Fibonacci

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Fundamental Theory of arithmetic

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The Fundamental Theorem of Arithmetic

One of the surprising things about mathematics is its insistence that every assertion needs a justification. Non-mathematicians are often surprised by the extent to which mathematicians enforce this dictum. For example, consider the following result, which is usually called the Fundamental Theorem of Arithmetic.

Theorem: Every integer N > 1 can be written uniquely as a product of finitely many prime numbers.

The whole issue of reducing a fraction to lowest terms doesn't make sense unless you can factor both numerator and denominator as a product of primes, and cancel the common factors.

To a mathematician, however, it is precisely because this result is so basic that it needs to be questioned. Is it so fundamental that it needs to be made an axiom? Or is it a consequence of other, simpler statements?

Why does the Theorem only apply to Integers greater than 1?

The main reason for this restriction is that the integer 1 is not a prime number. It meets the criteria presented in the usual careless definition (only divisible by 1 and itself), but careful mathematicians always explicitly exclude 1 from the list of primes. One reason for this exclusion is the uniqueness part of the Fundamental Theorem of Arithmetic. If one allows 1 to become a member of the body of primes, then it has the following prime factorizations

1 = 1.1 = 1.1.1 = 1.1.1.1 = ....

Now, having excluded 1 from the list of primes, 1 can't certainly be written as a product of other primes (all of which are greater than 1). Well, you could if you allowed a product of zero primes, but that seems too much like cheating.

Minus one has a similar problem, but one can handle other negative numbers by factoring out the -1 and then using this theorem. It would then be possible to state the theorem to include this case, but it would distort the main meaning so much that it's not worthwhile.

Factoriztion

Proposition: Every integer N > 1 is a product of finitely many prime numbers.

Proof: Proceed by mathematical induction. To start the induction, we observe that every prime number is such a finite product (consisting, in fact, of the product that only contains itself). We can, therefore, assume inductively that every integer less than N can be written as a product of finitely many primes. If N itself is prime, then we are done. If N is not prime, then by definition it can be written as a product

N = AB

where 1 < A < N and 1 < B < N. Applying the inductive hypothesis to both A and B, we can write each of them as a product of finitely many primes. But then N has also been written as such a product.

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