On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski

Question

In response to a comment posted under Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd primes. (Note the change in meaning of $O_n$; $P_n$ is the $n$th primorial.)

Before going into the questions, note that for integers $d$ of absolute value less than the largest prime factor of $O_n$ and coprime to $O_n$, such integers d have only 2 as a prime factor and $O_n + d$ has all its prime factors greater than the $n+1$st prime, unless $d$ has absolute value 1, in which case 2 is also a prime factor. Let this set of integers be called $C_n$.

1. (From the other question) How large a power of 2 divides $O_n^2 - 1$? As seen elsewhere, this quantity is not a power of 2 when $n>1$. More interestingly, the greater the power of 2 which divides this quantity, the further away one can demonstrate easily $O_n$ is from a power of 2.

2. Let $f(n)$ be that integer $e$ such that $2^e$ precisely divides $O_n^2 - 1$. So $f(1)=3, f(2)=5, f(3)=4.$ Is $f$ unbounded as a function of $n$? How fast does it grow? Is $f$ the sum of two nonconstant integer valued functions, one of them periodic and the other monotonic?

3. How many primes are in $C_n$?

4. Form $Q_n$, the product of the integers in $C_n$. This product is a multiple of 8, so is not squarefree. How far is it from being squarefree? In particular, is $Q_n$ the multiple of the square of any odd prime?

5. Given recent work on prime gaps (Zhang et al.), can we hope for a Generalized Dirichlet theorem? Specifically, translate the constellation $C_n$ by multiples of $O_n$: are there infinitely many translates which contain two or more primes in that translate?

I hope this makes my comment regarding related questions appear less vague and less broad.

Answer 1

Answer to question 2: obviously $f(n)\geq 3$ for all $n$, but I will show that $f(n)=3$ for infinitely many $n$, so we cannot give any other lower bound for the growth of $f$. To see this, notice that if $p_2...p_{n+1}\not \equiv \pm 1 \pmod 8$ (where $p_i$ is the $i$th prime), then the highest power of $2$ in $O_n^2-1$ is $2^3$. On the other hand, if $p_2...p_{n+1}\equiv\pm 1 \pmod 8$ and $n$ is chosen so that $p_{n+2}\equiv \pm 3 \pmod 8$ (by Dirichlet's theorem, there are infinitely many such $n$), we get $p_2...p_{n+2}\not \equiv \pm 1 \pmod 8$, so in any case there are infinitely many $n$ for which $f(n)=3$. We can even say something about the density of such $n$. By denoting $A=\{n:f(n)>3\}$, $B=\{n:f(n+1)=3\}$ and $C=\{n:p_{n+2}\equiv \pm 3 \pmod 8\}$, the natural denisties satisfy $d(A)+d(B)=1$ and we have $A\cap C \subset B$. Hence, using the PNT in aritmetic progressions, which tells that $d(C)=\frac{1}{2}$, we get $d(B)\geq d(A)-\frac{1}{2}=\frac{1}{2}-d(B)$, and from this we solve $d(B)\geq \frac{1}{4}$.

Answer to question 3: To clarify the definition, $C_n=\{O_n\pm d: d<p_{n+2} \hspace{0.1 cm}\text{is a power of two}\}.$ So the number of elements up to $x$ in $\bigcup_{n\geq 1} C_n$ grows like $\log^2 x$. As far as I know, no "natural" set that contains logarithmically few elements has been proved to contain infinitely many primes, so estimating the number of primes in $C_n$ seems hopeless. But one would expect that even $O_n+2$ is prime infinitely often.

Answer to question 4: first notice that the greatest common divisor of any two elements of $C_n$ is one. Indeed, if $q>2$ is a prime satisfying $q\mid O_n\pm 2^{\alpha}, q\mid O_n\pm 2^{\beta}$ and $2^{\alpha}\leq 2^{\beta}<p_{n+2}$, then $q\mid 2^{\beta-\alpha}-1,$ but this is a contradiction since $q\geq p_{n+2}$. So the question of $Q_n$ being divisible by an odd prime square reduces to an element of $C_n$ having this property. Again, this seems extremely hard since for example the following problem seems easier but is known to be open: Is $2^{p}-1$ ($p$ prime) always squarefree?

Answer to question 5: This would at least follow from Dickson's conjecture. According to it, there should even exist translates of $C_n$ that consist entirely (except $O_n\pm 1$) of primes. The constellation is admissible for this conjecture since if $q>2$ was a prime that divided one element of $kO_n+C_n$ for every $k$, then $q$ would be coprime to $O_n$ and bigger than $p_{n+1}$, but then $kO_n \pmod q$ would attain all the values $1,...,q-1$ as $k$ varies, which is a contradiction.