Complex identities
by t0rajir0u, Dec 5, 2006, 3:58 PM
Let's talk about a fun identity due to the theory of complex numbers.
Problem: Show that any integer solutions
to
Must satisfy
or 
.
Approach 1: This is the fun identity. Now, we know that
(By the way, a number of the form
where 
are integers is known as a Gaussian integer. This ring has many useful properties (for example, prime factorization) that are worth studying.) Let's calculate the "norm" of both sides. The norm 
is a multiplicative function (good lemma!), so
This is a very useful identity, especially for proving things about the properties of the integers representible as a sum of two squares. (Incidentally, this is a great way to prove Cauchy in Euclidean space.) Therefore, our given problem is equivalent to
Which is obviously absurd in the integers unless
. But then 
, and either or as desired. QED.
Approach 2: Has less to do with complex numbers, but is worth sharing anyway. It's related: given the vectors
, the above is equivalent to
Where
is the angle between the vectors. (Note that the RHS has an integer value, so the LHS also has an integer value.) But this means that
And of course, we know that
Where the LHS should have an integer value, but the RHS clearly cannot - unless
or 
. QED.
(Next entry: more applications of factoring over the complex numbers. And... beyond the Gaussian integers?
)
Practice Problem 1: (to take a leaf out of paladin8's book
)
Find all positive integer solutions
to
(There aren't many, but do this quickly and without a calculator.)
Practice Problem 2: Derive the Pythagorean Triple Generating Formulae in as few lines as possible.
Problem: Show that any integer solutions
to
Must satisfy
or 
.Approach 1: This is the fun identity. Now, we know that
(By the way, a number of the form
where 
are integers is known as a Gaussian integer. This ring has many useful properties (for example, prime factorization) that are worth studying.) Let's calculate the "norm" of both sides. The norm 
is a multiplicative function (good lemma!), so
This is a very useful identity, especially for proving things about the properties of the integers representible as a sum of two squares. (Incidentally, this is a great way to prove Cauchy in Euclidean space.) Therefore, our given problem is equivalent to
Which is obviously absurd in the integers unless
. But then 
, and either or as desired. QED.Approach 2: Has less to do with complex numbers, but is worth sharing anyway. It's related: given the vectors
, the above is equivalent to
Where
is the angle between the vectors. (Note that the RHS has an integer value, so the LHS also has an integer value.) But this means that
And of course, we know that
Where the LHS should have an integer value, but the RHS clearly cannot - unless
or 
. QED.(Next entry: more applications of factoring over the complex numbers. And... beyond the Gaussian integers?
)Practice Problem 1: (to take a leaf out of paladin8's book
)Find all positive integer solutions
to
(There aren't many, but do this quickly and without a calculator.)
Practice Problem 2: Derive the Pythagorean Triple Generating Formulae in as few lines as possible.
for positive integers 
with 
means 
.
is "prime" over the Gaussian integers, this is impossible because then either 
or one of the factors is a unit. So 
.
.
,
so 
and 
.
generates all primitive Pythagorean triples. Yeah... it goes something like that.
,
.
.
April 2009
March 2009