Why isn’t the fundamental theorem of arithmetic obvious? (2011)
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Why isn’t the fundamental theorem of arithmetic obvious? (2011)
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Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#2In math school we had a saying: "obvious means easy to prove". So the problem is about recognizing the difference between proofs and non-proofs. The hard but satisfying way to learn that difference is to start with axioms. Take some simple system of axioms that holds for Z, and try to prove the FTA from these axioms alone. Then check that the axioms aren't satisfied by Z[sqrt(-5)], or the even numbers, or some other ring where the FTA doesn't hold.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#3Why isn't 2+2==4 obvious?
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#4"The fundamental theorem of arithmetic can be approximately interpreted 3 * 5 * 13 and 3 * 13 * 5" - this is false. The fundamental theorem of arithmetic "can be more approximately interpreted as " 195 has one prime factorization and it only includes one 3, one 5, and one 13 and no other primes. Once we find one prime factorization we want to prove that 195 does not have a prime factorization that includes another 11 or other prime.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#5I've recently learned about Gödel numbering and I have to admit; it's my favourite application of this theorem. Sorry RSA.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#6"The fundamental theorem of arithmetic can be approximately interpreted 3 * 5 * 13 and 3 * 13 * 5" - this is false. The fundamental theorem of arithmetic "can be more approximately interpreted as " 195 has one prime factorization and it only includes one 3, one 5, and one 13 and no other primes. Once we find one prime factorization we want to prove that 195 does not have a prime factorization that includes another 11…
Where did you read that?
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#7"The fundamental theorem of arithmetic can be approximately interpreted 3 * 5 * 13 and 3 * 13 * 5" - this is false. The fundamental theorem of arithmetic "can be more approximately interpreted as " 195 has one prime factorization and it only includes one 3, one 5, and one 13 and no other primes. Once we find one prime factorization we want to prove that 195 does not have a prime factorization that includes another 11…
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Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#8"The fundamental theorem of arithmetic can be approximately interpreted 3 * 5 * 13 and 3 * 13 * 5" - this is false. The fundamental theorem of arithmetic "can be more approximately interpreted as " 195 has one prime factorization and it only includes one 3, one 5, and one 13 and no other primes. Once we find one prime factorization we want to prove that 195 does not have a prime factorization that includes another 11…
It doesn't say that. What it says is:
The fundamental theorem of arithmetic states that
every positive integer can be factorized in one
way as a product of prime numbers. This statement
has to be appropriately interpreted: we count the
factorizations 3x5x13 and 13x3x5 as the same, for
instance.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#9I remember reading a proof to the fundamental theorem of arithmetic (every number is composed of a unique multiple of primes) in a number theory book and really enjoying it. I would disagree with Gowers and say it is obvious intuitively, but I'd still argue it is worth writing a proof for.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#10"The fundamental theorem of arithmetic can be approximately interpreted 3 * 5 * 13 and 3 * 13 * 5" - this is false. The fundamental theorem of arithmetic "can be more approximately interpreted as " 195 has one prime factorization and it only includes one 3, one 5, and one 13 and no other primes. Once we find one prime factorization we want to prove that 195 does not have a prime factorization that includes another 11…
Where did you read that?
https://www.google.com/url?sa=t&source=web&rct=j&url=http://... note that prime factorization is unique up to order. We aren't talking about order when we use the term uniqueness.