Prove that there exists infinitely many primes of Digital root $2,5$ or $8$

Question

I am highly interested in properties of digital root.

Digital Root: Digital root of a number is a digit obtained by adding digits of number till a single digit is obtained.

It's clear that Digital root Partition the set of Natural number in 9 Equivalence Classes.

When I was reading the proof that Prime Numbers are Infinite. I pointed out something: Here is the famous Euclid's Proof:

Suppose that $p_1=2 < p_2 = 3 < ... < p_r$ are all of the primes. Let $P = p_1p_2...p_{r}+1$ and let $p$ be a prime dividing $P$; then $p$ can not be any of $p_1, p_2, ..., p_r$, otherwise $p$ would divide the difference $P-p_1p_2...p_r=1$, which is impossible. So this prime $p$ is still another prime, and $p_1, p_2, ..., p_r$ would not be all of the primes.

I noticed that all the primes generated in this way, has digital root =$4,7$ or $1$

Since $p_1\times p_2...$ is multiple of $3$ (Second prime is 3). Hence the digital root of $p_1, p_2, ..., p_r+1$ is $3+1, 6+1, 9+1$ ( Since digital root of multiple of 3 is 3,6,9) i.e., $4,7$ or $1$ So by this theorem we have proved that primes number of digital root=4,7 and 1 are infinite.

Is there is a way to prove that there are infinitely many primes of digital root $2,5$ or $8$.

A twin prime can have digital root =(2,4), (5,7) or (8,1). So if twin prime conjuncture is right then there must exists, infinitely many primes of digital root =2 or 5.

Answer 1

For digital root say $2$, use the fact that there are infinitely many primes of the form $2+9k$. This is a consequence of Dirichlet's Theorem on primes in arithmetic progressions.

Answer 2

The OP's proposed proof of infinitely many primes with digital roots $1,4,7$ is not complete (although the conclusion follows from Dirichlet's Theorem just as for digital roots $2,5,8$).

The idea was inspired by Euclid's proof of the infinitude of primes. We consider the expression:

$$ b = p_1\cdot p_2\cdot p_3\cdot \ldots\cdot p_n + 1 $$

where $\{p_1,p_2,\ldots,p_n\}$ is a (nonempty) finite set of prime numbers.

Now $b$ is obviously greater than $1$ and hence has a prime divisor $q$. But that prime divisor cannot equal any of the multiplied primes $p_1,p_2,\ldots,p_n$ since that would imply $1$ is divisible by $q$. So there is no finite set of all primes. We cannot say that $b$ itself is prime.

In the rest of this post we will restrict discussion to constructing $b$ by multiplying the $n$ smallest primes (and adding $+1$). In that case the prime factor $q$ must be greater than any of the $n$ smallest primes. But again there is no claim (or proof) that $b$ itself is always prime. Although it has been conjectured that $b$ is prime infinitely often, even this is not yet proven.

When such $b$ happens to be prime, it is called a Primorial Prime. For partial lists of such cases see the OEIS sequence A014545 and The Top Twenty Primorial Primes.

What one can argue (as the OP does) is that for $n\gt 2$, the residue of $b$ mod $3$ will be $1$, and therefore the possible digital roots (essentially the residues mod $9$) of $b$ are $1,4,7$.

Since it is not known that $b$ will itself be prime infinitely often, this does not guarantee infinitely many primes with even one of the digital roots $1,4,7$.

Further the prime factors $q$ of composite (non-prime) $b$ need not have residues $1,4,7$ mod $9$. The first composite example illustrates this:

$$ 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13 + 1 = 30031 = 59 \cdot 509 $$

Note that in this case both prime factors have digital root $5$.

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Dirichlet's Theorem, that any arithmetic progression $ka + b$ for fixed $a,b$ coprime to each other contains infinitely many primes, is very powerful. It is unfortunate that his proof uses the machinery of complex analysis, so various attempts have been made to produce "elementary" proofs of this. A. Selberg (1949) published a paper, An Elementary Proof of Dirichlet's Theorem about Primes in an Arithmetic Progression, which (building on his success in proving the Prime Number Theorem without complex analysis) is "elementary" in that sense.

On the other hand the "elementary" number theory used is pretty involved and unmistakably involves some difficult analysis estimating finite sums.

There is a well-known easy exercise to show there exist infinitely many primes of the form $4n+3$, but I will not repeat its proof here. A variety of cases can be proven similarly (show infinitely many primes of the form $an+b$ for coprime $a,b$), but also there are cases where this easy approach fails.

For the interested Reader I will suggest the problems of proving infinitely many primes of the form $9n+8$ and (not so easy) of the form $9n+1$.