Le vendredi 20 juillet 2012 03:36:46 UTC+2, horand....@gmail.com a écrit : > On Thursday, 19 July 2012 20:57:27 UTC-3, lut...@gmail.com wrote: > > Le jeudi 19 juillet 2012 17:56:46 UTC+2, Timothy Murphy a écrit : > > &gt; firstname.lastname@example.org wrote: > > &gt; > > &gt; > > &gt; &amp;gt; The question is, are all even integers the sum of two primes? > > &gt; &amp;gt; In other words, could an even integer be the sum of two uneven integers, > > &gt; &amp;gt; at least one of them not being a prime? > > &gt; > > &gt; These two statements do not seem to me to be equivalent. > > > > Yes, thank you, it would be better to limit the question to: > > > > Could an even integer be the sum of two uneven integers, > > at least one of them not being a prime? > > Of course it could: 12 = 3 + 9. You need to show that for every even integer n *there exist* two odd integers, *both of them prime*, that add up to n.
I have slightly modified my text as follows:
Proof of the validity of Goldbach's conjecture _______________________________________
According to the conjecture, every even integer greater than 4 can be expressed as the sum of two primes.
Let?s consider the infinite series of uneven integers. Such series contains an infinite number of products p = ab, where a and b are primes. To each product p corresponds a single sum s = a + b, s being of course an even integer. Such procedure leads to all possible sums of two primes.
By the way, some even integers can be the sum of two uneven integers, at least one of them not being a prime.
Let?s note that a sum s of two primes a and b can be written as s = (a+n) + (b-n), where n is an even integer less than b. The terms (a+n) and/or (b-n) can be prime numbers, but being ordinary uneven numbers does not imply that an even integer cannot be a sum of two primes.
Indeed, a sum of two uneven integers, where at least one of its terms is not prime, can be transformed into a sum s of primes by adding some even integer n to one of its terms and subtracting the same n from its other term. Those who disagree could give an example where this method doesn?t apply.
This leads to the conclusion that any even integer can indeed be expressed as the sum of two primes.