JFZ: how do we know there are infinitely many primes
ChatGPT: One way to show that there are infinitely many prime numbers is to use the fact that the set of prime numbers is infinite to show that there must be an infinite number of composite numbers as well. Composite numbers are positive integers that have at least one positive integer divisor other than 1 and itself. For example, 4 is a composite number because it can be divided by 2 (and is not prime), and 6 is a composite number because it can be divided by 2 and 3 (and is not prime).
To show that there are infinitely many composite numbers, consider the sequence of numbers 2, 4, 6, 8, 10, 12, 14, 16, and so on. Each of these numbers is composite because it can be divided by 2. Now consider the sequence of numbers 3, 6, 9, 12, 15, 18, 21, 24, and so on. Each of these numbers is composite because it can be divided by 3. And so on. For every prime number, there is an infinite sequence of composite numbers that can be generated by multiplying that prime number by other positive integers.
Since there are infinite sequences of composite numbers, and each composite number is the product of at least two prime numbers (for example, 12 is the product of 2 and 6, which are both prime numbers), there must be an infinite number of prime numbers as well. This is one way to show that there are infinitely many prime numbers, though there are other ways to prove it as well.
(seems hand wavey, and probably wrong though I can't figure out where the flaw is)
JFZ: is A5 a solvable group
Yes, the group A5 (also known as the alternating group on 5 elements) is a solvable group. A group is solvable if it can be constructed from simpler groups using only group extensions and group quotients, and if it has a composition series in which each factor is a simple group.
(wrong)
The group A5 is the group of even permutations of five objects. It has 60 elements, and it is isomorphic to the group of rotations of a regular dodecahedron. A5 is a simple group, which means that it has no proper normal subgroups (subgroups that are normal in the group but not equal to the group itself). This means that A5 cannot be constructed from simpler groups using group extensions or group quotients. However, it can be shown that A5 has a composition series in which each factor is a simple group, so it is solvable.
In general, many groups that arise in mathematics and physics are solvable, and the study of solvable groups is an important area of research in group theory.
> (seems hand wavey, and probably wrong though I can't figure out where the flaw is)
The flaw is here:
> Since there are infinite sequences of composite numbers, and each composite number is the product of at least two prime numbers (for example, 12 is the product of 2 and 6, which are both prime numbers), there must be an infinite number of prime numbers as well.
This doesn't establish that the prime numbers are distinct. The infinite sequence of even numbers has a single prime number in it, for example.
First it says I don't know the answer to 1+2. Then it says ofcourse 1+2 must be 1. In either case, its the same exact question, and if you assert closure on 3 elements, with 1+1=3 and 1+3=2, you are left with no choice other than 1+2=1 because closure and Cayley Hamilton. The very first property it lists is closure! So it knows what closure is, but doesn't know how to use it. But give it the answer to 2+1 and suddenly it uses associativity !
JFZ: how do we know there are infinitely many primes
ChatGPT: One way to show that there are infinitely many prime numbers is to use the fact that the set of prime numbers is infinite to show that there must be an infinite number of composite numbers as well. Composite numbers are positive integers that have at least one positive integer divisor other than 1 and itself. For example, 4 is a composite number because it can be divided by 2 (and is not prime), and 6 is a composite number because it can be divided by 2 and 3 (and is not prime).
To show that there are infinitely many composite numbers, consider the sequence of numbers 2, 4, 6, 8, 10, 12, 14, 16, and so on. Each of these numbers is composite because it can be divided by 2. Now consider the sequence of numbers 3, 6, 9, 12, 15, 18, 21, 24, and so on. Each of these numbers is composite because it can be divided by 3. And so on. For every prime number, there is an infinite sequence of composite numbers that can be generated by multiplying that prime number by other positive integers.
Since there are infinite sequences of composite numbers, and each composite number is the product of at least two prime numbers (for example, 12 is the product of 2 and 6, which are both prime numbers), there must be an infinite number of prime numbers as well. This is one way to show that there are infinitely many prime numbers, though there are other ways to prove it as well.
(seems hand wavey, and probably wrong though I can't figure out where the flaw is)
JFZ: is A5 a solvable group
Yes, the group A5 (also known as the alternating group on 5 elements) is a solvable group. A group is solvable if it can be constructed from simpler groups using only group extensions and group quotients, and if it has a composition series in which each factor is a simple group. (wrong)
The group A5 is the group of even permutations of five objects. It has 60 elements, and it is isomorphic to the group of rotations of a regular dodecahedron. A5 is a simple group, which means that it has no proper normal subgroups (subgroups that are normal in the group but not equal to the group itself). This means that A5 cannot be constructed from simpler groups using group extensions or group quotients. However, it can be shown that A5 has a composition series in which each factor is a simple group, so it is solvable.
In general, many groups that arise in mathematics and physics are solvable, and the study of solvable groups is an important area of research in group theory.