# Euclid's Theorem - Understanding Proofs & Induction Process

• nebbione
In summary, the conversation discusses the use of induction in proofs and specifically asks why it cannot be used in Euclid's Theorem. The response explains that there is no general formula for proving theorems and that understanding more math will make proofs clearer. It is also mentioned that induction cannot be used in Euclid's Theorem because it involves prime numbers distributed randomly in N, not contiguous numbers.
nebbione
Hi everyone! In this period I'm trying to understand how proofs work.
And i don't understand why the induction process is only used sometimes. When can i use it ?

For example in the euclid's theorem http://en.wikipedia.org/wiki/Euclid's_theorem
why this could not be proven with the induction ?

Hi. There is no general formula or template for proving something. You can't just take the "induction template" and use that to prove any old theorem.

I am curious how much math you actually know. Because frankly, I think you should focus on learning more math, because with that education, proofs will become clearer.Edit:
http://en.wikipedia.org/wiki/Mathematical_proof

Can you explain me why i can't use induction always when talking for examples of euclid's theorem ? we only have natural numbers in this context

Do you know what induction is? Can you tell me in your own words?

Do you know what Euclids Theorem is? Can you tell me in your own words?

sorry i re read the induction and the euclid's theorem and now i understand.
You are right.
I cannot use induction since I'm not considering contigous numbers, but thre prime numbers distributed randomly in N.

## 1. What is Euclid's Theorem?

Euclid's Theorem, also known as the Fundamental Theorem of Arithmetic, states that every positive integer can be expressed as a unique product of primes.

## 2. How did Euclid prove his theorem?

Euclid used a proof by contradiction, assuming that there exists a finite set of primes, and then showing that there must be at least one additional prime not in that set. This contradicts the assumption, proving that there is no finite set of primes.

## 3. What is the role of induction in Euclid's Theorem?

Induction is used to prove that every positive integer can be expressed as a product of primes. It involves showing that the theorem holds for a base case (such as 1) and then proving that if it holds for any positive integer, it also holds for the next integer. This process is repeated to cover all positive integers.

## 4. Can Euclid's Theorem be applied to negative numbers or non-integer numbers?

No, Euclid's Theorem only applies to positive integers. Negative numbers and non-integer numbers do not have a unique prime factorization.

## 5. What are some real-world applications of Euclid's Theorem?

Euclid's Theorem has many practical applications, including in cryptography, number theory, and computer science. It is also used in algorithms for finding the greatest common divisor of two numbers and determining whether a number is prime.

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