The crosshatch conjecture, on primes in $(p,p^2)$

Question

If the first $p^2$ integers are laid out in a $p\times p$ square, every row and column will have at least one prime. Easily visualized as so:

I recognize this should maybe be packaged as two conjectures, but hey, I like the aesthetics. Individually, we have:

1. Given some prime $p$, there will always be at least one prime in $[kp+1,kp+p-1]$ for all $1 \leq k \leq p-1$.

2. Given some prime $p$, there will always be at least one prime $q$ in $(p,p^2)$ such that $q \equiv k \pmod{p}$ for all $1 \leq k \leq p-1$.

I'm asking if a) this is somehow wrong or someone finds a counterexample, and b) if either of these conjectures already exist.

Answer 1

It's an interesting set of $2$ related conjectures. However, they are already known. When I did a Bing search for

"prime square" conjecture

I found Conjecture 26. The Calendar-like square Conjecture from Julio Cesar Aguilar in México. It's formal statement, along with an example picture for a $5$ x $5$ grid, is

In a calendar-like and square array of numbers from 1 to $p^2$, being $p$ any prime number:

a) there is always at least one prime per row

b) there is always at least one prime per column

The conjecture itself has no listed date, but the date of Nov. $11$, $2001$ for the first part of the "solution" by Luis Rodríguez shows this was conjectured by Aguilar at least by then. However, Luis' comment indicates these $2$ conjectures were made earlier separately by Schinzel & Sierpinski in $1958$. This is written in a book by Ribenboim who wrote they stated:

We do not know what will be the fate of our hypotheses, however we think that, even if they are refutes, this will not be without profit for number theory.

The rest of this Web page discusses various other related details and conjectures. Although neither of the $2$ conjectures have a formal proof stated or linked to, no counter-examples have apparently yet been found either.