Prime Numbers

4.7 Research problems 223
composite below 341550071728321 simultaneously passes the strong probable
prime test (Algorithm 3.5.2) for the prime bases below 20.
4.35. By consideration of the Diophantine equation
n k − 4 m =1,
prove that no Fermat number can be a power n k ,k>1. That much is known.
But unresolved to this day is this: Must a Fermat number be squarefree? Show
too that no Mersenne number Mn, withn a positive integer, is a nontrivial
power.
4.36. Recall the function M(p) definedinSection4.1.3asthenumberof
multiplications needed to prove p prime by traversing the Lucas tree for p.
Prove or disprove: For all primes p, M(p) =O(lg p).
4.37. (Broadhurst). The Fibonacci series (un) as defined in Exercise 2.5
yields, for certain n, some impressive primes. Work out an efficient primalitytesting
scheme for Fibonacci numbers, perhaps using publicly available
provers.
Incidentally, according to D. Broadhurst all indices are rigorously resolved,
in regard to the primality question on un, for all n through n = 35999
inclusive (and, yes, u35999 is prime). Furthermore, u81839 is known to be prime,
yet calculations are still needed to resolve two suspected (probable) primes,
namely the un for n ∈{50833, 104911}, and therefore to resolve the primality
question through n = 104911.
4.38. Given a positive nonsquare integer n, show that there is a prime r
with 1 + lg 2 n