Prove these two statements are logically equivalent

Question

Consider the following two statements about non-negative integers:

• Goldbach's conjecture: Every integer > 1 is the average of two primes

• SumOfThreePrimes: Every integer > 5 is the sum of three primes.

Show that both statements are logically equivalent.Namely,

1. Golabach's conjecture implies SumOfThreePrimes

2. SumOfThreePrimes implies Golabach's conjecture

I have tried direct proof, indirect proof and proof by contraposition.

But all of these don't work for me.

Any assistance will be appreciated. Thanks.

Answer 1

They are not logically equivalent. Goldbach's weak conjecture has been recently proved, while the strong conjecture is still open.

Answer 2

I have proved by myself.

Let Goldbach's conjecture denote as $P$,

and SumOfThreePrimes denote as $Q$.

Assume $P$ is correct.

For the first statement.

Using direct proof.

According to $P$,

an integer $n > 1$ is equal to the average of two primes $x, y $

Namely, $n= (x+y)/2$ where $n>1$

which can be written as $2n = x + y$ where $2n>2$

Adding $3$ on the equation,

we can get $2n + 3 = x + y + 3$ where $2n + 3 > 5$ for every integer n > 1

This equation is same as the statement $Q$ except for integer $6$.

We can solve it manually, $6 = (7+5)/2$ and $6 = 2 + 2 + 2$.

So we can conclude that Golabach's conjecture implies SumOfThreePrimes.

For the second statement.

According to Q.

Every integer n > 5 is the sum of three primes $x, y, z$

which can be written as $n = x + y + z$ for $n > 5$

equal to $(n - z)/2 = (x+y)/2$

since $n-4>=x,y,z>=2$

$(n-z)/2$ will in a range for $2<=(n-z)/2<=(n-2)/2$

which is same as every integer $n>1$

So we can conclude that SumOfThreePrimes implies Golabach's conjecture

We have shown that these two statements are logically equivalent.

If any step is wrong please tell me.

Answer 3

So the strong Goldbach conjecture states: Every even integer greater than $2$ can be expressed as the sum of two primes.

What if, rather than taking each even integer $N$ (of which there is an infinite number) and looking to find the two prime numbers $P_1$ and $P_2$ which, when added up, yield the original number $N$, we look at things another way.

Namely, we look to prove that by summing up all the possible pairs of prime numbers up to and including prime number $P$, we obtain ALL the even integers whose value is less than the next prime number after $P$.

For example, for $P = 13$, the sums of all the possible pairs of prime numbers up to an including $P$ are: $2+2=4$

$3+3=5$

$3+3=6$

$5+2=7$

$5+3=8$

$5+5=10$

$7+2=9$

$7+3=10$

$7+5=12$

$7+7=14$

$11+2=13$

$11+3=14$

$11+5=16$

$11+7=18$

$11+11=22$

$13+2=15$

$13+3=16$

$13+5=18$

$13+7=20$

$13+11=24$

$13+13=26$

(Note that P+P (like in 13+13) is considered as a valid pairing of primes.)

As can be seen in this example, by adding all the possible pairs of prime numbers up to and including $P=13$, we get ALL the even integers (i.e., 4, 6, 8, 10, 12, 13, 16) that are less than the next prime number, which is 17.

If this rule is generally applicable, regardless of the value of $P$, then this demonstrates the strong Goldbach conjecture, because it shows that any even integer can be obtained by adding up two prime numbers.

This rule still needs to be demonstrated - but I suspect it is easier to solve than the original strong Goldbach conjecture.