$1$ as difference of composites with same number of prime factors

Question

I noticed and found only first three cases:

We can write $1$ as difference of two composites that have one prime factor $$3^2-2^3=1$$

and as difference of two composites that have two prime factors $$3\cdot 5 - 7\cdot 2 = 1$$

and as difference of two composites that have three prime factors $$2^2 \cdot 3^2 \cdot 43-7 \cdot 13 \cdot 17=1$$

I believe that this holds for every $k \in \mathbb N$, that is, that for every $k \in \mathbb N$ there exist composites $a_k$ and $b_k$ that have exactly $k$ prime factors and are such that we have $a_k-b_k=1$.

Is my belief true? Is this known? What is known about all of this and similar problems? Can someone find solutions for some larger $k$´s?

There is a similar question here by Peter where he wants that all prime factors are different.

Answer 1

I had too many comments, so I will put it in a partial answer, referring to the special numbers in the OP's question as $(k, l)$ composituples:

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Definition: If $$a_{1,k},a_{2,k} = a_{1,k}+1,\ldots a_{l,k} = a_{(l-1),k}+1$$ are $l$ successive composites that have exactly $k$ prime factors, they are known as $(k,l)$ compo-situples. The set of these is denoted as $(k,l)_C\ni a_{l,k}$. I have investigated the following kind of set for prime pairs $p_n$ and $p_{n+1}$ such that I want to find the largest value of $l$ for a given $k$. $$(k,l)_C = \big\{\{p_n + 1,\ldots, p_{n+1} - 1\} : \text{card}(k,l) = l\,\land\,\Omega(a_{l,k}) = k\big\}.$$ Thus far, the largest value of $l$ has been $7$ such that $$(2, 7)_C =\{212, 213, 214, 215, 216, 217, 218, 219\}$$ for $k = 2$. Whether or not these are the smallest elements, I do not know.

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Questions in regards to composituples that I have found:

• The smallest value of $k$ and $l$ have to be $2$ by definition. Therefore, the smallest composituple is $(2,2)_C = \{14, 15\}$. But how many composituples are there, I do not know. My guess is in-finitely many.
• Let $(2,3)_C = \{403, 404, 405\}$, then are these the smallest elements?
• Let $(3, 3)_C = \{986,987,988\}$, then are these the smallest elements?
• Let $(2, 4)_C = \{445, 446, 447, 448\}$, then are these the smallest elements?
• Let $(2, 5)_C = \{54, 55, 56, 57, 58\}$, then these are the smallest elements.
• Let $(3, 5)_C = \{8214, 8215, 8216, 8217, 8218\}$, then are these the smallest elements?
• Let $(2, 6)_C = \{323, 324, 325, 326, 327, 328, 329\}$, then are these the smallest elements?
• Composituples $(3,4)_C,\,(3, 6)_C$ and $(3,7)_C$ have not been found yet.

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Oddly enough, I have not written a program to find these. I simply just went here to look at prime numbers, and then I went here to decompose the composites in between a given prime pair.

This is strictly a partial answer, or rather, a very long comment as opposed to an answer.