Unsolved Problem 36: prove nn+1 is not prime when n > 4

It seemed that this problem was solved, as the following flawed "proof" (from www.spiritual-coder.com/primepowers/) was offered.  Though it narrows down the possible values of n considerably, it doesn't prove every case.  In particular, it is not known whether there are more than five Fermat primes.

Theorem

nn+1 is never a prime number if n > 4.

Preliminary remark

All variables used below are positive integers.
n is always assumed to be greater than 4 unless indicated otherwise.

Proof

Case I: n is not a power of 2

If n is not a power of 2, it can always be written as (2m+1)p, where m and p are positive integers.

Then nn+1 = (((2m+1)p)p)(2m+1)+1 

which is a multiple of ((2m+1)p)p+1 

because of this factorization: x(2n+1)+1 = (x+1)(x2n - x2n-1 + ... + x2 - x + 1)  

Case II: n is a power of 2

Then nn+1 = (2p)(2^p)+1 = 2(p 2^p)+1 

Case IIa: p is not a power of 2

If p is not a power of 2, it can always be written as (2k+1)t, where k and t are positive integers.  Then, as in Case I, above, there is one plus a number raised to an odd power, which can be factorized:

nn+1 = (2(t2^p))(2k+1)+1

which is a multiple of 2(t2^p)+1

Case IIb: p is a power of 2

p is a power of 2, then p=2q.

Rewriting nn+1=(2p)(2^p)+1 results in the expression 2(2^(q+2^q))+1

Putting r=q+2q we recognize the expression as the Fermat number Fr.

For q=0 we have F1 = 5 and n=2, thus nn+1=5 which is prime.

For q=1 we have F3 = 257 and n=4, thus nn+1=257 which is prime.

For q > 1 we have Fermat numbers Fr with r = 6, 11, 20, 37, ... which all have a divisor of the form 2(r+1)L+1 with L integer (proven by Lucas in 1878 - see Mathworld: Fermat Number).

Alas, it is not known whether 2(r+1)L+1 is the only factor of some Fermat number thus making it prime.

Originally, the proof concluded that also in the case IIb nn+1=2(2^(q+2^q))+1 is not a prime for q > 1.

Final conclusion:

It has been shown in all but a small (but infinite) number of cases that nn+1 for n > 4 is never a prime.

The original "proof", upon which this page is based, is copyright © Bernard Jacobs – 16 March 2004, and used here with attribution under the "fair use" exception to copyright law.

Internet References

The "proof" (complete, except for case IIb) on this page was first published on www.spiritual-coder.com/primepowers/

Mathworld: Fermat Number.

Related pages in this website

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