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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)





The complete lack of any mention of Zero in this discussion has sucked out all meaning for me, and left me inside a total vacuum.
Since Zero multiplied, or divided (except of course Zero divided by Zero), by any number, natural perverted, or even fractional, will always be Zero: therefore Zero is the Prime of Primes, not to mention that Zero raised to any power remains Zero, not to mention that subtracting Zero from, or adding Zero to, any number leaves the number unchanged !
That any number divided by Zero is an infinity (whose ordinality, or Aleph, among other possible infinities: is ultimate ?) which cannot be conceptualized within linearly digital Turing/Von Neumann theoretical computational design, and must be expressed by some "placeholder" like "undefined," or "NaN," or will, on a practical level, in many circumstances crash a computer: is proof of its sacred power.
Zero is the unique singularity of the transition between positive and negative numbers, thus equivalent to the Omphalos, the stone of the navel of the geobody of the cosmos, which for the ancient Greeks was located at the shrine of the oracle at Delphi.
I propose to you that the infinite set of all possible prime numbers is contained within the infinity created by Zero divided by Zero like a tiny foot in a huge shoe: lots of wiggleroom no matter what #1 does, or does not, do.
best, Bill
"Takuan Sōhō died in Edo (presentday Tokyo) in December of 1645. At the moment before his death, Takuan painted the Chinese character 'meng' ("dream"), laid down his brush and died."





BillWoodruff wrote: That any number divided by Zero is an infinity
Normal mathematics defines that as undefined, not infinity.
A divisor that approaches an infinitely small value produces a value that approaches infinity.





I think he was talking about Lim(1/0), that is infinity.
for any other operation, 1/0 is undefined, just because no one can think what would be the result.
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)





Limit(lim) is still a convergence sequence. Which is the same as what I said.
And equating it to infinity is a misrepresentation of what it means mathematically.







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