From: Michael Bell [EMAIL PROTECTED]
Subject: Re: Mersenne: N and F
"Benny.VanHoudt" wrote:
I mentioned that this was true for all N if F is an odd factor,
it's a bit more complicated, it is only true for N even
if F is an odd factor and F is small enough.
This because we can't use negative numbers.
For example 12 = 3 + 4 + 5 (F=3, Q=4) works and
70 = 7+8+9+10+11+12+13 (F=7, Q=10),
but it fails for 14 because the only odd factor F=7 gives Q=2
and Q D.
But surely 14 = -1 + 0 + 1 + 2 + 3 + 4 + 5 (F=7, Q=2, D=3)
Why does it matter that the start of the sequence is negative?
Also: 81 =
- -10-9-8-7-6-5-4-3-2-1+0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16
(F=27, Q=3, D=13)
Indeed but,
The lemma of de la Rosa (1978) says that
a positive integer ( 1) is a prime or a power of 2 iff it cannot be expressed
as the sum of at least three consecutive positive integers.
Thus to use this observation as a proof (for one direction) we cannot
use negative numbers. Meanwhile I realised that you can still use this
properly. Because you can always eliminate the negative numbers, as the
sum of -k -(k-1) ... 0 1 2 ... k-1 k equals zero. You just have to show that
at least three numbers remain but this is a consequence of the fact that
the number in the middle (before the elimination of possible negatives) is
at least 2 (it equals N/F with F a factor N ).
The other direction of the lemma is just as easy to proof.
Benny
---
Benny Van Houdt,
University of Antwerp
Dept. Math. and Computer Science
PATS - Performance Analysis of Telecommunication
Systems Research Group
Universiteitsplein, 1
B-2610 Antwerp
Belgium
email: [EMAIL PROTECTED]
