Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,544

To Radic: I've looked over your paper and have a few comments.
You start out by giving a complicated proof, involving factorization into primes to show that the product of primes,plus one, cannot have any of those primes as a factor. I don't know why you did that, it is sufficient to argue (as Euclid did 2000 years ago) that, if you divide a_{1}a_{2}...a_{n}+ 1 by any of those primes, you get a remainder of 1. You conclusion, that there must be another prime, less than a_{1}a_{2}...a_{n}+1 that is a factor is not quite correct: it might be a prime itself.
You then go on to argue that the product of primes a_{1}, a_{2},..., a_{n} must be larger than a_{n+1}. I'm not completely convince by your argument, but, in fact, there is no reason to think that is not true.
(By the way, you define "a_{1},a_{2},...,a_{n} as "Set of primes such that a_{1}2< a_{2}<...< a_{n}. From what you do later, it's clear that you intend that a_{1}= 2, a_{2}= 3,... In other words that the set is the set of the first n primes. If that is what you mean, you should say so. The way you define the set it could be {3, 17, 37}. That is a set of primes such that 3< 17< 37.)
You arrive, finally, at the conclusion that n^{2}> a_{n}. Okay, that seems reasonable: 2^{2}= 4> 3, 3^{2}= 9> 5, 4^{2}= 16> 7 and the difference is getting larger. I'm willing to accept it.
You then assert "We have todetermine number of numbers that can be sum of k pairs of different primes. For example 24= 13+ 11= 17+ 7= 23+1. The number of that combinations is n(n+1)/2." This where I got lost.
First, a minor point: your example is incorrect. 23+1 is not a sum of primes because 1 is not a prime. However, you did not include 19+ 5 so your you could replace 23+1 with that: 24 can be written as a sum of two primes in 3 different ways.
However, my first reaction on reading that was "What is n? What happened to k?" You talked about the "number of numbers that can be sum of k pairs of different primes" and from your example I thought you meant "numbers that can be written as a sum of two primes in exactly k different ways". Even if we assume n was accidently typed for k, that formula doesn't work. In fact, I don't see any reason not to think that the "number of integers that can be written as the sum of two primes in exactly 3 different ways" isn't infinite.
In any case, I THINK what you actually meant was number of diffent possible combinations of the first n prime numbers, taken 2 at a time, and not considering permutations (so that we do not count both 17+ 7 and 7+ 17), is n(n1)/2. That's correct. (But does not count such possiblities as 17+17 or 7+ 7.)
Then you go on to say: "when we sub it from n^{2} we get that of n primes could be at least n(n1)/2 different even numbers.
Do you mean "subtract it from ... ". Where did you get the n^{2}. How did subtracting n(n1)/2 from n_{2} give you information about n(n1)/2? (n_{2} n(n1)/2= n(n+1)/2.)
In any case, it is clear that "from n primes" we CANNOT form
n(n1)/2 DIFFERENT even numbers. That is the figure for all possible sums of two distinct prime numbers of the first n. As your example showed they are not all different. Surely the correct figure is much smaller than n(n1)/2.
"now let sub of a_{n},n (because there are in interval a_{n} and divide it by 2 to eliminate odd numbers"
We may have a language problem here. I have absolutely no idea what you mean by "let sub of a_{n},n". Is "sub" here "subtract"? Are you subtracting n from a_{n} or exactly what?
I think, from what you say following, that you are subtracting n (the number of primes less than or equal to a_{n}) from a_{n} to determine how many "nonprimes" there are less than a_{n}. You then divide by 2 "to eliminate odd numbers". Excuse me? Are you assuming that of the numbers left exactly half are even and half are odd? How can that be when you have just removed a fair number of ODD numbers from the set?
For example, if n= 7, a_{7}= 17. There are, of course, 17 numbers less than that and 7 of them are prime. 17 7= 10 of them are composite. They are: 1,4, 6, 8, 9, 10, 12, 14, 15, 16.
Exactly 7 of those are even (4, 6, 8, 10, 12, 14, 16) and only 3 (1, 9, 15) are odd.
You end by declaring that the total number of "different even numbers" that can be formed by adding two primes less than or equal to a_{n} is n(n1)/2 while the total number of even numbers less than a_{n} is (a_{n}n)/2 and
n(n1)/2> (a_{n}n)/2. Finally you assert that since
n(n1)/2 > (a_{n}n)/2, Goldbach's conjecture follows.
Both of those numbers are incorrect. Since n(n1)/2 counts different pairs of primes that give the same sum, it is too large.
Since (a_{n}n)/2 assumes that the number of even and odd numbers in the composites less than a_{n} are the same (and they are not), it is too small.
By the way, if your assertions WERE true, those two numbers should have been exactly the same. Why did you think that
n(n1)/2 should be greater than (a_{n}n)/2 in order for Goldbach's conjecture to be true?
