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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?

Smallest examples found so far:

for $k=1$ $$3^2-2^3=1$$ for $k=2$ $$3 \cdot 5 - 7 \cdot 2=1$$ for $k=3$ $$3\cdot7\cdot11-2\cdot5\cdot23=1$$ for $k=4$ $$5 \cdot 7 \cdot 11 \cdot 19 - 2 \cdot 3 \cdot 23 \cdot 53=1$$

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?

Examples exist at least for $k=1,2,...11$ by this list .

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?

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?

Smallest examples found so far:

for $k=1$ $$3^2-2^3=1$$ for $k=2$ $$3 \cdot 5 - 7 \cdot 2=1$$ for $k=3$ $$3\cdot7\cdot11-2\cdot5\cdot23=1$$ for $k=4$ $$5 \cdot 7 \cdot 11 \cdot 19 - 2 \cdot 3 \cdot 23 \cdot 53=1$$

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.

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?