A comment on the equation $n!!=a_1!!\cdots a_t!!$

This paper investigates the equation $a_1!!\cdots a_t!!=n!!$ and demonstrates that, under the assumption of the explicit abc conjecture, the equation admits only finitely many nontrivial solutions in certain special cases.

The Gist

Imagine you are a detective trying to solve a very strange puzzle involving numbers. The puzzle is about factorials.

You know that $5!$ (5 factorial) means $5 \times 4 \times 3 \times 2 \times 1 = 120$.
But this paper is about double factorials (written as $n!!$). This is a slightly different game:

• If the number is even (like 6), you multiply all the even numbers down to 2: $6!! = 6 \times 4 \times 2 = 48$.
• If the number is odd (like 5), you multiply all the odd numbers down to 1: $5!! = 5 \times 3 \times 1 = 15$.

The Big Question

The paper asks a simple question: Can you multiply a bunch of these double factorials together to get another double factorial?

Mathematically, it looks like this:
$$a_1!! \times a_2!! \times \dots \times a_t!! = n!!$$

The researchers are looking for "non-trivial" solutions.

• Trivial solutions are like cheating. For example, if you have $10!!$, you can just say $10!! = 8!! \times (\text{something})$. It's like saying "I have a whole pizza, which is the same as a slice plus the rest of the pizza." There are infinitely many of these, and they aren't interesting.
• Non-trivial solutions are the "magic tricks" where the numbers don't just fit together obviously. The researchers want to know: Are there only a few of these magic tricks, or are there an infinite number of them?

The Detective's Tool: The "ABC" Conjecture

To solve this, the author uses a famous, unproven (but widely believed) rule in math called the ABC Conjecture.

Think of the ABC Conjecture as a universal law of conservation for prime numbers.

• Every number is built from "Lego bricks" called prime numbers (2, 3, 5, 7, 11...).
• The ABC Conjecture says: If you add two numbers together to get a third ($A + B = C$), the "Lego bricks" used to build the result ($C$) can't be too small compared to the size of the numbers themselves.
• In this paper, the author uses a specific, stronger version of this rule (Baker's explicit version) as a magnifying glass to inspect the numbers.

The Investigation

The author splits the investigation into two main scenarios based on whether the numbers in the puzzle are even or odd.

Scenario 1: All the numbers are Even

Imagine you are trying to build a tower using only even-numbered blocks.
The author proves that if you use the ABC Conjecture, you will eventually run out of room. The numbers get so big that the "Lego bricks" (primes) required to build them don't match up.

• The Result: There are only a finite number of these even-numbered magic tricks. You might find a few, but you won't find an infinite ocean of them.

Scenario 2: One number is Odd, the rest are Even

This is trickier. Imagine you have one odd block and a bunch of even blocks.
The author finds that there are still "cheating" ways to make this work (trivial solutions), but if you look for the real magic tricks:

• Case A: If the numbers are arranged in a specific way where no prime numbers are hiding in the middle of the sequence, the ABC Conjecture again proves there are only a finite number of solutions.
• Case B: If there are prime numbers hiding in the sequence, the author draws a boundary line. They show that if the numbers get too big, the equation breaks. Specifically, they prove that the first number ($a_1$) cannot be arbitrarily large unless the other numbers are also huge in a very specific, restricted way.

The "Why" (The Metaphor)

Why does this matter?
Imagine you are trying to balance a scale. On one side, you have a giant weight ($n!!$). On the other side, you have a pile of smaller weights ($a_1!! \dots a_t!!$).

The author is saying: "If you assume the laws of physics (the ABC Conjecture) hold true, you can't keep adding smaller weights to the pile forever to match the giant weight. Eventually, the pile becomes too 'dense' with prime factors, and the scale tips. The universe simply doesn't allow for an infinite number of these perfect balances."

The Conclusion

In plain English:

1. The Problem: Can you multiply double factorials to get another double factorial?
2. The Answer: Yes, but only a limited number of times (excluding the obvious, boring ways).
3. The Proof: By using a powerful mathematical rule (the ABC Conjecture), the author showed that the numbers involved are constrained. They can't grow infinitely large while still satisfying the equation.

It's like proving that while you can find a few rare, perfect snowflakes, you will never find an infinite number of them that are all exactly the same size and shape. The math just doesn't allow it.

Technical Summary

1. Problem Statement

The paper investigates the Diophantine equation involving double factorials:
$$\prod_{i=1}^t a_i!! = n!!$$
subject to the constraints $n > a_1 \ge a_2 \ge \dots \ge a_t \ge 3$ and $t > 1$.

The central question is whether this equation possesses finitely many nontrivial solutions.

• Double Factorial Definition: $m!! = 1 \cdot 3 \cdots m$ (if $m$ is odd) or $2 \cdot 4 \cdots m$ (if $m$ is even).
• Trivial Solutions: The paper identifies specific families of solutions that are considered "trivial" and thus infinite in number.
  • If all $a_i$ are even, setting $n - a_1 = 2$ yields infinite solutions.
  • If $a_1$ is odd and $a_2, \dots, a_t$ are even, solutions where $n - a_1$ is not fixed but follows a specific construction (e.g., $n = a_1! a_3!! \cdots$) are considered trivial.
• Context: This problem is a variation of the long-standing unsolved problem regarding single factorials ($\prod a_i! = n!$). Previous work by Nair and Shorey (2024) addressed the case where all $a_i$ and $n$ are odd. This paper focuses on the case where $a_2, \dots, a_t$ are all even, implying $n$ must also be even.

2. Methodology

The author employs a combination of analytic number theory and the Explicit $abc$ Conjecture (a strengthened version of the classical $abc$ conjecture proposed by Baker).

Key Mathematical Tools:

1. Explicit $abc$ Conjecture: The paper utilizes the inequality $c < N(abc)^{7/4}$, where $N(k)$ is the radical of $k$ (the product of distinct prime factors). This provides stronger bounds than the classical $abc$ conjecture.
2. Consecutive Integer Products ($\Delta$): The equation is transformed into forms involving products of consecutive integers, denoted as $\Delta(x, k) = x(x+1)\cdots(x+k-1)$.
3. Prime Factor Bounds:
   • Erdős's Theorem: Used to establish lower bounds for the largest prime factor $P(\Delta(x, k))$ when the terms in the product are composite. Specifically, $P(\Delta(x, k)) > \frac{2}{7}k \log k$ for sufficiently large $k$.
   • Bertrand's Postulate: Used to prove that under certain constraints, the terms in the product $\Delta(x, k)$ cannot contain primes, forcing them to be composite.
4. Stirling's Approximation: Used to estimate the logarithmic growth of factorials and double factorials to derive inequalities bounding the variables.
5. Case Analysis: The proof is divided based on the parity of $a_1$:
   • Case 1 ($r=0$): All $a_i$ are even.
   • Case 2 ($r=1$): $a_1$ is odd, and $a_2, \dots, a_t$ are even.

3. Key Contributions and Results

Theorem 1.1: All $a_i$ are Even ($r=0$)

• Statement: Under the explicit $abc$ conjecture, the equation has only finitely many nontrivial solutions when all $a_i$ are even.
• Mechanism:
  • The equation reduces to $\prod_{i=2}^t a_i!! = 2^k \Delta(m, k)$.
  • The author proves that for nontrivial solutions, the terms in $\Delta(m, k)$ must be composite.
  • By applying the explicit $abc$ conjecture to the difference of two terms in the product, the author derives an upper bound for $\log(m)$ in terms of $a_2$.
  • Combining this with lower bounds on the largest prime factor of the product, it is shown that $a_2$ (and consequently all other variables) must be bounded.

Theorem 1.2: $a_1$ is Odd, $a_2, \dots, a_t$ are Even ($r=1$)

• Statement: Let $a_1$ be odd and $a_2, \dots, a_t$ be even.
  • (i) If the product $\Delta(x_1, l_1)$ contains no prime factors, the explicit $abc$ conjecture implies finitely many nontrivial solutions.
  • (ii) If $\Delta(x_1, l_1)$ contains a prime:
    • If $x_1 > 4l_1$, there are finitely many solutions (except for a finite set of exceptions defined by a specific set $T$).
    • If $x_1 \le 4l_1$, $a_1$ is bounded by a function of $l_1$:
      $$0.99 a_1 \le 4l_1 + 4.01 + \frac{2\log(5) + 2\log(l_1)}{\log(2)}$$
• Mechanism:
  • Similar to Theorem 1.1, the proof uses the explicit $abc$ conjecture to bound $x_1$ relative to $a_1$.
  • For part (ii), the author analyzes the power of 2 on both sides of the equation. By comparing the valuation of 2 in the factorial terms (Legendre's formula) against the product terms, a linear inequality relating $a_1$ and $l_1$ is derived.
  • Theorem 2.4 (from Nair and Shorey) is used to handle cases where the product contains large primes, showing that $a_1$ is bounded unless $(x_1, l_1)$ belongs to a specific finite set $T$.

4. Significance and Limitations

Significance:

• Extension of Previous Work: This paper extends the results of Nair and Shorey (who handled the all-odd case) to the mixed-parity case where the smaller terms are even.
• Conditional Finiteness: It establishes that, assuming the explicit $abc$ conjecture, the equation has only finitely many nontrivial solutions in these specific parity configurations. This brings the understanding of double factorial Diophantine equations closer to the resolution of the single factorial case.
• Explicit Bounds: The paper provides explicit inequalities (e.g., $0.99a_1 \le \dots$) that bound the variables, which is a stronger result than merely proving existence.

Limitations and Open Problems:

• Dependence on Conjecture: The results are conditional on the Explicit $abc$ Conjecture. Without this unproven conjecture, the finiteness of solutions remains unproven.
• Restricted Cases: The methods used rely heavily on the properties of the product $\Delta(m, k)$ where terms are composite. The author notes that the current techniques do not work for cases where $2 \le r \le t-2$ (i.e., when there are multiple odd terms but not all are odd) or when $r=1$ but $a_t$ is odd.
• Obstacle: The main obstacle for the remaining cases is the inability to control the largest prime factor $P(\Delta(m, k))$ when the product contains prime numbers, as the "composite-only" assumption used in the proofs breaks down.

Conclusion

Saša Novaković's paper successfully applies the explicit $abc$ conjecture to prove the finiteness of nontrivial solutions for the double factorial equation in two specific parity configurations (all even, or one odd followed by evens). The work relies on transforming the equation into products of consecutive integers, utilizing bounds on prime factors, and deriving explicit inequalities to bound the variables. While it does not solve the general case, it significantly narrows the scope of the problem and highlights the specific structural barriers preventing a complete solution.