As I mentioned yesterday, I’d like to consider a proposed proof of the Goldbach Conjecture that has garnered some attention, at least some attention from people who ask me about things like the validity of proofs of the Goldbach Conjecture. I like this in particular because it illustrates how I look through some papers (those towards which I’m a bit skeptical) and it illustrates a problem I’ve seen before: switching between interpreting a number as an element of the integers and an element of $\mathbb{Z}/n\mathbb{Z}$. (There is a certain problem with this, in that although I ‘do number theory,’ were the conjecture proved it is almost certain that I would be not at all familiar with the methods of proof).
In particular, I’ll be looking at the 19 August 2012 preprint “The Goldbach’s conjecture proved” by Agostino Prastaro (the pdf is here).  The rest after the fold –

The Goldbach Conjecture states that every even number greater than $2$ can be written as the sum of two primes. It’s classically very well known, especially as it’s very easy to state but still unproven.

I think this will be a good time to pull out our good ol’ Ten Signs a Claimed Mathematical Breakthrough is Wrong from Scott Aaronson’s blog (which we have considered before). Have no fear – this paper passes the biggest ruinors: it’s in TeX and the author understands the problem. And let’s set our expectations by looking at the abstract:

We give a direct proof of the Goldbach’s conjecture in number theory, formulated in the Euler’s form. The proof is also constructive, since it gives a criterion to find two prime numbers $\ge 1$, such that their sum gives a fixed even number $\ge 2$ (A prime number is an integer that can be divided only for itself other than for 1. In this paper we consider 1 as a prime number). The proof is obtained by recasting the problem in the framework of the Commutative Algebra and Algebraic Topology.

Sure, so it tells us what a prime number is and mentions that $1$ is considered prime for the purposes of the proof. But it promises to give a constructive proof. Wow – that’s intense. It’s always easier to approach constructive proofs, as you see if the construction holds true on a few cases before muddling through theory. And we expect the proof to come from commutative algebra and algebraic topology.

But after the introduction, the paper quickly goes awry. There are pages and pages of Aaronson’s (8): The paper wastes a lot of space of standard material. The first twelve pagesare a summary of bits and pieces from Atiyah-MacDonald’s Commutative Algebra, or Ireland and Rosen’s Introduction to Modern Number Theory (both of which are great books). I need to be honest, though: I did not read the initial 12 pages through once I realized that they were so basic. It is on page 12 that the proof of the Goldbach Conjecture itself takes place. By this time, Aaronson’s (7) becomes apparent:The paper doesn’t build on (or in some cases even refer to) any previous work.

I will summarize the proof. He refers to two previous lemmas: one says that if $p \leq 2n$ and $p$ is coprime with all primes less than $2n$, then $p$ is a prime; the other says in essence that if $a$ generates $\mathbb{Z}_{2n}$, then $-a$ is also a generator. (He has an idea of strong generators, i.e. generators that are primes, but this is not a concept actually necessary in his proof).

The big idea, and the construction behind the constructive proof, is to start by considering $2n = 1 + (2n-1)$, and if that doesn’t work, then try $2 + (2n-2)$, and so on. That is, you just sort of try them all out. That’s disappointing, and nowhere is an actual constructive proof given. But back to the proof:

He claims that increasing $a$ in $p_2 + a = 2n - (p_1 - a)$ “must necessarily coincide with a prime number after a finite number of steps.” If we go along the $a$ such that $p_1 -a$ is always prime, then he claims that there must be a case when $p_2 + a$ is prime, as otherwise we will get to the case when $p_1 - a = 1$ (which, I’ll remind you, is currently considered a prime). He then proceeds to show that this necessarily means that $p_2 + a$ is prime.

Let’s draw special attention to this. The following proof in no way depends on anything previous. It is not based on commutative algebra, algebraic topology, nor any detail other than the fact that $p_1 - a = 1$ (and thus that $(p_2 + a) = 2n - 1$). In other words, the following ‘proof’ shows that every odd number is prime. This conflicts with Aaronson’s (3) and (4): The approach seems to yield something much stronger and maybe even false (but the authors never discuss that).

Back to the proof: If we call $p_2 + a =:p$ for ease of notation, then we are looking at $2n - p = 1$. Taking $b$ to be any prime less than $2n$, and starting with $2n - p = b b^{-1}$, the author writes $2np^{-1} - b b^{-1}p^{-1} = 1$. But then we have $1$ as a linear combination of $b$ and $p^{-1}$, meaning that they’re relatively prime. Herein lies the big error: on the one hand, when we’re considering each of these as cosets in the ring $\mathbb{Z}/2n\mathbb{Z}$, each of these inverses is just another coset. But the idea that the gcd is the smallest possible positive result of a linear combination of the numbers involved relies on integers, and absolutely does not work for arbitrary $\mathbb{Z}/n\mathbb{Z}$.

So the author went on to say that since $p$ is relatively prime to any prime smaller than $2n$ (he actually restricts to ‘strong prime generators’), and thus concludes that $p$ is prime. But alas, it’s not true that every odd number is prime, and this proof doesn’t hold. But the next time I teach number theory and I need an example of something that doesn’t hold in modular arithmetic that does hold in standard arithmetic, I will be ready.

To end, I emailed the author of the paper and mentioned the error, but he hasn’t yet gotten back to me.