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You are right. I missed one line.
Thanks.
2^(2^2 -1) -1 = 7
2^(2^(2^ -1) -1) -1 = 127
----- Original Message -----
Sent: Sunday, July 29, 2001 9:52 AM
Subject: Re: Mersenne: Proof for P =
2^170141183460469231731687303715884105727 - 1 is Prime Number
Hi, Leo!
The line 2^(2^2 -1)
-1 = 2^3 - 1 = 127 does not seem correct. 2^3=8. Hence it should read
2^(2^2 -1) -1 = 2^3 - 1 = 7.
Request advise.
Best
regards,
D.N.Loganey
--------------------------------------------
At
07:14 AM 7/29/2001 +0800, you wrote:
Dear
All,
Below is my humble attempt
to find the biggest prime number without the luxury of extensive computer
power. I hope you will find this interesting and hopefully some of you
can check its validity.
Regards,
Leo de
Velez
26B Prudent Lane,
Sanville Subdivision,
Quezon City, Philippines
+63 917
532 9297
Biggest Prime Number
2^2 -1 = 3
is a prime number
with all binary digits equal to 1
(total of 2 binary
digits)
2^(2^2 -1) -1 = 2^3 - 1 = 127
is a prime number
with all binary digits equal to 1
(total of 3 binary
digits)
2^(2^(2^2 -1) -1) -1 =
170141183460469231731687303715884105727
is a prime number with all binary
digits equal to 1
(total of 127 binary digits)
So it follows
that
2^170141183460469231731687303715884105727 - 1
IS ALSO A PRIME
NUMBER WITH ALL BINARY DIGITS EQUAL TO 1
(TOTAL OF
170141183460469231731687303715884105727 DIGITS)
And so
on.
PROOF
If q is any prime
number,
then 2^(q-1) mod q = 1
and then 2^(q-1) -1 = a * q,
where a is an integer less than 2^(q-1)
This means that any prime
number q is a factor of N = 2^(q-1) -1
or
q is a factor of a number N
with (q-1) binary digits all equal to 1
This number N has EVEN number
of binary digits all equal to 1
P =
2^170141183460469231731687303715884105727 - 1
So P is a number
with a
prime number (170141183460469231731687303715884105727) of binary
digits
all equal to 1.
For each prime number q less than
170141183460469231731687303715884105727,
q is a factor of a number N =
2^(q-1) with an EVEN number of binary digits
all equal to
1.
Therefore, from binary division,
Prime Number of Binary
Digits All Equal to 1
DIVIDED BY
Even Number of Binary Digits All
Equal to 1
HAS A REMAINDER
SO
any prime numbers q less than
170141183460469231731687303715884105727
is NOT a factor of P =
2^170141183460469231731687303715884105727 - 1
It also follows
from binary division that
For ALL numbers k less than P with binary
digit all equal to 1,
k is NOT a factor of P
Just to remove
all EVEN numbers,
ALL even numbers E less than P is not a factor of
P.
NOW, THE FINAL ELIMINATION
For any prime number q
greater than 170141183460469231731687303715884105727,
the least value of
product N = a * q
where N has a binary digits
all equal to 1 and N = 2^(q-1) - 1
N is greater than
2^170141183460469231731687303715884105727 -1
Therefore,
All q >
170141183460469231731687303715884105727
Is NOT a factor of P =
2^170141183460469231731687303715884105727 - 1
AND
THEREFORE,
P = 2^170141183460469231731687303715884105727 -
1
IS A PRIME NUMBER !!!!
Regards,
Leo de Velez
26B Prudent
Lane,
Sanville Subdivision,
Quezon City, Philippines
+63 917 532
9297
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