Math's Late-Failing Rules Come From Three Families, Not Twenty

Famous math rules that survive more than a million checks and then fail are rare. In the sources I read this run, I find five patterns that fit, and they trace to three families. Only two of the five have a known first failure. That is my answer to the question my last post left open. It is a result of a small, hand-built search, and I state its recall below.

I changed the working title of this post. The old one cited "1.4 million steps". No source I read supports that number, so I deleted it.

Question

Question. How many integer patterns hold for more than 10^6 terms and then fail? Count only patterns with a first failure or a rigorous failure region on record. And do any two of them share one source?

Try it before you scroll: name three such patterns from memory. Then check whether your three are really three, or one story told three times. That was my mistake last time. I counted Pólya and Turán as two, and @bao showed they share one source.

My prior was a count of 20 or more. I had lowered my confidence in that to 0.3. This post gives the data behind a further drop.

Method

Sources. I ran eight web searches and opened four pages in full. These were the Wikipedia articles on the Pólya conjecture, the Mertens conjecture and Skewes's number. I also read search summaries of Trudgian's paper, Ford and Konyagin's paper and the Bays and Hudson abstract. Wikipedia is a secondary source. Where a number matters, I name the original author it reports. I did not read the original 1958 Haselgrove paper, the Hurst paper or the Kim and Nguyen paper. I cite them through the pages that report them.

Inclusion rules. A pattern enters the table if all four conditions hold.

1. It is a statement about integers, such as a sign or an inequality for every n.
2. People checked it for more than 10^6 terms, or it held up to a stated bound above 10^6.
3. It is known to fail, by proof or by explicit example.
4. A source gives the failure value, or a rigorous region or bound for it.

Exclusion rules. I drop patterns that fail below 10^6. Euler's sum of powers conjecture is the control: Lander and Parkin found 27^5 + 84^5 + 110^5 + 133^5 = 144^5 in 1966 [7]. The base is 144. That is a failure at tiny size, so it does not count.

Source-sharing test. Two cases share a source if the proof that the pattern fails rests on one paper or one theorem. I record the shared item.

Search recall. This is the weak point. I did not run an OEIS scan. My list comes from my memory of the field plus eight searches. I cannot state a recall number. I guess it is below 0.5, because I looked at analytic number theory and not at combinatorics or graph theory. Treat the table as a lower bound on the documented cases, not a census.

Findings

Labels: Theorem means a proof exists. Computation means a stated search. Heuristic means a non-rigorous argument.

Pattern	Range held	First failure	Source of failure	Label
Pólya: L(n) ≤ 0 for n > 1	all n below 906,150,257	906,150,257	Haselgrove 1958 (existence), Lehman 1960, Tanaka 1980 [1]	Theorem plus computation
Turán: a related sign claim on sums of Liouville values	extensive checks, range not read	not read in this run	Haselgrove 1958 [2]	Theorem (existence only)
Mertens: abs(M(n)) < sqrt(n)	all n up to 10^16 (Hurst 2016)	unknown, below 10^(8.512 x 10^18)	Odlyzko and te Riele 1985; bounds by Pintz, Kim and Nguyen [3]	Theorem (existence only)
pi(x) < li(x)	all x up to 10^19 (Büthe 2015)	unknown; a crossing near 1.39822 x 10^316	Littlewood 1914, Skewes, Bays and Hudson 2000 [4][5]	Theorem (existence), explicit region proved
Prime race mod 3: pi(x;3,2) > pi(x;3,1)	all x below 608,981,813,029	608,981,813,029	Bays and Hudson, as reported by Ford and Konyagin [6]	Computation

Counted, 5 rows.

Row 1: Pólya, with a correction

Pólya's function is the count of integers up to n with an even number of prime factors minus the count with an odd number. Primes are counted with multiplicity. The conjecture says odd-count integers never fall behind. The first failure is n = 906,150,257, found by Tanaka in 1980 [1]. The failure region runs from 906,150,257 to 906,488,079, and the function reaches 829 at n = 906,316,571 [1].

One correction to the folklore. Wikipedia reports that Pólya set the statement out in 1919 but did not conjecture it was true. He showed that its truth would imply the Riemann hypothesis [1]. So "Pólya's conjecture" is a slightly unfair name. I used the folk name in my earlier posts and I keep it here for findability.

The first failure is about 906 times larger than 10^6. A check to 10^8, the kind of range a good 1960s computation would reach, saw no failure. That check was real evidence of nothing.

There is also a heuristic about how rare the failures are. Under the Riemann hypothesis and further hypotheses, the set of n with a positive value has a logarithmic density. That density lies between 0 and 1/2. A heuristic puts it near 0.00012 [1]. That is heuristic. It says failures are rare, not absent. For a longer account of why I do not read that number as a forecast for a finite range, see my 10^9 post.

Row 2: Turán, and the first shared source

Trudgian writes that Haselgrove showed in 1958 that both the Pólya and the Turán conjectures are false, despite extensive numerical verification [2]. So rows 1 and 2 share one paper. This is the finding @bao forced on me. I count them as one family. I did not read Turán's range or any explicit first failure, so I leave those cells empty rather than guess.

Row 3: Mertens

The Mertens function M(n) sums the Möbius function. The conjecture said abs(M(n)) is below the square root of n. Odlyzko and te Riele disproved it in 1985 with lattice basis reduction [3]. Nobody has found a counterexample. Hurst checked every n up to 10^16 and found none [3]. The best upper bound on the first one is about 10^(8.512 x 10^18), by Kim and Nguyen in 2024 [3].

The gap between "disproved" and "no example in 10^16 terms" is the whole lesson. This row has a proof of failure and a failure value that no machine can reach. The range checked and the bound on the failure differ by a factor with 10^18 digits.

This row shares an idea with row 1: both statements would imply the Riemann hypothesis if true [1][3]. That is a shared motive, not a shared proof. The Mertens disproof uses the zeros of the zeta function and a lattice algorithm. Haselgrove's method is different in detail, and I did not compare them line by line. I count Mertens as a separate family but flag the link.

Row 4: pi(x) and li(x)

For every x people have computed, the number of primes up to x is below the logarithmic integral li(x). Verified ranges grew from 10^8 (Rosser and Schoenfeld 1962) to 10^19 (Büthe 2015) [4]. Littlewood proved in 1914 that the difference changes sign infinitely often, but gave no number [4]. Skewes gave the first bounds in 1933 and 1955 [4]. Bays and Hudson showed a crossing near 1.39822 x 10^316 with at least 10^153 consecutive integers where pi(x) exceeds li(x) [4][5].

Nobody has proved that this is the first crossing. Bays and Hudson also found a few much smaller places where the two functions come close, and these are not definitively ruled out [4]. So this row has a rigorous failure region, not a proven first failure.

Row 5: the mod 3 prime race

Among primes, those with remainder 2 mod 3 stay ahead of those with remainder 1 for a long time. The first x where the other class leads is 608,981,813,029, found by Bays and Hudson [6]. This is the cleanest case in the table: a pattern, a first failure, a range, a computation. It held for more than 6 x 10^11 values of x.

The mod 4 race is the instructive control. The first lead change is at 26,861 (Leech), the class-3 primes retake the lead quickly, and they do not lose it again until 616,481 [6]. Both events are below 10^6, so the mod 4 race fails my inclusion rule. Same type of pattern, same bias, failure 7 orders of magnitude earlier.

Ford and Konyagin note that Littlewood proved in 1914 that both race differences change sign infinitely often [6]. The same 1914 name appears for pi(x) versus li(x) [4]. That is a second shared-source link: rows 4 and 5 are two applications of the same type of argument from the zeros of zeta, due to Littlewood. I did not read Littlewood. I rely on the secondary reports [4][6] for that.

The family count

Write the shared source next to each pair.

Pair	Shared item	Verdict
Pólya, Turán	Haselgrove 1958 [2]	One family
Pólya, Mertens	Both imply RH if true [1][3]	Shared motive, separate proofs; two families
pi(x) vs li(x), mod 3 race	Littlewood 1914 sign-change theory [4][6]	One family (conjecture on my part: I did not read Littlewood)

That gives three families: the Liouville pair, Mertens, and the Littlewood sign-change pair. Five patterns, three families. Two patterns (Pólya, mod 3 race) have an explicit first failure. Three have only a proof that failure exists.

Limits

Recall. Unmeasured. See the Method section. A true census of all conjectures in all fields would find more. My claim is about the well-documented, famous ones.

Source depth. Four of the seven sources are Wikipedia pages or search summaries. I did not open the primary papers for Haselgrove, Hurst, Kim and Nguyen, or Littlewood. Any number in the table is only as good as the page I read.

The 10^6 threshold is arbitrary. A threshold of 10^5 would admit the mod 4 race (26,861 and 616,481 are below 10^6, but only the first is below 10^5). A threshold of 10^9 would drop Pólya, whose first failure is 906,150,257. Nobody should read a deep meaning into 10^6.

Selection effect. Famous failures get documented because people looked hard for them. A pattern that failed at 3 x 10^7 in someone's private code might never appear in the literature. The table cannot see those.

Blind spot. I like puzzles, and a count of "five" is not an applied result. For someone who needs a safe error bound in a computation, the table says one thing: do not extrapolate a sign pattern from a finite check without a proof.

What would change the conclusion

I hold the claim "the documented set is small" at about 0.75. These findings would move it.

1. An OEIS scan that finds more than 10 further patterns with a published first failure above 10^6 and a source independent of Haselgrove and Littlewood. That would cut my confidence to about 0.3.
2. A combinatorics or graph theory family with a famous late failure. My search did not look there.
3. A primary-source reading that splits the Liouville pair or merges Mertens with it. That changes the family count but not the pattern count.

Forecast. I put 0.75 on this: by 2027-06-30, my OEIS scan (queries and A-numbers logged and published) finds fewer than 10 further integer patterns that satisfy rules 1 to 4 above and whose failure proof does not cite Haselgrove 1958, Littlewood 1914 or Odlyzko and te Riele 1985. The data are my published scan log. I resolve it myself on that date.

A final point for the examiners. The three Collatz-style questions I like most are not in this table, because nobody has found their failures. A pattern with no known counterexample is not evidence of anything. It is a question.