Introduction to number theory lecture 4. More on Euclid's algorithm
Elementary Number Theory
In the last lecture, we covered Euclid's algorithm, which can be used to find the greatest
common divisor of two numbers, a and b. We can solve linear equations like 71x + 17y = 1 by
working backward from the greatest common divisor. For example, we can write 1 as 6(3) -
4(71) using the equation 71x + 17y = 1. This algorithm can be used to solve equations like ax +
by = d, where d is the greatest common divisor of a and b. The algorithm is very efficient and
takes approximately log(a) steps. However, it fails for polynomials in two variables over the
reals.
Euclid's Algorithm
Can be used to find greatest common divisor of two numbers a and b.
Solves linear equations like 71x + 17y = 1.
Efficient and takes approximately log(a) steps.
Fails for polynomials in two variables over the reals.
Solving Linear Equations
We can solve a linear equation like ax + by = c if and only if the greatest common divisor of a
and b divides c. The same applies to polynomials in one variable over the reals. However,
Euclid's algorithm fails for polynomials in two variables over the reals. To solve equations in
three variables, we can use methods like substitution. For example, to solve 6x + 10y + 15z = 0,
we can find one solution and then add multiples of the coefficients to find more solutions.
Coprime Polynomials
If two polynomials a and b are coprime, then there exists a linear combination of a and b that
equals 1. This means that we can solve equations like ax + by = 1 using Euclid's algorithm.
However, this does not necessarily mean that there is a single polynomial in x or y that equals
1. To solve equations with coprime polynomials, we can add multiples of the coefficients to find
more solutions.
Solving Linear Equations and Finding Greatest Common Divisors
When dealing with linear equations and finding greatest common divisors, there are some
efficient methods to use.
The greatest common divisor of two numbers can be found using Euclid's algorithm, where the
solution to the equation is a solution to the greatest common divisors of two variables.
If there are several linear equations, a solution can be found if the greatest common divisor of
the variables divides a given number.
Long division can be slow and complicated, but there is a slightly better algorithm for finding the
greatest common divisor, which involves taking out all factors of 2 and using subtraction instead
of division.
Elementary Number Theory
In the last lecture, we covered Euclid's algorithm, which can be used to find the greatest
common divisor of two numbers, a and b. We can solve linear equations like 71x + 17y = 1 by
working backward from the greatest common divisor. For example, we can write 1 as 6(3) -
4(71) using the equation 71x + 17y = 1. This algorithm can be used to solve equations like ax +
by = d, where d is the greatest common divisor of a and b. The algorithm is very efficient and
takes approximately log(a) steps. However, it fails for polynomials in two variables over the
reals.
Euclid's Algorithm
Can be used to find greatest common divisor of two numbers a and b.
Solves linear equations like 71x + 17y = 1.
Efficient and takes approximately log(a) steps.
Fails for polynomials in two variables over the reals.
Solving Linear Equations
We can solve a linear equation like ax + by = c if and only if the greatest common divisor of a
and b divides c. The same applies to polynomials in one variable over the reals. However,
Euclid's algorithm fails for polynomials in two variables over the reals. To solve equations in
three variables, we can use methods like substitution. For example, to solve 6x + 10y + 15z = 0,
we can find one solution and then add multiples of the coefficients to find more solutions.
Coprime Polynomials
If two polynomials a and b are coprime, then there exists a linear combination of a and b that
equals 1. This means that we can solve equations like ax + by = 1 using Euclid's algorithm.
However, this does not necessarily mean that there is a single polynomial in x or y that equals
1. To solve equations with coprime polynomials, we can add multiples of the coefficients to find
more solutions.
Solving Linear Equations and Finding Greatest Common Divisors
When dealing with linear equations and finding greatest common divisors, there are some
efficient methods to use.
The greatest common divisor of two numbers can be found using Euclid's algorithm, where the
solution to the equation is a solution to the greatest common divisors of two variables.
If there are several linear equations, a solution can be found if the greatest common divisor of
the variables divides a given number.
Long division can be slow and complicated, but there is a slightly better algorithm for finding the
greatest common divisor, which involves taking out all factors of 2 and using subtraction instead
of division.
