Let n be an integer greater than one. Prove that n is a perfect square i.f.f. n's prime factorization, n=p_1*p_2*...*p_r, has only even exponents.
So I think I'm doing an analytical proof that establishes a biconditional with reference to the above statement.
By definition a perfect square is a number, n, such that n=m^2x=(m^x)^2 and after substituting,
I need to show that each exponent on a prime is even and I'm not entirely sure how to do this at the moment.
I don't think I can just go straight to
since I would need some kind of condition that requires each exponent to be an integer (maybe?) and that 2 divides any a with no remainder?
The proof of the other direction, from each prime factor having even exponents to n being a perfect square, is done already.
OK I think I can work it out this way;
If n is a perfect square then it has the form of some integer, m, times itself.
By the Fundamental Theorem of Arithmetic n has a unique factorization into distinct primes up to order.
Since by definition a perfect square is some number times itself I have to be able to group the factors if n into two equal groups.
For any given a, if that a is even then it is perfectly divisible by two and so can be split into two equal groups.
For any given a, if that a is odd then it is not perfectly divisible into two groups. From which there are two cases;
Either you have integer exponents and unequal groupings, in which case A does not equal A and there is a contradiction with the definition of a perfect square.
Or, you have fractional exponents in equal groupings. In other words each grouping, A, contains at least one root and possibly more. Since each A is a grouping of distinct primes it is not that case that,
Where N is an integer.
Right here I want to say "As a perfect square is the square of an integer and as A can not be an integer, there is a contradiction if there is at least one odd exponentiation on a prime factor of n." I am however not entirely sure that the product of distinct primes and some variable number of roots of primes is not an integer.
Like the standard counter example to the claim that "The product of two irrational numbers is irrational" is that of,
which is of course not the case here since we're taking about the product of distinct primes with fractional exponents.
If we have:
(it is typical to insist that: to avoid questions of "re-ordering" and also to leave out the special case n= 1),
then for each which is prime, we certainly have and then because , and is PRIME, it divides one of m or m, that is to say, it divides m.
Since divides m, some highest positive power of it divides m. let's say this power is . Thus:
is an integer, which does NOT divide.
Thus is likewise an integer that does not divide.
By the uniqueness of the prime factorization for this integer, we have:
and cross-multiplication gives:
so again by the uniqueness of prime factorization for n, we have:
and uniqueness of factorization for THIS number forces:
that is, is even, for every j.