I think it's just a cse of me being more careful of how I define the prime number. I simply don't think I've mentioned positive and negative before, just assumed that they would only consider positive factors.
     
  10. A prime number is very specifically determined.
    Each number is a unique product of primes.
    For example, 100 = 2 x 2 x 5 x 5
    If you let 1 be prime then 100 (or any other number for that matter) can no longer be determined by a unique product of primes.
    The numbers that cannot be determined in this way (e.g. 17) are prime.
    The explanation that a number is prime if it has only 2 factors - 1 and itself - is very unhelpful and just adds to the dumbing down of Maths.
    The negative number question is fascinating though. However, they cannot be prime.
    -7 cannot be made up from a unique product of prime numbers. (-7 x 1 or -1 x7). As there are at least two ways to make this number it cannot be prime.
     
  11. DM

    DM New commenter

    Would you like an avatar Standard?
    [​IMG]
     
  12. In fairness, of all the various cool theorems involving primes (e.g. the Prime Number Theorem is a good'un! http://en.wikipedia.org/wiki/Prime_number_theorem), this is perhaps one of the less interesting ones, even if the consequences of the theorem are massively important. [​IMG]
     
  13. Sorry DM, Number Theory in general has to be one of the most boring, sorry, "un-inspiring" things in the world, let alone in mathematics.
    Students do look at me in dismay when I get excited about circle theorems and other geometry and algebra though - now these really are cool.
     
  14. Are you kidding me? What about RSA cryptography, or the race to solve Fermat's Last Theorem that took half a millenium. Or the general aura of mystery that surrounds the field - whether all perfect numbers end with 6 or 8, and work towards proving Goldbach's Conjecture or the Riemann Hypothesis. For me it's probably one of the most awesome fields in mathematics!



     
  15. I can appreciate the things you mentioned and I recognise their significance and relevance but they don't 'spark' my interest really. Actually I did enjoy learning about cryptography at uni. I make up for my lack of interest in number in algebra, geometry, trigonometry, calculus and probability!
     

  16. I
    forgive you then, particularly as I'm a stickler for statistics/probability. I'd
    love to teach my brighter students some 'lottery maths', even if the
    hypergeometric distribution is outside the syllabus. [​IMG]
    And to 'Standard', you rightly say that the fundamental theorem of arithmetic can be used to show that certain numbers can't be considered prime (a proof by contradiction), but this is predicated on already having a definition of a prime; i.e. a consequence not a definition. In other words, you can show by this method a number is not prime, but not that it is prime. Otherwise we could arbitrarily add say 4.237 to our list of primes, since its only prime factorisation (using our extended list of prime numbers) is 1 x 4.237. Sorry to 'dumb maths down'.
     
  17. Sorry, let me backtrack on the last sentence since I included 1. But my point still stands: 4.237 could arbitrarily be considered a prime since it doesn't violate the fundamental law of arithmetic according to your definition.
     
  18. DM

    DM New commenter

    Edited because I failed to read the previous post properly.
     
  19. ' usually mention the Fundamental Theorem of Arithmetic as well but am routinely disappointed by students who fail to find this awe-inspiring.'
    I must admit, I have never found Mathematics awe-inspiring. My jaw never drops, but I sometimes laugh out loud when I come across an unexpected result - foir example that the sum of consecutive odd numbers, starting with 1, is a perfect square.
    It is not until one encounters a system of numbers for which the FTA does not hold that it becomes in any way interesting. Otherwise it is just an 'obvious' tool that we all 'know' to be true for the positive integers.

     
  20. How can a person who assiociates themselve with Mathematics Teaching say such a controversial thing. Without Number Theorems there would be no Circle Theorems or Algebra. It is the foundations of mathematics which everything else is built on.
    Joe

     

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