Does the Sylvester-Schur theorem apply to complete residue systems?

Question

Let $p_k$ be the $k$th prime.

If $k \ge 3$, the following sequence forms a complete residue system modulo $p_k$:

$$3(1)+1, 3(2)+1, 3(3)+1, \dots, 3(p_k)+1$$

If $k=3, p_3=5$, $7 = 3(2)+1$

If $k=4, p_4=7$, $13 = 3(4)+1$

Does it always follow that for any integer $x\ge 1,n > 3$, for a sequence of $3(x)+1, \dots,3(x+n-1)+1$, there will always be a prime $p > n$ and an integer $i$ such that $x \le i \le x+n-1$ and $p | (3i+1)$?

Can Sylvester-Schur be generalized to apply to complete residue systems?

Answer 1

Your sequence of

$$3(x)+1, \dots,3(x+n-1)+1 \tag{1}\label{eq1A}$$

is an arithmetic progression with a first term $a = 3x + 1$, difference $d = 3$, and $n$ terms.

Note James Joseph Sylvester proved a useful generalization to the Sylvester-Schur theorem, with page #$19$ of On Arithmetical Series, Messenger Math. $21$ ($1892$), stating

... it will be seen we have proved that whatever $n$ and whatever $i$ may be, provided $m$ is relatively prime to $i$ and not less than $n$, the product $$(m + i)(m + 2i)\ldots(m + ni)$$ must contain some prime number by which $2.3\ldots n$ is not divisible, ...

In your case, $m = 3x - 2$ and $i = 3$ (which are coprime, as required), with the requirement for $m$ to "not be less than $n$" meaning $3x - 2 \ge n \implies x \ge \left\lceil \frac{n + 2}{3} \right\rceil$, so it doesn't cover some of the cases you asked about.

However, in Prime Factors of Arithmetic Progressions and Binomial Coefficients, by T. N. Shorey and R. Tijdeman, in "$0.1$ Notation", the top of page $2$ states

For an integer $ν$ with $|ν| \gt 1$, we denote by $P(ν)$ the greatest prime factor of $ν$ ...

Next, "Part $2$. Arithmetic Progressions", about the middle of page $6$ states

For positive integers $x$ and $d \ge 2$ and $k \ge 3$ with $x$ and $d$ coprime, put $$\Delta := \Delta(x, d, k) := x(x + d)\cdots(x + (k − 1)d)$$

As you can see, your conditions meet those for $\Delta$. Just further down the page, in "$2.1$ The greatest prime factor",

Sylvester [Syl] proved a result that implies that $P(\Delta) \gt k$ if $x \ge d + k$. Langevin [Lan$77$] replaced the assumption $x \ge d + k$ by $x \gt k$. Thereafter the authors [ST$90$b] showed that, for all $x$, $d$ and $k$, $$P(\Delta) \gt k \; \text{ unless } \; (x, d, k) = (2, 7, 3)$$

Since the one restriction doesn't apply to your case, this shows your conjecture is always true. Note [Syl] is the article I mention at the top, while the "References" section on page $10$ gives [Lan$77$] as being

M. Langevin, Plus grand facteur premier d’entiers en progression arithmétique, Sém. Delange - Pisot - Poitou, $18$e année, $1976/77$, Exp. No. $3$, $6$ pp., Paris, $1977$.

and, on page $11$, their [ST$90$b] as being

T.N. Shorey and R. Tijdeman, On the greatest prime factor of an arithmetical progression, A Tribute to Paul Erdős, ed. A. Baker, B. Bollobas and A. Hajnal, Cambridge University Press ($1990$), $385$-$389$.

with an online version from Acta Arithmetica. However, I suspect it's an earlier version than the referenced one since there's no explicit mention of the stated result, but with this result being implicit based on determining a constant & proving all smaller values (if any) still work, such as finding $C_4$ and using their inequality $(6)$.

Regarding your final question of

Can Sylvester-Schur be generalized to apply to complete residue systems?

I assume you mean generalize it to prove your conjecture that at least one integer in \eqref{eq1A} has a prime factor larger than $n$. Since I haven't looked at the original proof, I can't comment on that, but with Paul Erdős' version you linked to, I believe it will at least be quite difficult. This is mainly because several parts of it are intrinsically linked with using all of the consecutive integers, with no way I know of to change to them being $3$ apart instead but still keep the same techniques' functionality. For example, the second page uses Legendre's formula to determine the largest power of a prime factor of a binomial term. I don't know of any corresponding precise, general formula one can use with the product of integers where each are $3$ apart instead of them all being consecutive.