Sri SaiRam engineering College
Introduction to number theory lecture 1.
Introduction to Prime Numbers
A prime number is a number bigger than one that's only divisible by one and itself. The first few
primes are:
2
3
5
7
11
13
17
19
We can ask some basic questions about prime numbers. The first question is how do we find
primes.
Finding Primes
The first question is how do we find primes. Well, if we look at this, one way to do it is to use the
sieve of Eratosthenes. The first number we haven't crossed out is 2, so we write down 2 as a
prime and we then cross off all the multiples of 2. Because these can't be prime, we move on to
the next number that hasn't been crossed off, which is 3. We write down 3 as a prime and then
cross off all the multiples of 3. We continue in this manner until we have found all the primes up
to a given number.
The course will cover learning all the primes less than the square root of 50, and if you're doing
number three, you will very soon learn all primes up to about a hundred.
Number of Primes
The next question is to see how many primes there are. If we look and see what numbers we've
got left, we have the prime numbers. The number of prime numbers is actually infinite according
to number theory due to Euclid. Table primes seem to be fairly common, and there's no sign of
them suddenly stopping. This suggests that the number of primes should be infinite. This is the
first theorem of number theory.
Finding Large Primes
The next question is how to find large primes. Number theorists compete to see who can find
the biggest prime. Number theorists look at things called Mersenne primes. These are primes of
the form 2 to the n minus 1. The largest known prime is actually a Mersenne prime because
they're particularly easy to find, and so people have used computers to find large numbers of
Mersenne primes. In fact, there are very good tests to see if a very large number is a Mersenne
prime, and you can actually test numbers with millions of digits to see whether they're Mersenne
primes. So, we should also look at primes of the form 2 to n plus 1.
Introduction to number theory lecture 1.
Introduction to Prime Numbers
A prime number is a number bigger than one that's only divisible by one and itself. The first few
primes are:
2
3
5
7
11
13
17
19
We can ask some basic questions about prime numbers. The first question is how do we find
primes.
Finding Primes
The first question is how do we find primes. Well, if we look at this, one way to do it is to use the
sieve of Eratosthenes. The first number we haven't crossed out is 2, so we write down 2 as a
prime and we then cross off all the multiples of 2. Because these can't be prime, we move on to
the next number that hasn't been crossed off, which is 3. We write down 3 as a prime and then
cross off all the multiples of 3. We continue in this manner until we have found all the primes up
to a given number.
The course will cover learning all the primes less than the square root of 50, and if you're doing
number three, you will very soon learn all primes up to about a hundred.
Number of Primes
The next question is to see how many primes there are. If we look and see what numbers we've
got left, we have the prime numbers. The number of prime numbers is actually infinite according
to number theory due to Euclid. Table primes seem to be fairly common, and there's no sign of
them suddenly stopping. This suggests that the number of primes should be infinite. This is the
first theorem of number theory.
Finding Large Primes
The next question is how to find large primes. Number theorists compete to see who can find
the biggest prime. Number theorists look at things called Mersenne primes. These are primes of
the form 2 to the n minus 1. The largest known prime is actually a Mersenne prime because
they're particularly easy to find, and so people have used computers to find large numbers of
Mersenne primes. In fact, there are very good tests to see if a very large number is a Mersenne
prime, and you can actually test numbers with millions of digits to see whether they're Mersenne
primes. So, we should also look at primes of the form 2 to n plus 1.
