When Numbers Become Strangers: Three Erdős Conjectures Fall to Probability
A unified probabilistic method has toppled three of Erdős’s long-standing conjectures on the behavior of prime factors in consecutive integers.
26 Apr 2026, Yanjiang
Paul Erdős spent much of his career planting questions like seeds, knowing he might never see them flower. Some of those questions lay dormant for decades, waiting for the right mathematical climate. Now, a remarkable preprint from Terence Tao at UCLA and Joni Teräväinen at the University of Cambridge has coaxed three of them into bloom simultaneously.
The paper (arXiv:2512.01739) tackles old problems about the behavior of prime factors in consecutive integers—problems that sound deceptively simple but have resisted solution for generations. To understand what they’ve done, we first need to understand what they were looking at.
The function ω(n) counts how many distinct prime factors a number has. For n=12 (which factors as 2² × 3), ω(12)=2. For n=30 (2 × 3 × 5), ω(30)=3. It’s a simple tally, but its behavior across consecutive numbers—how ω(n) relates to ω(n+1)—turns out to be extraordinarily subtle. Erdős and his collaborators suspected patterns in this behavior, but proving them required tools that didn’t exist when the conjectures were first posed.
The first problem concerns a conjecture of Erdős and Straus: are there infinitely many integers n such that ω(n+k) is bounded by a constant multiple of k for every positive integer k? In other words, can we find numbers that are “prime-factor-sparse” across all their neighbors simultaneously? Think of it like a town where every nearby house has at most a few types of trees in its garden—no matter how far you walk. Tao and Teräväinen prove that such numbers exist infinitely often, using a high-dimensional sieve of the type developed by Maynard for work on prime gaps.
The second problem is perhaps the most elegant. Erdős conjectured that the infinite sum ∑ ω(n)/2ⁿ is irrational. This series—add up ω(1)/2 + ω(2)/4 + ω(3)/8 + …—converges to some number between 0 and 2. Is that number a rational fraction, or does it escape the neat patterns of ratios? The team proves it is irrational, settling a question Erdős posed decades ago. The proof uses a probabilistic method: treat the prime factor counts as random variables, then show that the sum’s behavior cannot be captured by any rational number.
The third problem addresses a more subtle pattern. Erdős, Pomerance, and Sárközy conjectured an asymptotic formula for how often ω(n) equals ω(n+1) among numbers up to x. Tao and Teräväinen prove this formula holds for “almost all” x—meaning the exceptions, if any, are vanishingly rare. They achieve similar results for the related functions Ω(n) (which counts prime factors with multiplicity) and τ(n) (which counts all divisors).
What makes this work remarkable is not just the results, but the method. All three proofs share a common thread: the probabilistic method, which treats number-theoretic objects as random variables governed by statistical laws. For the first problem, this is combined with a sophisticated sieve that carefully manages the contributions from different size ranges of primes. For the second and third problems, the team developed a general quantitative estimate for two-point correlations of multiplicative functions—a technical achievement that may prove as valuable as the conjectures it resolves.
To understand the power of this correlation estimate, consider a simple analogy. Imagine you’re trying to predict whether it will rain on two consecutive days. If the days were independent, you’d just multiply the individual probabilities. But weather patterns create correlations: a rainy Tuesday makes Wednesday more likely to be wet. The same principle applies to prime factors—knowing ω(n) tells you something about ω(n+1), but quantifying that “something” is extraordinarily difficult. The team’s estimate, building on recent work of Pilatte, provides a precise bound on how strongly these values correlate, with a “logarithmic saving” that makes the calculations tractable.
The paper’s numerical experiments are worth pausing over. Figure 2 tracks the probability that τ(n+1)/τ(n) is a power of 2, which converges to about 0.4888—a number that emerges from the return probabilities of certain random walks. Figure 5 shows histograms of ω(n+1)−ω(n) and Ω(n+1)−Ω(n) for n up to 10⁷, compared against predicted Gaussian distributions. The agreement is striking, though the convergence is slow enough to suggest hidden correction terms.
The probability that two consecutive integers share the same number of prime factors rises toward 1 as numbers grow larger, though the climb is extremely slow. This slow convergence hints that hidden correction terms are needed to fully predict this behavior, refining our understanding of prime number patterns.
There is something deeply satisfying about watching decades-old conjectures fall, one after another, to a unified approach. Erdős, who died in 1996, would likely have appreciated the irony: problems about the deterministic structure of integers, solved by treating them as random variables. The probabilistic method, which he himself championed, has become so refined that it can now address questions that once seemed purely combinatorial.
What remains open? The asymptotic formula for ω(n)=ω(n+1) is proven for “almost all” x, but proving it for every x would require even finer control over the correlation terms. The undetermined constants B₅, B₆, and B₇ that appear in the heuristic approximations (Figure 4 suggests B₇ ≈ −1) hint at deeper structure waiting to be uncovered.
For a field that has long treated prime factor counts as a playground for elementary methods, this paper opens a door. Whether anyone walks through it depends on the next generation of number theorists—theorists who are, even now, learning the probabilistic tools that Tao and Teräväinen have forged.
Yanjiang is an online editor of Loom Science
References
- Terence Tao and Joni Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739