Le 09/08/2012 12:15, Invisible a écrit :
> Warp's challenge on prime numbers got me thinking...
> 
> The product of two integers is always an integer. However, the quotient
> is not always an integer. When it is, we say that the integers are
> "divisible". This idea leads directly to the concepts of "factors" and
> "divisors", and eventually to the "prime numbers". Famously, every
> natural number is uniquely expressible as a product of prime numbers.
> 
> Addition has no such properly. For any given integer, there are an
> unlimited number of integer sums which produce it. And that's because
> unlike a quotient, the difference between two integers /is/ always an
> integer.


Actually there is a conjecture about the sum:

every even number greater than 2 is the sum of two prime numbers.
(Goldbach 's conjecture)

every odd number is at most the sum of three prime numbers.
(easy: add any odd prime to an even number... if you can prove Goldbach!)

So, every positive number is the sum of at most three prime numbers.


On similar lines, every positive integers integer is the sum of 4
squares. (Lagrange, proven).

Another conjecture: Every large odd number (n > 5) is the sum of a prime
and the double of a prime.

Caveat about defining prime number, the definition might or not include
1. (holy war predicted...)
(notice that Goldbach does not state if the sum is unique: it is not. 18
= 13+5 = 11+7)

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