Is the sum of two distinct primes an injective function on integers?

Is the sum of two unique primes always unique? I.e is it an injective function on integers? For example, $89 + 2 = 91$ and there's no other sum of two different primes that's equal to $91$.

It's been a long time since I did a proper proof but I checked it programmatically for the first $10^4$ primes and it seems to hold true. How can I prove that though?

PS: This was true by definition if it was multiplication, but I don't see the connection from the definition to addition.

Solution 1:

Consider $3,5$ and $11,13$.

Notice $3+13=5+11$

Solution 2:

On the back of an envelope, so to speak, I get

\begin{array}{|r|r|r|r|r|r|r|r|r} & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 \\ \hline 2 & 5 & 7 & 9 & 13 & 15 & 19 & 21 & 25 & 31 \\ 3 & & 8 & 10 & 14 & 16 & 20 & 22 & 26 & 32 \\ 5 & & & 12 & 16 & 18 & 22 & 24 & 28 & 34 \\ 7 & & & & 18 & 20 & 24 & 26 & 30 & 36 \\ 11 & & & & & 24 & 28 & 30 & 34 & 40 \\ 13 & & & & & & 30 & 32 & 36 & 42 \\ 17 & & & & & & & 36 & 40 & 46 \\ 19 & & & & & & & & 42 & 48 \\ 23 & & & & & & & & & 52 \\ \end{array}

So, the odd numbers seem to be unique. By contrast, the even numbers show a lot of repetition: 16 appears twice in the table above, and so do 18, 20 and 22. And 24 appears thrice, as does 30. With just one more column, 36 would appear four times.

Before giving a proper proof of the apparent uniqueness of the even numbers in the table above, we need to agree on our definitions.

If $n$ is an odd number and there exist distinct positive primes $p$ and $q$ such that $p + q = n$, then that pair of primes is the only pair of primes with that sum. Either $p = 2$ or $q = 2$.

Um... you know, as I write that, a formal proof feels like overkill to me.

Solution 3:

Huh? What!? Ever hear of the Goldbach conjecture?

Is the sum of two unique primes always unique?

No, not at all. In fact, if $n$ is an even integer greater than $38$, it's probably the sum of two distinct primes in more than one way. For example, $$40 = 37 + 3 = 29 + 11 = 23 + 17.$$

The situation is different for odd numbers, of course. If $n$ is odd, there is only one way, or no way at all, to represent it as a sum of two distinct primes. That is if $n = p + 2$, where $p$ is an odd prime. So with $91$, you observed that $89 + 2 = 91$. But there's no such expression for $87$, since $85$ is obviously not prime.

However, if you allow $-2$, then $87 = 89 + (-2)$. This neither bolsters nor undermines my faith in the Goldbach conjecture.

Solution 4:

Note that $3+17=7+13$. And that $7+23=13+17$.

Solution 5:

Oh, just to be....

Let $p_1,q_2 = p_1 + 2$ be a pair of twin primes. Let $p_2,q_2 = p_2 + 2$ be another pair.

Then $p_1 + q_2 = p_2 + q_1=p_1 + p_2 + 2$.

First twin prime is $3,5$ and the second is $5,7$ so $3+7 = 5+5$.

It is conjectured that there are infinite twin primes so....