A question about numbers

Question

Let $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ be the prime decomposition of the integer $n$. Define $$n' = n \sum_{i=1}^r \frac{\alpha_i}{p_i}\quad\text{and}\quad\Omega(n) = \sum_{i=1}^r \alpha_i\quad\text{and}\quad\omega(n) = r.$$ Let $a,b$ be relatively prime, i.e., $\gcd(a,b)=1$, and let $c = a+b$. Suppose that $$\Omega(c) = \min\bigl\{\Omega(a),\Omega(b),\Omega(c)\bigr\}.$$ Is it true that $$\Omega(\gcd(a,a'))+\Omega(\gcd(b,b'))+\Omega(\gcd(c,c')) \le \Omega(ab)-1?$$ From this it would follow that $$\min\bigl\{\Omega(a),\Omega(b),\Omega(c)\bigr\} \le \omega(abc) - 1.$$

Edit: From the answer given by Kevin Buzzard, one can see, that the first inequality is wrong. It is unclear to me however, if the second inequality is also wrong.

Answer 1

No. How about $a=9$ and $b=16$? Then $c=25$ so $\Omega(a)=\Omega(c)=2\leq\Omega(b)$, the gcd's are $3,16,5$ so the left hand side is 6 and the right hand side only 5.

Edit: if $a=316$ and $b=27$ then we even have the left hand side being greater than $\Omega(ab)$, answering a question in the comments.

Editedit: if $a=544$ and $b=81$ then indeed your "from this it would follow" inequality is wrong.

Answer 2

Suppose there are prime numbers $p, q, r$, such that $2^k p+3^kq=5^k r$. Then the left hand side of the second inequality becomes $k+1$, while the right hand side is 5. Thus, in this case, the second inequality does not hold, even if you replace $\omega(abc)-1$ by any function of $\omega$.

To show that such prime numbers actually exist, you use the circle method. The major arcs get a little complicated, as $\sum_{n=1}^N\Lambda(n)e(\alpha 2^k n)$ gets very large at $\alpha=\frac{u}{2^k}$, $u\in\{0, \ldots, 2^k-1\}$, but, thanks to the fact that $2^k$ is not sqaurefree, the other exponential sums become quite small at these points which means that the region close to these points do not contribute significantly.

I would assume that someone already showed that under the obvious conditions on $a, b, c$, the equation $ap+bq=cr$ has the expected number of solutions in primes $p,q,r$, but could not find a reference.