49 questions linked to/from Examples of patterns that eventually fail
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### Statements with rare counter-examples [duplicate]

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 ...
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### Best basic algebra examples to show students that proof by example is not sufficient [duplicate]

Often, students will try to 'prove' a propositon by checking some examples and 'concluding' that it will be true for all $n \in N$. I'm looking for some good, non-trivial examples from highschool ...
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### The largest number to break a conjecture [duplicate]

There are several conjectures in Mathematics that seem to be true but have not been proved. Of course, as computing power increased, folks have expanded their search for counterexamples ever and ever ...
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### Are there theorems in number theory which are true for a large interval, but are known to be false for an arbitrarily large number outside that range? [duplicate]

As an example, it is certainly trivial to verify that Fermat's Last Theorem is true for all numbers up to 10^10 using a reasonably powerful computer. We now have mathematical proof of that theorem for ...
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### Conjectures that have been disproved with extremely large counterexamples?

I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture. I'm sure that everyone here is familiar with it; it describes an operation on a ...
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### Has lack of mathematical rigour killed anybody before?

One of my friends was asking me about tertiary level mathematics as opposed to high school mathematics, and naturally the topic of rigour came up. To provide him with a brief glimpse as to the ...
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### Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
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### What is an example of a sequence which “thins out” and is finite?

When I talk about my research with non-mathematicians who are, however, interested in what I do, I always start by asking them basic questions about the primes. Usually, they start getting reeled in ...
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### Accidents of small $n$

In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that ...
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### Percentage of primes among the natural numbers

How high is the percentage of primes in $\mathbb{N}$? ($\mathbb{N} := \lbrace { 1, 2, 3, \ldots \rbrace }$ ; a prime is only divisible by itself and 1 in $\mathbb{N}$) The percentage has to be lower ...
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### Examples of simple but highly unintuitive results? [closed]

QUESTION: What are some simple math problems whose answers are highly unintuitive, and what makes them so? There are plenty of unintuitive and frankly baffling results in math, like the Banach-Tarski ...
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### Two curious “identities” on $x^x$,$e$,and $\pi$

A numerical calculation on Mathematica shows that $$I_1=\int_0^1 x^x(1-x)^{1-x}\sin\pi x\,\mathrm dx\approx0.355822$$ and $$I_2=\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x\,\mathrm dx\approx1.15573$$ A ...
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### A non-mathematician’s (programmer’s) question on infinity?

I apologize for my total ignorance in the sphere of mathematics and the possibly very silly question I'm about to ask. My mathematical knowledge level is quite limited (pretty much finished with some ...
Here is another dice roll question. The rules You start with $n$ dice, and roll all of them. You select one or more dice and fix them, i.e. their value will not change any more. You re-roll the other ...
### If a property in $\mathbb{N}$ is true up to $10^{47}$ are there reasons to think it is probably true in all $\mathbb{N}$?
You have probably heard at some point statements like that the twin prime conjecture (namely that $2$ is an infinitely ocurring prime gap) is "probably" or "almost certainly" true. Same goes for a ...