The French mathematician Pierre de Fermat (1601-1665) considered the sequence 5,

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.

Explanation / Answer

I think the given conjecture may be proved by principle of Induction. Assume it is true for n=m, 2^(2^m)+1 is a prime no.(say p)
We have to prove it is true for n=m+1
2^(2^(m+1))+1 = 2^(2.2^m)+1=2^(2.log(p-1)/log2)+1 = (p-1)^2 + 1
which is prime

Hence can be said that it holds for all natural no.s

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