Disproof by counter example...

[apple2357]

Can anyone offer me examples of results in mathematics that seem to be true for many cases but then fail?
I have been searching the web and found one but its not as accessible as i would like it to be, so the simplest the result the better. I am working with my 17 year old and want to get across the idea that just because something may be true for many values it doesn't mean it is always true.
The example i have found is this one:

?^17+9 and (?+1)^17+9 are relatively prime until n= 8424432925592889329288197322308900672459420460792433

 

[yoscar04]

Some examples are given here: https://en.wikipedia.org/wiki/List_of_disproved_mathematical_ideas

 

[apple2357]

Thanks for that, some interesting stuff on that. I am after something particular which is one of those results that looks true for many test cases but fails when you go very large.

 

[JeffM]

Thanks for that, some interesting stuff on that. I am after something particular which is one of those results that looks true for many test cases but fails when you go very large.

I'm sure there are many such instances, but I doubt that many are both interesting and accessible.

"All odd numbers are prime" seems to me to fail too early for your purposes.

 

[Deleted member 4993]

Look at:

Counterexamples

If n is a whole number, then n^2+ n + 11 is a prime number. Give me counterexamples of it

www.freemathhelp.com

 

[Dr.Peterson]

Can anyone offer me examples of results in mathematics that seem to be true for many cases but then fail?
I have been searching the web and found one but its not as accessible as i would like it to be, so the simplest the result the better. I am working with my 17 year old and want to get across the idea that just because something may be true for many values it doesn't mean it is always true.

A classic example is counting the maximum number of regions into which you can divide a circle by taking n points on the circle and joining each point with every other. With 1 point, you get 1 region; with 2 points, you get 2 regions; with 3, you get 4; and it continues with 8, then 16, and you expect that in general it is 2^(n-1). But then the apparent pattern fails.

It's not a huge number of cases, but enough that you want to stop without counting more.