Problems & Puzzles:
Conjectures
Conjecture
24. Paul
Underwood's conjecture
Paul Underwood sent to these pages (22/7/2001) the following
conjecture
f=x^n-x^k-1 is prime if f divides (x^f-x)
given:
(1) x in N and x>1
(2) n in N and n>2
(3) k in N and n>k>0
(4) except x=2 and k=n-1
Question: Can you prove it or give a
counterexample?
Solution
Rudolph Knjzek sent (28/10/01) the following
comment to this conjecture:
This is no proof or counterexample, only a contribution. The given
formula for f generates a value, coprime to x, because f(mod x)=1. f divides
x^f-x is the (weak) primality test by Fermats little theorem. This
means, if f divides x^f-x then f is prime or a pseudoprime base x. Now
it's to proof, that the given formula for f only produces primes and
composites, but no pseudoprimes base x. To proof this is a hard work. I
have inspected the records of known pseudoprimes for a counterexample
but I could not find any pseudoprime of the given form. I think, this is
a heavy argument for the truth of Pauls conjecture.
***
T. D. Noe wrote (Feb. 2006):
I tested this conjecture for all x<1000 and all {n,k} such that
f<10^200.
I found no counterexamples. The requirement n>2 is necessary, otherwise it
is possible to find exceptions
***
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