Linked Questions

56 questions linked to/from Examples of patterns that eventually fail
26 votes
8 answers
4k views

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 ...
• 1,783
1 vote
4 answers
289 views

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 ...
• 111
3 votes
1 answer
612 views

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 ...
• 35.6k
0 votes
1 answer
242 views

Examples of conjectures that were widely believed to be true but turned out to be false [duplicate]

Note: This is the same question, but it does not have enough answers and it is almost a year old. This question is similar, with many answers, but a conjecture is similar to, but not the same as a ...
0 votes
0 answers
77 views

Theorems only valid for huge numbers [duplicate]

I am trying to find a theorem that is valid only for very large numbers. Example: There are numbers which have more than 100 distinct factors. Above theorem satisfies this condition, but it is a ...
• 131
1 vote
0 answers
55 views

What are some examples where a sequence follows a regular pattern out to very large numbers, but not forever? [duplicate]

This question is inspired by this old paper by Euler: http://eulerarchive.maa.org/docs/translations/E175en.pdf. In it, Euler considers a particular sequence, and finds that the next number in the ...
• 253
279 votes
20 answers
57k views

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 ...
• 14.4k
155 votes
15 answers
15k views

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 ...
• 10.3k
61 votes
16 answers
14k views

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 ...
• 1,037
81 votes
13 answers
8k views

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 ...
• 1,640
32 votes
20 answers
2k views

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 ...
• 940
44 votes
6 answers
18k views

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 ...
• 9,791
24 votes
7 answers
3k views

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 ...
79 votes
3 answers
2k views

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 ...
• 2,951
20 votes
6 answers
1k views

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 ...

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