Examples of eventual counterexamples

Question

Define an "eventual counterexample" to be

• $P(a) = T $ for $a < n$

• $P(n) = F$

• $n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to make.

where 'reasonable' is open to interpretation, and similar statements for rational, real, or more abstractly ordered sets for $n$ to belong to are acceptable answers.

What are some examples of eventual counterexamples, famous or otherwise, and do different eventual counterexamples share any common features? Could we build an 'early warning system' set of heuristics for seemingly plausible theorems?

edit: The Polya conjecture is a good example of what I was trying to get at, but answers are not restricted to number theory or any one area.

Answer 1

It was once conjectured that factors of $x^n-1$ over the rationals had no coefficient exceeding 1 in absolute value. The first counterexample comes at $n=105$.

Answer 2

The least positive integer for which the equality $$ \left\lceil \frac{2}{2^{1/n}-1}\right\rceil = \left\lfloor \frac{2n}{\log 2} \right\rfloor $$ fails is $n=777,\!451,\!915,\!729,\!368$. See https://oeis.org/A129935.

Another example that I like is the number $f(n)$ of inequivalent differentiable structures on $\mathbb{R}^n$. We have $f(n)=1$ if $n\neq 4$, while $f(4)=c$, the cardinality of the continuum.

Answer 3

The essence of the phenomenon of eventual counterexamples is that a certain pattern that holds among small numbers, turns out not to be universal. In the very best examples, such as the examples provided in the other answers, which I have enjoyed very much, what we have is an easily described property $P(n)$, whose first failing instance is very large in comparison. Indeed, the quality of answer might be measured by the difference between the size of the description of the property and the size of the first failing instance of it. When an easily described property holds for a very long time and then suddenly fails at some very large number, we are surprised. Therefore, to my mind the phenomenon of eventual counterexamples is intimately wrapped up with the possibility of providing very short descriptions of enormous numbers.

Surely we are all able easily to provide short descriptions of some very large numbers, such as $2^{100}$ or $2^{2^{100!}}$. In order to go beyond exponentiation and factorials, we might make use of other easily described functions exhibiting even more enormous growth. The Ackermann function, for example, defined by a simple one-line recursion, has diagonal values 1, 3, 7, 61, $2^{2^{2^{65536}}}$, with the next value $A(5)$ mind-bogglingly huge.

All such examples, short descriptions of large numbers, can be systematically transformed into instances of eventual counterexamples. For if $d$ is a short description of an enormous number $N$, then the property $P(k)=$"$k$ does not exhibit $d$" is easily described and holds for all values $k$ below $N$, but not of $N$ itself. Thus, it does very well by the quality measure I mentioned above.

So to my mind, the real issue is: what are the largest numbers that you can describe by a very short description?

This question can be made precise by requiring the description to be expressible in a particular formal language. Once the language is rich enough, however, this problem will certainly wade into interesting foundational waters, for the question of whether a given description actually succeeds in describing a number---for example, "the length of the shortest proof of a contradiction in ZFC"---may be independent of our basic axioms, even if it is enormous.

Answer 4

Strong Law of Small Numbers by Guy.

Steve

Answer 5

In reference to the Prime Number Theorem (then Conjecture) both Gauss and Riemann further conjectured that $\pi(n) < \operatorname{Li}(n)$ (where $\pi(n)$ is the number of primes from $1$ to $n$ and $\operatorname{Li}(n)$ is the logarithmic integral, $\int_2^n \frac{1}{\ln(t)} \, dt$).

Although it has been proven that this does not hold (Littlewood), that there exists some $n$ such that $\pi(n) \geq \operatorname{Li}(n)$, the first $n$ where this takes place is so huge no-one has worked it out yet (allegedly). The number is known as Skewes' Number. It is known to be between $10^{14}$ and $1.39822\times 10^{316}$, and strongly believed to be about $1.397162914\times 10^{316}$. (References at the foregoing link.)

Answer 6

I'm trying to reconstruct an example I saw somewhere some years back. It goes something like this: $\gcd(n^5-5,(n+1)^5-5)=1$ is true for $n=1,2,\dots,1435389$ but fails for $n=1435390$ (when the gcd is 1968751).

Answer 7

Freeman Dyson observed in my presence that the sequence with initial condition $a_0=3,a_1=0,a_2=2$, and recurrence $a_{n+3}=a_{n+1}+a_{n}$ almost has the property that $n\mid a_n$ if and only if $n$ is prime or 1, except that it doesn't.

He challenged us (grad students) to explain this ``near-phenomenon'', as it seems too close to being too good to be true to be coincidence. I've never seen an explanation.

Since this is Math Overflow, I'll give the spoiler, the first counterexample is $n=521^2=271441$.

Answer 8

A famous example is the isomorphism problem for integral group rings: suppose $G$ and $H$ are two finite groups of order $n$ such that $\mathbb{Z}G \cong \mathbb{Z}H$ does it mean that $G \cong H$? It was proved to be true for many cases and for many $n$'s and I think it was believed to be true in all cases. Nonetheless, eventually a counter example was found, see Hertweck, Martin. A Counterexample to the Isomorphism Problem for Integral Group Rings. Annals of Mathematics, vol. 154, no. 1, 2001, pp. 115–138. https://www.jstor.org/stable/3062112.

The groups constructed in this paper have order $n=2^{21}\cdot 97^{28}\approx 9\cdot 10^{61}$. It doesn't seem like a smaller counterexample has been found yet.

Answer 9

The numbers 12, 121, 1211, 12111, 121111, etc., are all composite - until you get to the one with 138 digits, that's a prime. Saw this in a talk Lenny Jones gave at the New Orleans meeting earlier this month.

Answer 10

Any finite loop space has the rational cohomology of a Lie group -- up to rank 65. From then on, there are counterexamples in every dimension. The smallest known dimension of a counterexampe is 1250, but whatever the actual smallest dimension is, counterexamples will occur in every dimension after that.

Answer 11

The Borwein Integrals are integrals of products of the sinc function. They exhibit certain "apparent patterns" which, while eventually breaking down, are actually indicative of something larger at work. (The example given on the Wikipedia page is a good one.)

Answer 12

In answering another MathOverflow question on Graham's number, I quoted from Harvey Friedman's Enormous Numbers in Real Life. Perhaps eventual counterexamples bear some relation to proof strength in certain systems of logic? Anyway, that example there could be rephrased to fit the current question.

Suppose I look at strings on three symbols, and given a word w of length n I look at subwords of the form (forgive the AWK notation) spc[i] = substr(w,i,i+1), i.e. those substrings starting at the $i$th character going for length $i+1$ characters. So spc[1] gets the first two characters of $w$, spc[2] == w[2]w[3]w[4], and so on.

I manage to find, for every $n$ that I can compute, a string $w_n$ that I use for $w$ above such that for $0 < i < j \leq \frac{n}{2}$, spc[i] is not a subsequence of spc[j]. Others find such examples for even larger values of $n$. It would be reasonable for me to believe I could find arbitrarily long strings with this property.

Enter Harvey Friedman:

"Theorem 8.1. Let $k \geq 1$. There is a longest finite sequence $x_1, \dots ,x_n$ from $\{ 1, \dots ,k \}$ such that for no $i < j \leq \frac{n}{2}$ is $x_i, \dots ,x_{2i}$ a subsequence of $x_j , \dots ,x_{2j}$.

For $k \geq 1$, let $n(k)$ be the length of this longest finite sequence.

Paul Sally runs a program for gifted high school students at the University of Chicago.

He asked them to find $n(1), n(2), n(3)$. They all got $n(1) = 3$. One got $n(2) = 11$. Nobody reported much on $n(3)$. I then started to ask several mathematicians to give an estimate on $n(3)$, some of them very famous. I got guesses like this: $60, 100, 150, 200, 300$.

They were not in combinatorics. Recently I asked Lovasz, telling him about these five guesses. He guessed $20000$.

Theorem 8.2. $n(3) > A(7,184)$.

Lovasz wins, as his guess is closer to $A(7,184)$ than the other guesses.

Recall the discussion about $A(5,5)$ being incomprehensibly large. With the help of computer investigations (with R. Dougherty), I got:

Theorem 8.3. $n(3) > A(7198, 158386)$.

A good upper bound for $n(3)$ is work in progress. Crude result: $A(n,n)$, where $n = A(5,5).$"

Here $A(n,n)$ is defined earlier in Friedman's paper as an Ackermann-like sequence.

I suspect $n(3)$ squishes Graham's number quite unlike a galactic black hole absorbing a prion or even a quark.

EDIT: I have been corrected; in the squishing hierarchy, $n(4)$ squishes Graham's number, which squishes $n(3)$. Again, unlike any physical realization I can imagine. END EDIT

The moral here is: "Don't jump to conclusions without a sufficiently strong proof system as back up".

Gerhard "Ask Me About System Design" Paseman, 2010.02.17

Answer 13

The Mertens conjecture.

Answer 14

Let $a_1=1$, $a_{n+1}=(1+a_1^2+a_2^2+\dots+a_n^2)/n$. Are all terms integer? No, the first non-integer is $a_{44}$. I do not know neither reference (my source is private communication by Dmitry Rostovsky, and he does not remember where is it from), nor deep reason (if they exist) why first 43 terms are integer.

Answer 15

Here's another one, maybe mostly of historical interest. Fermat once conjectured that all numbers of the form $$ p=2^{2^n}+1 $$ are prime, which he had the means to verify up to $n=4$. It took more than 100 years until Euler showed that this fails at $n=5$. Today we still don't know if there are any other Fermat primes, so quite possibly Fermat's conjecture fails in the worst possible way.

Answer 16

Smallest counterexample to "There is no positive integer $n$ such that the concatenation of (the decimal representation of) $n$ with itself is a square" is $n=13223140496$, according to https://oeis.org/A102567; $1322314049613223140496 = 36363636364^2$.

Answer 17

Recently I saw that in any $2,3,4,5,\ldots$ consecutive integers, one of them is comprime to the rest, then I conjectured that it should be trivially true for any $k$ consecutive integers, but I didn't able to prove this and I asked this question in MSE, and I surprised by Noah Schweber answer! It's true only for $1,2,\ldots,16$ and the first counterexample is the sequence of length $17$ beginning with $2184$.
There are infinitely many counterexamples for $17\leq k $.
https://oeis.org/A090318/internal

Answer 18

Shapiro inequality: Let $x_1,x_2,\dots, x_n,x_{n+1},x_{n+2}$ be positive real numbers with $x_{n+1}=x_1$ and $x_{n+2}=x_2$. Now the inequality $\sum_{i=1}^{n} \dfrac{x_i}{x_{i+1}+x_{i+2}} \geq \dfrac{n}{2}$ must be true if $n<14$ or if $n\leq 23$ and $n$ is odd. So $n=14$ is the first $n$ where a counterexample can be found. I know that 14 is not that large a number, but remember that for each $n$ we have a problem with a lot of freedom.

Answer 19

One of my favourite examples in this context is the following: Define a sequence $(s_n)$ by $s_1=8$, $s_2=55$ and for $n\geq3$ $s_n$ the smallest integer such that $s_n/s_{n-1}>s_{n-1}/s_{n-2}$ so that $s_3=379$ as $379/55>55/8$. Then we have $s_n=6s_{n-1}+7s_{n-2}-5s_{n-3}-6s_{n-4}$ for $5\leq n\leq11056$ but not for $n=11057$ (I have lost track of the name of the person to whom this is due, but it is, nowadays, easily verified on a computer).

Answer 20

This is a bit tongue-in-cheek, but what about Special Relativity? In this case let property $P(x), x\in \mathbb{R}$ be the property that a given velocity $x$ is attainable. After all, Galilean Transforms allow one to change to a frame moving at an arbitrary velocity. Only Einstein's interpretation of the discoveries of Lorenz and Poincaré allowed for us to realize that property $P$ is only true if $x \in [-3 \times 10^8, 3 \times 10^8]$

Answer 21

I'm late to the party, but here's one from algebraic number theory.

The ring of integers of $\mathbb Q(\sqrt[n]2)$ is exactly $\mathbb Z[\sqrt[n]2]$ for $2\leq n \leq 1092$. At $n =1093$, the ring of integers is bigger. One can show that $\displaystyle\frac{(\sqrt[1093]{2}-2)^{1092}}{1093}$ is an algebraic integer, but is not in $\mathbb Z[\sqrt[1093]2]$.

Keith Conrad has a nice paper on this: https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf

Answer 22

How about this paper?

Answer 23

I'm surprised no one has mentioned Graeco-Latin Squares https://en.wikipedia.org/wiki/Graeco-Latin_square

Euler showed these exist for $n$ odd, or any multiple of 4. As none exist for $n=2$ or $6$, he conjectured that none exist for any $n\equiv 2 (mod 4)$.

As it happens, such exist for any $n\geq 3$ except $6$. This is quite a famous example, if small.

Answer 24

I had a conjecture that for any two natural numbers with the same least prime factor, there must be at least one number in between them with a higher least prime factor. It seemed very robust for reasonably-sized numbers and empirical trends suggested it would hold for arbitrarily large numbers as well.

Months later, I found a paper giving some freshly computed large terms for the Jacobsthal primorial function $h(n)$, and using those, ferreted out a counterexample starting at $$724968762211953720363081773921156853174119094876349.$$

I think this may be the smallest counterexample; even if not, you can show that if there is one smaller, it can't be by much. Until I found this, I would have said I was certain my conjecture was correct. Lesson learned!

Answer 25

I've had fun showing $1,2,4,8,16,31$ to people, both math and non-math people, actually. (OEIS)

Answer 26

Here's a recent one I didn't see on either page. The following are true statements:

\begin{align} & \int_0^\infty \frac{\sin t}{t} \, dt = \frac{\pi}{2} \\[8pt] & \int_0^\infty \frac{\sin t}{t} \, \frac{\sin \left(\frac{t}{101}\right)}{\frac{t}{101}} \, dt = \frac{\pi}{2} \\[8pt] & \int_0^\infty \frac{\sin t}{t} \, \frac{\sin \left(\frac{t}{101}\right)}{\frac{t}{101}} \, \frac{\sin \left(\frac{t}{201}\right)}{\frac{t}{201}} \, dt = \frac{\pi}{2} \\[8pt] & \int_0^\infty \frac{\sin t}{t} \, \frac{\sin \left(\frac{t}{101}\right)}{\frac{t}{101}} \, \frac{\sin \left(\frac{t}{201}\right)}{\frac{t}{201}} \, \frac{\sin \left(\frac{t}{301}\right)}{\frac{t}{301}} \, dt = \frac{\pi}{2} \end{align}

Is it true that

$$\forall n,\, \int_0^\infty \prod_0^n\frac{\sin \left(\frac{t}{100n+1}\right)}{\frac{t}{100n+1}} \, dt = \frac{\pi}{2} \text{?} $$

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No, it's not. However, you could have a computer calculating this for the rest of your life and never find a counter-example. The first counter-example for $n$ has $43$ digits.

I found this example here. It was specially constructed to have a large counter-example - the page gives a way to construct similar statements with arbitrarily large counter-examples.

Answer 27

D. H. Lehmer showed that the first prime value of the Ramanujan tau-function, defined by $$\sum_{n=1}^\infty \tau(n) q^n = q \prod_{n=1}^\infty (1-q^n)^{24} = q - 24q^2 + 252q^3 - 1472q^4 + \dots,$$ occurs at the 63001st term. This is slightly less surprising when one knows that prime values can only occur for odd square inputs.

Answer 28

This came up a few years ago from an error I noticed in the OEIS database. For all $0 \leq n \leq 58$, the numerator of $\sum_{k=0}^n \dfrac{2^{k+1}-1}{k+1}$ is equal to the numerator of $\sum_{k=0}^n \dfrac{\binom{n}{k}}{(k+1)^2}$. This fails first at $n=59$ and then at $n=1519, 7814, \ldots$. See A134652.

Answer 29

The first counterexample to the second Hardy-Littlewood conjecture is expected to occur somewhere between $10^{174}$ and $10^{1199}$ (at least, according to the references from the Wikipedia page), though it has not yet been definitively established that such a counterexample exists.

Answer 30

This one is a little bit a joke. If you calculate the powers $2^k$, it seems that the leading (decimal) digit can never be $7$.

Actually, the first digit happens to be $7$ not before $2^{46}$.