In the 1980s, a programme called Graffiti started asking questions cipher had thought to ask, and 1 of them caught the attraction of Paul Erdős. Almost forty years later, Randy Davila handed that aforesaid mobility to Theo-Conjecture, an automated find strategy supported by a ample connection exemplary that proposes, tests and revises mathematical ideas successful a loop. What came backmost was a proof of the answer Erdős and his collaborators had guessed at, an unexpected other word cipher had predicted, and a glimpse of what it looks for illustration erstwhile an AI supplier and a quality mathematician activity a problem together.
Here's the image to clasp successful your head. Take the integers from 2 done 30 and tie each 1 arsenic a dot. Draw a statement betwixt 2 dots whenever the numbers stock a facet greater than one. So 6 connects to 10, since some are divisible by 2, and 15 connects to 25, since some are divisible by 5.
What you extremity up pinch is simply a graph, successful the mathematical sense, of dots (vertices) joined by lines (edges). Color the primes gold. None of the golden dots touch each other, because 2 different primes ne'er stock a factor.
The astonishing part? The primes aren't conscionable immoderate postulation of unconnected dots. They shape the largest imaginable group of dots pinch nary connections betwixt them astatine all. Mathematicians person a sanction for groups for illustration these - an independent set. It's worthy seeing why the primes triumph this contest.
Pick immoderate independent group successful the graph. Every number successful it has to beryllium coprime to each different number successful it (that's what “no shared factors” means). Now propulsion retired 1 premier facet from each number. Since nary of those numbers stock factors, the primes you conscionable pulled retired person to each beryllium different from each other. Which intends you tin ne'er stitchery much numbers into your independent group than location are primes to tie from successful the first place. Written arsenic a formula:
α(Gₙ) = π(n)
Here Gₙ is the chart built from the integers 2 done n, α(Gₙ) is the size of its largest independent set, and π(n) is the classical prime-counting function, the number of primes up to n.
That's a neat personality connected its own; it turns a mobility astir counting primes into a mobility astir a graph. But it besides opens a doorway to thing much strange, a number you tin compute quickly utilizing thing but arithmetic connected a graph's structure, pinch nary evident relationship to primes astatine all.
A shortcut that shouldn't activity (but does)
Finding the largest independent group successful a chart is, successful general, a difficult computational problem because it doesn't standard well. But there's a overmuch easier number to get your hands on, called the grade of a vertex, meaning really galore edges touch it.
There's a instrumentality for turning degrees into a little bound connected independence, called the Havel-Hakimi procedure. List each vertex's degree, largest to smallest. Take the biggest number, telephone it d, transverse it off, and subtract 1 from each of the adjacent d entries successful the list. Sort again. Repeat.
Eventually you're near pinch thing but zeros. Count them. That count is called the residue of the graph, written R(G).
Computing the residue is fast, and it ever undersells the truth successful a predictable direction:
R(G) ≤ α(G)
In different words, the residue is simply a certificate you tin compute successful seconds that guarantees a little bound connected the size of the largest independent set, without you ever having to find that set. Applied to our premier graph, that becomes:
R(Gₙ) ≤ π(n)
So here's the existent question. This residue calculation only ever looks astatine degrees. It ne'er looks astatine which vertex connects to which. Does that stripped-down position still seizure the correct bid of magnitude for counting primes? Or does throwing distant each that structural item propulsion distant the reply too?
That mobility has a amazingly agelong history, and it starts pinch 1 of the first superior attempts to get a instrumentality to propose mathematics connected its own.
Graffiti, Erdős, and a mobility near connected the wall
Siemion Fajtlowicz, a mathematician astatine the University of Houston, built a programme called Graffiti successful the mid-1980s. It stored a room of graphs on pinch numerical properties of each one, generated campaigner inequalities relating those properties, and utilized a group of heuristics to select retired thing existent but boring. Fajtlowicz wasn't conscionable trying to make formulas, he was trying to understand what makes a mathematical connection worthy a mathematician's time.
Graffiti's conjectures ended up inspiring hundreds of papers and drew the attraction of mathematicians for illustration Paul Erdős, Fan Chung, László Lovász and Paul Seymour. Erdős, successful particular, was a earthy fit. He spent overmuch of his life bouncing betwixt collaborators, actual examples and unfastened problems successful number mentation and combinatorics, and Fajtlowicz erstwhile summed up the resemblance betwixt the man and the instrumentality successful a azygous line:
“Whatever the differences, Erdős and Graffiti had 1 constituent successful common: their ideas were profoundly rooted successful examples.”
In November 1987, Fajtlowicz introduced the residue arsenic an invariant into Graffiti and conjectured that it tin ne'er transcend the independency number. Favaron, Mahéo, and Saclé proved that the pursuing year, and the proof was published successful 1991. Graffiti past turned its attraction to these common-divisor graphs connected the integers, and the mobility made it into the moving database of unfastened problems called Written connected the Wall, arsenic Conjecture 448.
Erdős showed that the residue grows astatine slightest arsenic accelerated arsenic n / log n, the aforesaid complaint arsenic the prime-counting usability itself. Working pinch the chart theorist William Staton, he pinned down the starring changeless much precisely:
ζ(2) − 1 = π² / 6 − 1 = 0.644934...
That ζ(2) is simply a well-known worth of the Riemann zeta function, the aforesaid usability astatine the halfway of 1 of math's astir celebrated unfastened problems. Erdős past asked whether anyone could find a matching precocious bound of the aforesaid order. Fajtlowicz guessed the changeless mightiness beryllium 2/3. Staton, however, had a different hunch. He hypothesized that the Erdős-Staton little bound wasn't conscionable a little bound, it was the nonstop answer. Written connected the Wall records his position plainly: “Bill Staton thinks that the residue ... is asymptotically (π²/6 − 1) n / log n.”
Numerical checks up done n = 10,000 backed him up. But cipher produced a proof, astatine slightest not 1 that made it into immoderate grounds anyone could find.
The problem comes backmost around
Randy Davila is simply a chart theorist and Member of Technical Staff astatine FirstPrinciples, pinch an world affiliation astatine Rice University. Since 2016 he's been building TxGraffiti, a modern successor to Fajtlowicz's original program. The halfway thought carried complete intact, wherever each illustration and each invariant should beryllium laid retired intelligibly capable that a mathematician tin look astatine it, flick astatine it, and push back.
Davila rediscovered the common-divisor chart independently and rebuilt its information from scratch. In 2025 he sent his computations and land to the number theorist Jeffrey Lagarias, who traced the mobility backmost done a group of hard-to-find Fajtlowicz documents and mentioned related activity he'd done pinch Fan Chung. Chung's ain publication database now lists an undated preprint called Bounds for Havel-Hakimi residues of non-coprime graphs, crediting Chung alongside Erdős, Lagarias and Staton, though there's nary linked manuscript aliases theorem connection attached to it. Worth flagging directly: this chronology comes from backstage correspondence, and it should beryllium independently confirmed pinch Lagarias and Chung earlier anyone treats it arsenic settled history.
Still, the reconstruction did 2 useful things. It confirmed precisely what the conjecture claimed, and it clarified what a existent beforehand would request to look like. That is, an precocious bound that matches the Erdős-Staton constant, ideally pinch thing beyond conscionable the starring term.
That's the style of the problem Davila fed into the Theo-Conjecture loop. Theo-Conjecture isn't a chatbot, and it isn't a formula-spitting instrumentality either. It's a find strategy pinch a quality advisor successful the loop, wherever a ample connection exemplary acts arsenic an agent, many times calling a combinatorial conjecturing engine, moving nonstop computations, and updating an evolving mathematical representation arsenic it goes.
Every effort gets kept (including the ones that fail) successful a moving grounds the exemplary tin hunt backmost done and build connected later. Importantly, thing counts arsenic proven conscionable because the exemplary said so, aliases because it matched a heap of examples. A quality mathematician has to motion disconnected earlier thing counts arsenic a existent result.
That's what made the failures useful alternatively of wasted. When an thought didn't clasp up, it stayed connected the grounds and shaped what sewage tried next, alternatively than getting softly dropped.
What the failures really taught it
The system's first thought was wrong. It tried to explicate the residue utilizing really galore numbers successful a scope person nary repeated premier factors, and the prediction missed by a wide and increasing margin, disconnected by 739 astatine larger values. That ruled the thought retired completely.
A amended lucifer came from a amount called the Caro-Wei sum, which looks astatine really galore connections each number has alternatively than which circumstantial numbers are connected. Using it, the aged ζ(2) − 1 changeless fell retired naturally, on pinch a 2nd word cipher had predicted. That still only proved half the story, a little bound. Closing the spread took a existent insight, and realizing the residue only cares astir really galore connections each number has, ne'er which ones specifically. So the full chart tin beryllium redrawn, arsenic agelong arsenic each number keeps the aforesaid number of connections it started with.
Here's what that redrawing looks like. Group the primes that stock the aforesaid number of connections into a tight cluster wherever they're each connected to each other, and fto mean composite numbers sorb the connections that sewage displaced. Once primes are clustered that way, it's easy to spot you tin ne'er prime much than 1 of them astatine a time, which is precisely the bound Staton predicted.
The result
Putting each of that together proves Staton's original prediction and adds a second-order word cipher had recovered before:
R(Gₙ) = c₀ · n/log n + (c₀ − A) · n/log²n + O(n/log³n)
where
c₀ = ζ(2) − 1 and A = Σ(k=2 to ∞) log k / [k²(k−1)] = 0.3201986326...
Which gives us, successful the limit:
R(Gₙ) · log n / n → ζ(2) − 1
Put successful plain terms, that theorem says the quick, degree-only calculation captures a fixed, predictable fraction of the existent premier count arsenic n grows. Flip the ratio astir and it sounds conscionable arsenic cleanly:
π(n) / R(Gₙ) → 1.550546...
To beryllium clear astir what this isn't, it's not a faster measurement to count primes, and it doesn't touch the Riemann hypothesis. But the quality of the zeta usability present isn't a coincidence either. It falls retired people from summing up the layered, multiplicative building created by premier divisors, and the graph-theoretic impervious explains precisely why the residue, a number that only ever looks astatine degrees, ends up capturing that aforesaid starring behavior.
The second-order word is the genuinely caller part. It doesn't conscionable show you wherever the residue is heading asymptotically, but really accelerated the finite chart closes successful connected that destination.
A tighter mystery, still open
One particularly cleanable mobility came retired of each this and hasn't been settled yet. Across each worth tested consecutively up done n = 10,000, and astatine scattered checkpoints up to a million, the residue ne'er strayed much than 2 integers from the rounded-up Caro-Wei sum:
⌈CW(Gₙ)⌉ ≤ R(Gₙ) ≤ ⌈CW(Gₙ)⌉ + 2
Nobody has proven that yet. If it holds, it would switch the theorem's existent asymptotic correction word pinch a hard, absolute bound of astatine astir two, and it would explicate the fine-grained behaviour of the Havel-Hakimi process acold much precisely than the degree-switching statement presently does.
There's besides a bigger constituent sitting underneath each of this. Graffiti's domiciled was to propose a narration and manus it to a mathematician for judgment. Theo-Conjecture turns that aforesaid speech into a running, iterative loop. Its registry holds onto examples and formulas, but it besides holds onto the grounded conjectures, the counterexamples, the abandoned representations, and the impervious obligations nobody's vanished yet. The connection exemplary gets to reorganize the problem and determine what to effort next. The conjecturing motor supplies disciplined breadth. The nonstop computations get to opportunity no. And the quality advisor stays responsible for the intent down the hunt and the last proof.
Trace the statement from Fajtlowicz done Erdős and Staton, connected to Davila, Lagarias and Chung, and yet into the Theo-Conjecture loop, and what you get isn't a communicative astir machines replacing mathematicians. It's a communicative astir mathematical representation becoming thing you tin really run, and astir a connection exemplary gaining existent leverage by leaning connected a specialized find strategy again and again, alternatively of trying to logic its measurement done alone. A mobility near connected a wall for decades yet recovered a procreation of devices that could publication it differently.
Further reading
Ermelinda DeLaViña, Some History of the Development of Graffiti (uhd.edu)
Siemion Fajtlowicz, “On Conjectures of Graffiti,” Discrete Mathematics (1988)
Randy Davila, “Automated conjecturing successful mathematics pinch TxGraffiti” (arXiv:2409.19379)
Paul Erdős biography, MacTutor History of Mathematics
Jeffrey Lagarias, University of Michigan
Fan Chung Graham's publication list, including the undated common-divisor residue preprint listing
Odile Favaron, Maryvonne Mahéo, and Jean-Francois Saclé, On the residue of a chart (Journal of Graph Theory, 15(1):39–64, 1991)
The theorem and impervious described present are contained successful a research manuscript nether mentation and person not yet completed adjacent review. Historical claims based connected backstage correspondence are identified explicitly successful the editorial root notes.
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