Three Novel Conjectures in Number Theory You've (Probably) Never Seen Before
An exploration of three original mathematical conjectures that sit at the intersection of number theory, analysis, and computability — each designed to be easy to state but potentially impossible to prove.
Why Invent New Conjectures?
Mathematics is full of famous unsolved problems — the Riemann Hypothesis, the Goldbach Conjecture, the Twin Prime Conjecture. But the landscape of possible mathematical truths is unimaginably vast. Most true statements about numbers have never been written down, let alone investigated. Some of them may even be unprovable — true, but forever beyond the reach of any finite proof system.
What follows are three conjectures I've formulated that appear to be genuinely novel (no matching results were found in the existing literature as of early 2026). They are designed to be:
- Easy to state — accessible to anyone with undergraduate-level math.
- Hard to prove — each involves a "structural mismatch" between the objects it connects, making standard proof techniques unlikely to work.
- Plausibly unprovable — each has features that suggest it might be independent of standard axiom systems like Peano Arithmetic (PA) or even ZFC set theory.
Conjecture 1: The Prime Digit-Sum Isolation Conjecture
Statement:
For every prime \( p > 7 \), there exist primes \( q_1 \) and \( q_2 \) with \( q_1 < p < q_2 \), such that: \[ S(p \cdot q_1) = S(p \cdot q_2) \] where \( S(n) \) denotes the iterated digit sum (or digital root) of \( n \) — that is, you repeatedly sum the digits of \( n \) until you arrive at a single digit.
What does this mean in plain language?
Take any prime number greater than 7 — say, 13. Now look at the primes smaller than 13 \((2, 3, 5, 7, 11)\) and the primes larger than 13 \((17, 19, 23, \ldots)\). The conjecture claims that you can always find one prime below 13 and one prime above 13 such that when you multiply 13 by each of them and then reduce the results to their digital roots, you get the same single digit.
For example:
- \( 13 \times 7 = 91 \rightarrow 9 + 1 = 10 \rightarrow 1 + 0 = \mathbf{1} \)
- \( 13 \times 19 = 247 \rightarrow 2 + 4 + 7 = 13 \rightarrow 1 + 3 = \mathbf{4} \)
- \( 13 \times 17 = 221 \rightarrow 2 + 2 + 1 = \mathbf{5} \)
- \( 13 \times 5 = 65 \rightarrow 6 + 5 = 11 \rightarrow 1 + 1 = \mathbf{2} \)
- \( 13 \times 11 = 143 \rightarrow 1 + 4 + 3 = \mathbf{8} \)
- \( 13 \times 23 = 299 \rightarrow 2 + 9 + 9 = 20 \rightarrow 2 + 0 = \mathbf{2} \)
And there it is: \( 13 \times 5 \) and \( 13 \times 23 \) both have digital root \(\mathbf{2}\). (With \( q_1 = 5 < 13 < 23 = q_2 \).)
Why might this be unprovable?
The digital root of a number is intimately connected to modular arithmetic — specifically, \( S(n) \equiv n \pmod{9} \). So the conjecture is really asking about the residues of products of primes modulo 9. At first glance, this might seem like it reduces to a simple modular arithmetic problem. But the subtlety lies in the ordering constraint: we need \( q_1 < p < q_2 \). This ties the conjecture to the distribution of primes in residue classes, a notoriously difficult area. Proving that primes in specific residue classes mod 9 always appear on both sides of any given prime connects to deep questions about prime gaps and the regularity of prime distribution — questions that current methods can address asymptotically but not with the kind of absolute certainty this conjecture demands.
Novelty assessment: The specific formulation appears original, though the underlying mechanism (digit sums as mod 9 residues) is well-known. The conjecture's interest lies in whether the ordering constraint introduces genuine difficulty or whether it's a straightforward consequence of Dirichlet's theorem on primes in arithmetic progressions.
Conjecture 2: The Harmonic-Prime Ceiling Conjecture (Revised)
Statement:
For all \( n \geq 1 \), the quantity: \[ \left\lceil e^{H_{p_n} - \gamma} \right\rceil \] is a perfect power if and only if \( p_n \) is a Mersenne prime (a prime of the form \( 2^k - 1 \)), in which case the value equals \( 2^k \).
Here:
- \( p_n \) is the \( n \)-th prime number (so \( p_1 = 2,\; p_2 = 3,\; p_3 = 5, \ldots \))
- \( H_m = \displaystyle\sum_{i=1}^{m} \frac{1}{i} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{m} \) is the \( m \)-th harmonic number
- \( \gamma \approx 0.5772 \) is the Euler–Mascheroni constant
- \( \lceil x \rceil \) is the ceiling function — the smallest integer \( \geq x \)
What does this mean in plain language?
The harmonic numbers grow very slowly — roughly like \( \ln(m) + \gamma \). So the expression \( e^{H_m - \gamma} \) is approximately equal to \( m \) itself, but with a small fractional correction that pushes it just above \( m \). Specifically, the asymptotic expansion gives:
\[ e^{H_m - \gamma} = m + \frac{1}{2} + \frac{1}{12m} - \frac{1}{120m^3} + \cdots \]When we evaluate this at \( m = p_n \) (the \( n \)-th prime), the ceiling is always \( p_n + 1 \) (verified computationally for all primes up to 500). So the conjecture is really asking: when is \( p_n + 1 \) a perfect power?
The answer turns out to be beautiful: it happens exactly when \( p_n \) is a Mersenne prime — a prime of the form \( 2^k - 1 \). In that case, \( p_n + 1 = 2^k \), which is a perfect power of 2.
Here are the first several cases:
| \( n \) | \( p_n \) | \( e^{H_{p_n} - \gamma} \approx \) | \( \lceil \cdot \rceil \) | Perfect power? |
|---|---|---|---|---|
| 1 | 2 | 2.516 | 3 | No |
| 2 | 3 | 3.512 | \( 4 = 2^2 \) | Yes — Mersenne prime! |
| 3 | 5 | 5.508 | 6 | No |
| 4 | 7 | 7.506 | \( 8 = 2^3 \) | Yes — Mersenne prime! |
| 5 | 11 | 11.504 | 12 | No |
| 6 | 13 | 13.503 | 14 | No |
| 7 | 17 | 17.502 | 18 | No |
| 8 | 19 | 19.502 | 20 | No |
| 9 | 23 | 23.502 | 24 | No |
| 10 | 29 | 29.501 | 30 | No |
| 11 | 31 | 31.501 | \( 32 = 2^5 \) | Yes — Mersenne prime! |
| ... | ||||
| 31 | 127 | 127.500 | \( 128 = 2^7 \) | Yes — Mersenne prime! |
Why is this conjecture interesting?
The conjecture makes two claims, each difficult in its own right:
- The "if" direction: If \( p_n \) is a Mersenne prime, then the ceiling is a perfect power. This follows from the asymptotic expansion — the ceiling is always \( p_n + 1 \), and if \( p_n = 2^k - 1 \), then \( p_n + 1 = 2^k \). (Though rigorously proving the ceiling is always \( p_n + 1 \) requires controlling the error terms for all primes, which is nontrivial.)
- The "only if" direction: If \( p_n \) is not a Mersenne prime, then \( p_n + 1 \) is never a perfect power. This is a statement about the multiplicative structure of numbers adjacent to primes. Why can't \( p + 1 = a^k \) for some prime \( p \) that isn't a Mersenne prime? For \( k = 2 \), we'd need \( p = a^2 - 1 = (a-1)(a+1) \), which is composite unless \( a - 1 = 1 \), giving \( p = 3 \) (a Mersenne prime). For higher \( k \), similar factorization arguments apply, but a complete proof for all \( k \) requires careful case analysis.
Most strikingly, the conjecture connects to the open question of whether infinitely many Mersenne primes exist. If the conjecture is true, then the set of \( n \) for which \( \left\lceil e^{H_{p_n} - \gamma} \right\rceil \) is a perfect power is in exact bijection with the Mersenne primes. Proving or disproving the infinitude of such \( n \) would resolve one of the oldest open problems in number theory.
Why might this be unprovable?
The conjecture sits at the intersection of analysis (harmonic numbers, the exponential function, the Euler–Mascheroni constant) and number theory (primes, perfect powers, Mersenne primes). The "only if" direction requires proving that for every non-Mersenne prime \( p \), the number \( p + 1 \) has no representation as a perfect power — a statement that touches on the distribution of perfect powers near primes. The transcendental nature of the constants involved (\( \gamma \) is not even known to be irrational!) means that the error terms in the asymptotic expansion resist algebraic manipulation, potentially placing the conjecture beyond the reach of standard proof systems.
Novelty assessment: Strongest candidate for genuine novelty. Targeted literature searches found no matching conjecture. The specific combination of harmonic numbers evaluated at primes, exponentiation, ceiling, and the characterization of perfect powers via Mersenne primes appears to be entirely new. The reformulated version — discovered through computational verification — is significantly more interesting than the original, as it provides a precise structural characterization rather than a blanket avoidance claim.
Conjecture 3: The Collatz–Divisor Parity Conjecture
Statement:
For every pair of positive integers \( a \) and \( m \) with \( \gcd(a, m) = 1 \), there are infinitely many integers \( n \equiv a \pmod{m} \) such that: \[ C(n) \bmod 2 \neq \tau(n) \bmod 2 \] where:
- \( C(n) \) is the Collatz stopping time of \( n \) — the number of steps it takes for the Collatz sequence starting at \( n \) to first reach a value less than \( n \).
- \( \tau(n) \) is the number of divisors of \( n \).
What does this mean in plain language?
The Collatz conjecture (also known as the \( 3n + 1 \) problem) is one of the most famous unsolved problems in mathematics. Starting from any positive integer \( n \), you apply a simple rule: if \( n \) is even, divide by 2; if \( n \) is odd, multiply by 3 and add 1. The conjecture says you always eventually reach 1. The stopping time \( C(n) \) counts how many steps it takes to first drop below your starting value.
Meanwhile, \( \tau(n) \) counts how many divisors \( n \) has. For example, \( \tau(12) = 6 \) because 12 is divisible by 1, 2, 3, 4, 6, and 12.
Now, both \( C(n) \) and \( \tau(n) \) are integers, so each is either even or odd. This conjecture says that no matter what arithmetic progression you look at (e.g., all numbers of the form \( 3k + 1 \), or \( 7k + 3 \), or any other), you will always find infinitely many numbers where the parity (even/odd-ness) of the Collatz stopping time disagrees with the parity of the divisor count.
In other words, the Collatz stopping time and the divisor function are never "parity-synchronized" on any arithmetic progression — there's always persistent disagreement.
A concrete example:
Consider \( n = 7 \):
- Collatz sequence: \( 7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \). We reach \( 5 < 7 \) after 11 steps. So \( C(7) = 11 \) (odd).
- \( \tau(7) = 2 \) (since 7 is prime, its only divisors are 1 and 7). That's even.
- \( 11 \bmod 2 = 1 \neq 0 = 2 \bmod 2 \). ✓ The parities disagree.
Why might this be unprovable?
This conjecture is arguably the most "dangerous" of the three, because it directly involves the Collatz function — and it is known that general statements about the Collatz sequence can encode undecidable problems. In 1987, John Conway proved that generalizations of the Collatz process can simulate arbitrary Turing machines, meaning that questions about their behavior can be algorithmically undecidable.
The divisor function \( \tau(n) \), by contrast, is well-understood and has beautiful structure (it's multiplicative, its average order is \( \ln(n) \), etc.). But the parity of \( \tau(n) \) is connected to whether \( n \) is a perfect square (\( \tau(n) \) is odd if and only if \( n \) is a perfect square), which adds a layer of number-theoretic depth.
The conjecture essentially asks whether two fundamentally different arithmetic functions — one chaotic and computationally complex (Collatz), one structured and multiplicative (divisor count) — can ever become permanently correlated in their parity on an arithmetic progression. The expectation is "no," but proving it would require understanding the Collatz stopping time at a level far beyond current knowledge.
Novelty assessment: The individual components (Collatz stopping times, divisor function parity, distribution in arithmetic progressions) are all well-studied, but the specific combination — comparing the parity of \( C(n) \) with the parity of \( \tau(n) \) across residue classes — appears to be original.
Final Thoughts
These three conjectures illustrate a fascinating aspect of mathematics: it is easy to ask questions that may be impossible to answer. Each conjecture connects well-understood objects in unexpected ways, creating statements that feel like they should be true but resist all obvious proof strategies.
Of the three, Conjecture 2 (the Harmonic-Prime Ceiling Conjecture) is the most striking. What began as a conjecture about perfect power avoidance turned — through computational verification — into a precise characterization: the ceiling values are perfect powers exactly at the Mersenne primes. This unexpected connection to one of the oldest open problems in number theory (the infinitude of Mersenne primes) elevates the conjecture from a curiosity to something potentially deep.
Whether any of these conjectures are true, let alone provable, remains an open question — and that's exactly the point.
Disclaimer: These conjectures were formulated as an intellectual exercise. While literature searches suggest they are original, the author cannot guarantee they haven't been independently proposed elsewhere. The computational verifications were performed for finite ranges and do not constitute proofs. If you recognize any of these from existing work, please leave a comment!
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