Question

The French mathematician Pierre de Fermat (1601-1665) considered the sequence 5, 17, 257, 65537, ... , 2^(2^n) + 1, ... He observed that the first four terms of the sequence (here given) corresponding to n = 1, 2, 3, 4, are primes. (A prime number is a natural number other than one divisible only by itself and one. A natural number is divisible by another natural number if the ratio of numbers is itself a natural number.) He conjectured that the following terms were also primes and, being confident his induction was correct, challenged the prominent English mathematicians of the day to prove his conjecture. (Fermat was unable to do so himself.) Discuss the validity of the conjecture and the efforts placed in proving or disproving this conjecture.

Answer 1

\[2^{2^{5}}+1\]
Sadly, Fermat's time period did not allow such a calculation.

Answer 2

\(2^{2^6}+1\) is also not prime.

Answer 3

So far only the first 4 terms that Fermat calculated are prime.

Answer 4

@KingGeorge, I kept remembering there was some really fatal irony here...

Now I know what that fatal irony was.

Answer 5

less than 100 years later, Euler found a counter-example. (so not much time spent on this problem as compared to his "Last Theorem"
According to
http://www.sciencenews.org/view/generic/id/3615/title/Math_Trek__Cracking_Fermat_Numbers
people have spent their time looking for divisors of these numbers.

Answer 6

However, there are still many interesting properties to these numbers. Such as if \(m\neq n\) are two positive integers, then \[\large 2^{2^m}+1\]and \[\large 2^{2^n}+1\]share no common divisors.

Answer 7

Alrighty, Thanks guys :))))))) All use all this info to create my discussion post Thanks :)))))))

Answer 8

You're welcome, Sam! Goodluck.

Answer 9

Y cant i give all of u guys medals. NO FAIR :(

Answer 10

Here's a solution: Give me a medal. I'll give KG a medal. KG will give Phi a medal. WE ALL WIN!

Answer 11

hahah i was thinking abt that lol

Answer 12

(I'm being serious.)

Answer 13

Any objections, @Phi and @KingGeorge ?

Answer 14

kinnnnnggggg medal limitless lol

Answer 15

I've got no objections

Answer 16

and limitless medal phi

Answer 17

Works

Answer 18

oh wait, what's the plan?

Answer 19

and then we r set to go :)

Answer 20

KG, medal me. Then I'll medal Phi.

Answer 21

SUCCESS

Answer 22

Chrome lion approves.

Answer 23

YAY it worked :D