RE: [PrimeNumbers] no primes with this forms
> you tell me two state but by PrimeForm for the firstI would be interested in a proof or disproof of this conjecture. Several
> 1000Digits you can find there are an infinite Prime f the form 2^n+n^2
> can you proof There is an infinite primes with n=3*k
years ago I started examining primes of the form x^y + y^x, of which yours
is just a special case. There are clearly an infinite number of such primes
(take x=1, y=p-1, p prime to see this) but the question of whether there are
an infinite number with x>y>1 is still open as far as I know.
It's easy to put some constraints on x and y, for instance they have to be
of opposite parity and co-prime, but I haven't yet found any way of
determining the number of primes. I believe there are an infinite number of
them but it may be, for example, that for any finite value of y there may be
only a finite number of prime values x^y+y^x with x>y>1
Two larger examples of primes of this form have appeared in Chris Caldwell's
tables. The largest of these is 364^405+405^364 with 1038 digits. I have a
table of other discoveries, but not easily available at the moment, alas.
If there's general interest I'll dig out the information.