What is the famous prime number theorem?
Could you please explain to me in simple terms what is the famous Prime Number Theorem? I've heard it's related to the distribution of prime numbers, but I'm not quite sure how it works or what its significance is in the field of mathematics. Could you provide a concise overview of the theorem, including its main points and why it's considered so important?
What is the fundamental theorem of abelian groups?
Could you please elaborate on the fundamental theorem of abelian groups? Specifically, how does it relate to the structure and properties of abelian groups? Is there a concise way to summarize its key insights, and how does it help in understanding the behavior of these mathematical structures? I'm particularly interested in how this theorem can be applied in the context of cryptography and blockchain technology, if there are any relevant implications.