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Vijay Sringeri wrote: The link for "Euclid's lemma" was helpful, however it builds other theorems based on the fundamental fact that " Any non prime number can be expressed as product of prime numbers".
There is nothing in mathematics that is not based on other proofs, assumptions and/or definitions.
And there is a proof for that lemma.






Hey, this is my standard answer to my children's questions.
~RaGE();
I think words like 'destiny' are a way of trying to find order where none exists.  Christian Graus
Do not feed the troll !  Common proverb





Thanks all, for showing keen interest in answering/trying to answer this questions.
But my question still remains unanswered
However, I just wanted to put some info here.
Prime number : Numbers > 1, and which has 1 and itself as it factor is prime nuber.
Composite number : All non prime numbers are composite numbers.
What about 1 then ?
1 is neither prime nor composite.





Wheter or not to include 1 in the list of prime numbers is debated among mathematicians. There are arguments to include it, and argument to not include it.
1 cant be written as a product of smaller primes except 1*1
However 1*N = N so you could always write any nyumber as a product of two primes if that was the case.






Quote: Why this unique ability for prime numbers ?How is it possible that, any number can be expressed as product of prime factors ?
Both of these two questions could be answered by the fundamental theorem of aritmatic.
Quote: What is it, which makes these prime numbers special ?
You could read my article, and there are lots of referances there.
Finding prime numbers[^]





Thanks a lot, your article is very informative.





Vijay Sringeri wrote: Why this unique ability for prime numbers ?
It is not a unique ability for prime numbers, it is just that these numbers can not be subdivided any further. You can find the LCM and HCF using any numbers, but they will always be a combination of prime factorials, so using prime numbers is far easier.
Vijay Sringeri wrote: How is it possible that, any number can be expressed as product of prime factors ?
Essentially because a prime number can not be divided and a non prime number can be.
Any number n that is not prime has at least two divisors that are not 1 and n. These divisors are either prime or non prime. If they are non prime then by definition they follow the same rule as n. These numbers are smaller then n, so repeating this rule will always result in only prime divisors.
Vijay Sringeri wrote: What is it, which makes these prime numbers special ?
The fact that they are prime and can not be divided.





Member 2053006 wrote: Any number n that is not prime has at least two divisors that are not 1 and n.
what about 4 ? AFAIK, for has only one divisor thats not 1 or 4, and it's 2...
I'm brazilian and english (well, human languages in general) aren't my best skill, so, sorry by my english. (if you want we can speak in C# or VB.Net =p)







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