Hello,
I came up with the following theorem on composite
Mersennes and wondering whether it was familliar
to anyone and if so did they know of any interesting
corollaries, relations between the terms or properties
of the terms (I could not find any).
If p is a prime and Mp=uv with v>1 and u=7 mod 8
then there exists integers x and w and integers
n, i and j such that 2^n=x^i-(2^j)xw+w, where
p=n+ij, i>1, j>1, j>n>0 and (2^j)x-1=u.
Further v=1+(2^j)x+((2^j)x)^2+...+((2^j)x)^(i-1)-(2^(ij))w,
or v=(((u+1)^i)-1)/u-(2^(ij))w.
e.g. M11=23*89, j=3, i=3, n=2, x=2, w=1:
check: u=(2^3)3-1, v=(((23+1)^3)-1)/23-2^9=89.
There is always a solution where j represents the
the number of lsb's before the first zero any divisor
congruent to 7 mod 8 (that was my original
motivation for representing the divisors in this form).
Is it yet another useless theorem?
What is the highest value for j that has been discovered?
Thanks,
----------------------------------------------------------
Daniel
e-mail: [EMAIL PROTECTED]
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