Htykuut

Someone could help me find some error in the reasoning:

We know, that the canonical decomposition of $n!$ is:

$n!=prod_{p_{i}leq n}p_{i}^{alpha_{i}(n)}$, where:

$alpha_{i}(n)=sum_{t=1}^{r}[frac{n}{p_{i}^{t}}]$.

It's possible use the canonical decomposition of $n!$ to found the canonical decomposition of $n$, let's see:

$n=frac{n!}{(n-1)!}=frac{prod_{p_{i}leq n}p_{i}^{alpha_{i}(n)}}{prod_{p_{i}leq n-1}p_{i}^{alpha_{i}(n-1)}}=prod_{p_{i}leq n}p_{i}^{alpha_{i}(n)-alpha_{i}(n-1)}=prod_{p_{i}leq n}p_{i}^{beta_{i}(n)}$, where:

$beta_{i}(n)=alpha_{i}(n)-alpha_{i}(n-1)$.

Note an important propertie of $beta_{i}(n)$:

$beta_{i}(p_{j})=delta_{i,j}$, this is because the unicity of the canonical decomposition:

$n=p_{j}=prod_{p_{i}leq p_{j}}p_{i}^{beta_{i}(p_{j})}$.

Let's use this resoults to probe the followin hypothesis:

$forall N > 1$, $exists kin {0,1,2,...,N-1}$ such that:

$N^{2}-k^{2}=prod_{p_{i}leq N^{2}-k^{2}}p_{i}^{gamma_{i}(N,k)}$, and:

$gamma_{i}=begin{cases}{1}&text{if}& i=l\1 & text{if}& i=m\0 & text{if}& other-caseend{cases}$

This is where I need help to find some error in the reasoning:

we know that:

$(N-k)=prod_{p_{i}leq N-k}p_{i}^{beta_{i}(N-k)}$

$(N+k)=prod_{p_{i}leq N+k}p_{i}^{beta_{i}(N+k)}$

then:

$(N-k)(N+k)=prod_{p_{i}leq N+k}p_{i}^{beta_{i}(N-k)+beta_{i}(N+k)}$

$N^{2}-k^{2}=prod_{p_{i}leq N+k}p_{i}^{gamma_{i}(N,k)}$, where:

$gamma_{i}(N,k)=beta_{i}(N-k)+beta_{i}(N+k)$, then:

$gamma_{i}(N,k)=alpha_{i}(N-k)-alpha_{i}(N-k-1)+alpha_{i}(N+k)-alpha_{i}(N+k-1)$

The only way (due to the uniqueness of the decomposition), to satisfy:

$gamma_{i}=begin{cases}{1}&text{if}& i=l\1 & text{if}& i=m\0 & text{if}& other-caseend{cases}$

is to suppose that $N-k=p_{l}$ and $N+k=p_{m}$ for some $p_{l}< 2N$, $p_{m}< 2N$ primes, lets see:

$gamma_{i}(N,k)=alpha_{i}(N-k)-alpha_{i}(N-k-1)+alpha_{i}(N+k)-alpha_{i}(N+k-1)$

$gamma_{i}(N,k)=alpha_{i}(p_{l})-alpha_{i}(p_{l}-1)+alpha_{i}(p_{m})-alpha_{i}(p_{m}-1)$

$gamma_{i}(N,k)=beta_{i}(p_{l})+beta_{i}(p_{m})$

$gamma_{i}(N,k)=delta_{i,l}+delta_{i,m}=begin{cases}{1}&text{if}& i=l\1 & text{if}& i=m\0 & text{if}& other-caseend{cases}$

(Q.E.D.)

Then:

$forall N > 1$, $exists kin {0,1,2,...,N-1}$ such that:

$N^{2}-k^{2}=prod_{p_{i}leq N^{2}-k^{2}}p_{i}^{gamma_{i}(N,k)}$, and:

$gamma_{i}=begin{cases}{1}&text{if}& i=l\1 & text{if}& i=m\0 & text{if}& other caseend{cases}$

That is to say:

$N^{2}-k^{2}=prod_{p_{i}leq N^{2}-k^{2}}p_{i}^{gamma_{i}(N,k)}$

$N^{2}-k^{2}=p_{l}^{gamma_{l}}p_{m}^{gamma_{m}}$

$N^{2}-k^{2}=p_{l}p_{m}$

$(N-k)(N+k)=p_{l}p_{m}$, then:

$N-k=p_{l}$ and $N+k=p_{m}$

Finanlly, adding the equations:

$2N=p_{l}+p_{m}$

That is to say: "Any even number greater than two can be expressed as the sum of two prime numbers."

Thanks for your help.