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What's going on everybody! Welcome
to today's tutorial on Elementary
Number Theory. My name is Jack Jenkins
and this is Academy of One. Today I'm
gonna be talking about diophantine equations.
In this video, I'm gonna be talking about
what is a diophantine equation.
Then I'm going to be talking about pythagorean
triplets. And that's it.
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[Music]. So a diophantine
equation is an intermediate
polynomial equation where variables are
integers. Let's break this down.
Intermediate in mathematics means an
expression with more than one solution. For
instance, 3x + y = 0.
x could be 2 and y could be
negative six or x could be zero and
y could be zero. You can have many
different solutions.
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For a recap. A polynomial
is an expression with both variables a
and constants. It's mostly
numbers. For instance, 3x + 9.
That is a polynomial.
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An equation is an expression that has an
equal sign. For instance, 4 + 1
equals 5. And yes, I did this just
so I can make the slide number complete.
So, an example of a diophantine equation
is ax + by = 1.
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The pythagorean triplets are the
most popular diophantine equation.
The pythagorean triples are
A^n + B^n = C^n.
And the most popular version of this are
the pythagorean theorem. Which states that A^2 + b^2
equals c^2. For instance,
3 + 4 = 5.
This matches the pythagorean theorm.
Now, I want to point out that A^n
+ B^n = C^n.
doesn't work for anything that is greater than 3.
exponents (A^n if n>2). And that is
fermat's last theorm. And that is something we'll cover
in algebraic number theory.
That's it for today's video on elementary number theory.
Tune in next time for when I'm gonna be talking about
the greatest common divisor. Time
for the plugs. You can go ahead and
subscribe to us on youtube. Go ahead and
visit academyofone.org. Go ahead and
donate and ask me questions at jack@academyofone.
where I'll answer them either in the conclusion video or
on hangout thursdays. Also, if you have any questions
for the podcast, you can email them at
podcast@academyofone.org.
And remember, the podcast questions are supposed to be
more open ended. Like how do I find
a girlfriend. Let's say that one.
The correct answer is,
just be yourself. [Music]
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