Examples of Perceived Patterns Proven Wrong

[erszega]

Dear All,

Could you give me examples of conjectures based on perceived patterns but proved to be wrong? Fermat numbers, with Fermat's conjecture that all Fermat numbers are primes, would be one example that I know of. I would appreciate elementary examples which are easy to understand.

The reason that I am asking for this is that I am doing some business studies, and I would like to persuade fellow students (and maybe the tutors), with examples, that it is very easy to jump to the wrong conclusions from perceived patterns.

Besides maths, any science example would also be welcome.

Regards

 

[murshid_islam]

here is one:
31, 331, 3331, 33331, 333331, 3333331, 33333331 are all prime numbers. but 333333331 (this one has eight 3's) is composite.

333333331 = 17*19607843

 

[arildno]

here is one:
31, 331, 3331, 33331, 333331, 3333331, 33333331 are all prime numbers. but 333333331 (this one has eight 3's) is composite.

Cool, haven't seen that before!
(Of course, what I found surprising with this, was that the pattern didn't break down earlier).

 

[Curious3141]

I had a little conjecture (easily proved false) when I was a schoolkid. The product of consecutive primes from 2 to any prime PLUS one was prime.

Pattern seemed true for :

1) 2 + 1 = 3
2) 2*3 + 1 = 7
3) 2*3*5+1 = 31
4) 2*3*5*7+1 = 211
5) 2*3*5*7*11+1 = 2311

but broke down for

2*3*5*7*11*13+1 = 30031 = 59*509

Higher order terms broke the pattern too (are the remainder all composite? That would be equally fascinating if true).

Oh well, it was fun for the day or so of excitement it afforded my young mind!

(BTW, the similar sequence for product of primes MINUS one breaks down much earlier).

 

[CRGreathouse]

Curious3141, Sloane's http://www.research.att.com/~njas/sequences/A018239 = {2, 3, 5, 7, 11, 31, 379, 1019, ...} is the list of primes such that the product of that prime and all lower primes, plus one, is itself prime.

 

[Curious3141]

Curious3141, Sloane's http://www.research.att.com/~njas/sequences/A018239 = {2, 3, 5, 7, 11, 31, 379, 1019, ...} is the list of primes such that the product of that prime and all lower primes, plus one, is itself prime.

Thanks for that. I never followed up on the sequence properly.