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This is a soft question. I'm searching for examples of mathmatical statements (preferably in number theory, but other topics are also fine), that seem to be true, but are actually not. Statements where observing some examples would let you think it is always true, but then there is a well hidden counter-example.

If Riemann's $\zeta$ function had a zero beside the critical line, this would be such an example I'm looking for. Or if Fermat's Last Theorem would be false.

Do you know any such surprising counter-exmaples?

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    $\begingroup$ See the prime generating polynomials: mathworld.wolfram.com/Prime-GeneratingPolynomial.html $\endgroup$
    – Amzoti
    May 26, 2014 at 12:56
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    $\begingroup$ The number of primes $\le x$ of the form $4k+3$ is never greater than the number of primes $\le x$ of the form $4k+1$. Please see Prime Number Races. Counterexamples are actually not rare, but the first one is big. $\endgroup$ May 26, 2014 at 13:15
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    $\begingroup$ Fermat's Last Theorem is proven true... $\endgroup$
    – Platonix
    May 26, 2014 at 13:21
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    $\begingroup$ There are some nice examples in this thread. $\endgroup$ May 26, 2014 at 14:05
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    $\begingroup$ All prime numbers are odd :-) $\endgroup$
    – gnasher729
    May 26, 2014 at 16:32

8 Answers 8

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The following spikedmath is cute.

SM

The books "Counterexamples in Topology" by Steen and Seebach as well as "Counterexamples in Analysis" by Gelbaum and Olmstead have some that are surprising when you first see them.

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For every $n$ $$\int_0^\infty 2 \cos(x) \prod_{i=0}^n\frac{\sin\frac{x}{2i+1}}{\frac{x}{2i+1}}\,dx=\pi/2$$

Is it true? Well, for every $n$ less than 56, but after...

An example of Borwein integral.

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  • $\begingroup$ I.e. provided $\prod_{i=0}^n \dfrac1{2i+1}<3$. $\endgroup$
    – Rosie F
    Jan 29, 2017 at 20:26
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Pierre de Fermat conjectured that all Fermat numbers were prime, and a similar mistaken conjecture can be made for most pseudoprimes (Catalan, Fibonacci, Euler, Wieferich, etc.). Also see Euler's sum of Powers conjecture , and the Polya conjecture.

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  • $\begingroup$ Oh, and Merten's conjecture. $\endgroup$
    – Platonix
    May 26, 2014 at 13:49
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Fermat's ‘little’ theorem states that if $n$ is prime, then $$a^n\equiv a\pmod n\tag{$\ast$}$$ holds for all $a$. The converse, which is false, states that if $(\ast)$ holds for all $a$, then $n$ is prime.

Counterexamples to this converse are uncommon; the smallest is $n=561$.

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The largest odd value of $n$ such that regular $n$-gons are constructable by compass and straight edge is $n=1~431~655~765$.

There is one known exception: $n=4~294~967~295$.

All in all, there are currently only $31$ known constructable regular polygons with an odd number of sides. Prior to Gauss, the largest odd-sided constructable regular polygon was just the pentagon. No doubt much of the numerological and mystical lore surrounding the pentagon and pentagram can be traced back to observations by the ancients that this five sided shape represented some fundamental boundary between the world of finite imperfect man on one side, and the perfect and infinite on the other.

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    $\begingroup$ I'd expect the ancients were smart enough to construct the regular $15$-gon from the triangle and pentagon. $\endgroup$ May 26, 2014 at 17:51
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    $\begingroup$ @DanielFischer You're right. Book IV, Proposition 16, of Elements in fact gives the construction. Now editing. $\endgroup$
    – David H
    May 26, 2014 at 17:56
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    $\begingroup$ Is there something missing in the first paragraphs, like "for all $n\leq N$"? The first sentence seems to say that the largest odd $n$ for which $P_n$ is true is $n=1\,431\,655\,765$. What is the exception? That $P_m$ is true for $m=4\,294\,967\,295$? If so, then what is the point of the first sentence? $\endgroup$
    – JiK
    May 27, 2014 at 13:43
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    $\begingroup$ @JiK Exactly. If the first number was $15$ or something it would make more sense, but the fact that the first number itself is so large makes the point of the first example pointless. All it's saying is that the largest known example of a constructable regular polygon with odd sides is $n = 4294967295$, and the second largest example is $n = 1431655765$.. $\endgroup$
    – MT_
    May 28, 2014 at 4:22
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Euler's Sum of Powers Conjecture:

$$a_1^k+\ldots+a_n^k=b^k\qquad\text{with }k>n>1$$

has no solutions in positive integers.

The smallest counterexample is

$$95800^4 + 217519^4 + 414560^4 = 422481^4$$

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    $\begingroup$ smallest in what sense? $\endgroup$ May 27, 2014 at 23:39
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    $\begingroup$ @BennettGardiner Smallest in lexicographic order $(k,a_1,a_2,\dots,a_n)$. $\endgroup$
    – Toscho
    May 28, 2014 at 14:09
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If $\ n\ $ that $\ ◎(n)=\frac{n+1}{2^x}$ or $\ ◎(n)=\frac{n-1}{2^x}, \ n \in \mathbb{Z^+},\ x \in \mathbb{N}_{\gt 0},\ $then $\ n\ $ is prime.
First counter-example is $\ 92673$.
$◎(n)\ $ is called the cycle length of $n$ , detail see: How to prove these two ways give the same numbers?

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  • $\begingroup$ If change 'then $\ n\ $ is prime' to 'then $\ n\ $ is prime or semiprime' ,then $\ 92673\ $ is the only known counter-example. $\endgroup$
    – miket
    May 28, 2014 at 1:33
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Lucke numbers of Euler:

For every natural number $n$, the following number is prime: $$n^2-n+41$$

The smallest $n$ for which this conjecture is wrong, is 41 -- naturally.

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