# Perceived patterns

1. Sep 17, 2006

### 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

2. Sep 17, 2006

### 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

Last edited: Sep 17, 2006
3. Sep 17, 2006

### arildno

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

4. Sep 17, 2006

### 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).

Last edited: Sep 17, 2006
5. Sep 17, 2006

### CRGreathouse

Curious3141, Sloane's A018239 shows that the remainder are not all composite. The next two primes are 200 560 490 131 and 1 719 620 105 458 406 433 483 340 568 317 543 019 584 575 635 895 742 560 438 771 105 058 321 655 238 562 613 083 979 651 479 555 788 009 994 557 822 024 565 226 932 906 295 208 262 756 822 275 663 694 111. A005234 = {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.

Last edited: Sep 17, 2006
6. Sep 23, 2006

### Curious3141

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