关于“对任意n，⌊e^(p_{n²}#/p_{n²+1})⌋为平方数”的猜想技术问询

Hey everyone,

I've been digging into relationships between primes, primorials, and exponential functions lately, and stumbled on a couple of eye-catching examples that led me to a conjecture. Let me walk through the details:

Key Observations #

First, here's a neat identity I uncovered:
$$\left\lfloor e^{\frac{p_4#}{p_5}}\right\rfloor=\left\lfloor e^{\frac{210}{11}}\right\rfloor =13981^2,$$

To clarify the notation used here:

• $\lfloor x \rfloor$ is the floor function, formally defined as $\lfloor x \rfloor := \max{m\in\mathbb{Z} : m\leqslant x}$
• $p_n$ refers to the $n$-th prime number (so $p_1=2$, $p_2=3$, $p_4=7$, $p_5=11$, etc.)
• I'm using a generalized primorial definition:
  $$p_n#c := \prod{i=1}^n (p_i + c).\tag{$p_n#_0 = p_n#$}$$
  (Note: $p_n#$ is the standard primorial—the product of the first $n$ primes.)

A smaller, simpler case also checks out perfectly:
$$\left\lfloor e^{\frac{p_1#}{p_2}}\right\rfloor = 1^2.$$

The Conjecture #

Based on these verified cases, I'm putting forward this hypothesis:

For every positive integer $n$, the value $\lfloor e^{\frac{p_{n2}#}{p_{n^2 + 1}}}\rfloor$ is a perfect square.

I'm keen to dive into technical discussion around this. Some questions I'm curious about:

• Are there any existing number theory results that could explain why this pattern might hold (or break) for larger $n$?
• What's the best approach to test this conjecture for bigger values of $n$—whether computationally or theoretically?
• Could there be an underlying number-theoretic structure linking primorial ratios, exponentials, and perfect squares in this specific way?

Looking forward to hearing your insights!

内容的提问来源于stack exchange，提问作者Mr Pie