+++ title = "Checked to Four Quintillion" description = "Goldbach's conjecture has been confirmed for every even number up to 4×10¹⁸ and is still a conjecture. Number theory has an unusually exact account of what a checked range is evidence about, and it is not the length of the range." date = 2026-08-22 draft = false [taxonomies] topics = ["famous-problems", "analytic-number-theory"] kinds = ["verification", "scale", "reversibility"] [extra] tier = "public" schema_type = "Article" canonical = "" hero = "articles/checked-to-four-quintillion/hero.webp" hero_alt = "Computer’s notebook checked through a ruled stopping line" podcast = "" mechanism = "Exhaustive checking is evidence about the range checked. It becomes evidence about the rest only where something outside the observations generates both the rule and its departures, and can therefore say at what scale a departure is due." apparatus = "Number theory, which records how far a conjecture has been checked as a bound with a paper and a year attached, refuses the word true for it, and keeps disproof-with-a-counterexample separate from disproof-by-existence." discriminating_test = "Ask what produces the exceptions. If the account that generates the rule cannot say at what scale the first exception should appear, the tested range carries no information about the untested range, however long it is." false_friend = "A proved result with a bounded error term. The prime number theorem is not a conjecture confirmed to a large number, and in a summary table it sits in the same column as one." test = "For every rule in production, record the tested range and, separately, the smallest scale at which the mechanism that would break it can act. Where the second is larger than the first, the length of the backtest is decoration." symptoms = [ "A strategy is called robust on the strength of how much history it survived", "Confidence in a rule is stated as a span of years rather than as a set of conditions covered", "A relationship holds in every period in the sample and the sample contains no instance of the condition that would break it", "A result is described as almost always true with no account of when the exception is due", "A tested bound and a proved theorem appear in the same table at the same strength", ] sources_verified = false [[extra.faq]] q = "A rule has held for twenty years of data. How much is that worth?" a = "It is worth exactly as much as the range covers, and the range is measured in conditions rather than in years. Number theorists write these results as checked up to a bound, never as true up to a bound, because the two are different claims: Goldbach's conjecture has been confirmed for every even integer below 4×10¹⁸ and is still an open question. A rule that can only fail when a funding market seizes has been tested for zero days in a sample containing no seizure, whatever the sample's length." [[extra.faq]] q = "Has a well-tested mathematical conjecture ever turned out to be false?" a = "Pólya's conjecture is the standard case. It survives any computation a person would do by hand, and its smallest known counterexample is 906,150,257 — found by Minoru Tanaka in 1980, twenty-two years after C. Brian Haselgrove had proved that one must exist. Haselgrove's own argument estimated the failure near 1.845×10³⁶¹, which is roughly 350 orders of magnitude away from where the smallest one actually sits. The argument that killed the conjecture could not locate the failure it had proved was there." [[extra.faq]] q = "Can something be known to be false and never observed to fail?" a = "Yes, and the discipline keeps a name for it. Andrew Odlyzko and Herman te Riele disproved the Mertens conjecture in 1985 using lattice-basis reduction, establishing that a counterexample exists below a stated bound without producing one. Later work has tightened the bound to below exp(1.59×10⁴⁰). No paper has ever printed a number that violates the conjecture. Number theory calls this an existence-only disproof and does not let it share a column with the disproofs that come with a number attached." [[extra.faq]] q = "What makes a large sample actually informative about what has not been sampled?" a = "Something outside the sample that produces both the pattern and its exceptions. Riemann's 1859 explicit formula writes the prime-counting function as a smooth main term minus a sum over the zeros of the zeta function, so the deviations have the same generator as the rule and their expected scale can be compared against the range already checked. Where no such generator exists, the checked range is evidence about the checked range." [[extra.faq]] q = "Is a pattern that holds ninety-nine per cent of the time a stable pattern?" a = "Not necessarily, and prime counting is the clean demonstration. Primes of the form 4k+3 lead those of the form 4k+1 about 99.59 per cent of the time by logarithmic density, on the 1994 Rubinstein–Sarnak result. J. E. Littlewood had already proved the lead changes sides infinitely often. The first crossover is at 26,861, small enough to have been found by hand. A frequency and a guarantee are separate measurements, and only one of them was available here." +++ In November 2013, three researchers reported that Goldbach's conjecture holds for every even number up to four quintillion. Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi spent roughly 770 single-core CPU-years confirming that each even integer below 4×10¹⁸ is a sum of two primes. Every one of them. No exceptions found. It is still a conjecture, and nobody in the field calls it nearly proved. The phrasing is a discipline rather than a hedge. Checked up to a bound, with a paper and a year attached. Never true up to a bound. It exists because the field holds a collection of cases where the checking ran a very long way and the rule was false anyway. ## The first failure sits past the search Pólya's conjecture says that most of the integers below any given bound have an odd number of prime factors. It survives every computation a person would think to do by hand. C. Brian Haselgrove disproved it in 1958, on EDSAC. His method produced no violating number. It established that one had to exist, somewhere in the region of 1.845×10³⁶¹. R. S. Lehman found an explicit counterexample in 1960. Minoru Tanaka found the smallest known one in 1980. It is 906,150,257. Nine hundred and six million. Reachable on a laptop over lunch, and unreached for two decades because nothing indicated that stretch of the number line rather than any other. The distance between Haselgrove's estimate and Tanaka's number is around 350 orders of magnitude. The argument that killed the conjecture could not locate the failure it had proved must be there. ## False, and never observed to fail The Mertens conjecture holds that the Mertens function never exceeds the square root of its argument in absolute value. Andrew Odlyzko and Herman te Riele disproved it in 1985, using the Lenstra–Lenstra–Lovász lattice-basis reduction algorithm. What they proved is that a counterexample exists below a specific bound, far past anything a search reaches. They did not produce one. Subsequent papers have tightened the bound, currently to below exp(1.59×10⁴⁰). None of them has printed a number that violates the conjecture. Four decades on, it is known false with certainty and has never been observed to fail. Number theory files this separately from the Pólya case, under existence-only disproof, because the two are not the same strength of claim and it would be easy to record them as though they were. ## Almost always is a measured quantity Pafnuty Chebyshev noticed in 1853 that counting up to nearly any limit gives more primes of the form 4k+3 than of the form 4k+1. The prime number theorem for arithmetic progressions says the two counts are asymptotically equal. The lead is a real thing that the asymptotics do not predict. J. E. Littlewood proved the lead changes sides infinitely often. The first crossover is at 26,861. Michael Rubinstein and Peter Sarnak showed in 1994 that measured by logarithmic density, the 4k+3 side leads about 99.59 per cent of the time — conditional on the generalized Riemann Hypothesis and on a linear-independence assumption about the relevant L-function's zeros. Three separate statements about one pattern. How often it holds. Where it first fails. What produces it, which is the low-lying zeros of the Dirichlet L-function attached to the character mod 4. ## The generator makes the range informative Riemann's 1859 paper gives the general form. The prime-counting function equals a smooth main term, essentially the logarithmic integral, minus a correction that is a sum over the non-trivial zeros of the zeta function, each zero contributing an oscillation of size roughly x raised to that zero. The departures from the rule have the same source as the rule. Move a zero and both move. That is the property doing the work, and it is not the size of the search. Where one object produces both the pattern and its exceptions, the scale at which an exception is due can be computed and compared against the range already covered, and the comparison is the evidence. Where there is no such object, a checked range is evidence about the checked range. ## A backtest is a verification bound Twenty years of daily data is five thousand observations, and it gets quoted the way four quintillion gets quoted, as a quantity of confirmation. The prior question is the one the number theorists ask. What would have to occur for this to fail, and does the tested range contain it? A rule that can only break when a funding market seizes has been tested across zero such events in a sample with none, whatever its span in years. The length of the record is not the coverage of the record, and only one of the two is usually reported. There is a further distinction the field maintains and a results table does not. The prime number theorem is not a conjecture confirmed to a large number. Hadamard and de la Vallée Poussin proved it in 1896, de la Vallée Poussin bounded its error in 1899, and that bound has been sharpened since, most recently in a 2023 paper. Goldbach has a search. Chebyshev's bias has a proof that the exceptions are infinite. The prime number theorem has a theorem. All three would read as supported in a summary. ## The integers were holding still Goldbach's conjecture in 2013 concerns the same integers Goldbach wrote to Euler about in 1742. The 770 CPU-years bought coverage of an object that does not move. A rule tested across twenty years of prices is not tested against one object twenty times. Participants arrive and leave, the instrument is redefined, and the rule is itself traded once enough people hold it, which edits the thing it describes. Coverage does not accumulate the way it does over the integers, because the domain is not waiting to be covered. Nor does a failure settle anything. After 906,150,257 nobody reopens Pólya. A rule that broke in March is not disproved in June; it is available again, and it will be described as having been out of favour for a while. The Mertens conjecture has been false for four decades and no one has ever watched it break. A rule that has never broken is in the opposite position, and from the inside the two are indistinguishable. What separates them is a proof, and there is not going to be one.