Sri SaiRam engineering College
Introduction to number theory lecture 2: Survey.
This lecture is a continuation of the previous one and serves as a survey of some of the topics
that will be covered later in the course.
Congruences
We often use congruences to solve problems in number theory. For example, to determine if
1234567 is a perfect square, we can look at it modulo 10. If a number is congruent to another
number modulo n, it means their difference is divisible by n. Using this, we can quickly rule out
1234567 being a perfect square. Similarly, we can use congruences to check if large numbers
are prime without finding their factors.
Fermat's and Euler's Theorems
Fermat's and Euler's theorems are some of the most useful theorems in number theory. They
are often used to solve Diophantine equations.
Quadratic Equations
For quadratic equations of the form x^2 = a (mod p), where p is prime, we can use the Legendre
symbol to determine if a has a square root modulo p. The Legendre symbol is defined as
follows:
If a is divisible by p, the Legendre symbol is 1.
If a is not divisible by p and has a square root modulo p, the Legendre symbol is 1.
If a is not divisible by p and does not have a square root modulo p, the Legendre symbol is -1.
Using the Legendre symbol, we can quickly calculate if a is a quadratic residue modulo p. This
is useful for finding prime numbers.
The Hasse Principle
The Hasse principle states that if a function has a solution modulo m for all m, then it has a
solution in the integers. This principle is useful for solving Diophantine equations.
Number Theory and Prime Numbers
Number theory involves studying the properties of numbers, specifically integers. Additive
number theory involves questions about adding numbers together, such as Goldbach's
conjecture, which asks if every even number greater than or equal to four is the sum of two
primes. Multiplicative number theory involves questions about multiplying numbers together,
such as finding the product of two primes.
The law of quadratic reciprocity determines whether numbers are squares or not. It is a simple
calculation based on modular arithmetic. A prime can be expressed as the sum of two squares if
and only if it is congruent to 1 modulo 4.
Introduction to number theory lecture 2: Survey.
This lecture is a continuation of the previous one and serves as a survey of some of the topics
that will be covered later in the course.
Congruences
We often use congruences to solve problems in number theory. For example, to determine if
1234567 is a perfect square, we can look at it modulo 10. If a number is congruent to another
number modulo n, it means their difference is divisible by n. Using this, we can quickly rule out
1234567 being a perfect square. Similarly, we can use congruences to check if large numbers
are prime without finding their factors.
Fermat's and Euler's Theorems
Fermat's and Euler's theorems are some of the most useful theorems in number theory. They
are often used to solve Diophantine equations.
Quadratic Equations
For quadratic equations of the form x^2 = a (mod p), where p is prime, we can use the Legendre
symbol to determine if a has a square root modulo p. The Legendre symbol is defined as
follows:
If a is divisible by p, the Legendre symbol is 1.
If a is not divisible by p and has a square root modulo p, the Legendre symbol is 1.
If a is not divisible by p and does not have a square root modulo p, the Legendre symbol is -1.
Using the Legendre symbol, we can quickly calculate if a is a quadratic residue modulo p. This
is useful for finding prime numbers.
The Hasse Principle
The Hasse principle states that if a function has a solution modulo m for all m, then it has a
solution in the integers. This principle is useful for solving Diophantine equations.
Number Theory and Prime Numbers
Number theory involves studying the properties of numbers, specifically integers. Additive
number theory involves questions about adding numbers together, such as Goldbach's
conjecture, which asks if every even number greater than or equal to four is the sum of two
primes. Multiplicative number theory involves questions about multiplying numbers together,
such as finding the product of two primes.
The law of quadratic reciprocity determines whether numbers are squares or not. It is a simple
calculation based on modular arithmetic. A prime can be expressed as the sum of two squares if
and only if it is congruent to 1 modulo 4.
