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Science Forum Index » Mathematics Forum » Mersenne numbers, Mp163
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| Author |
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| Don McDonald |
 Posted: Sun Dec 28, 2003 9:17 pm |
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myfile.> DON02. Calc.Factors.FermatMers.Mersenne.CARMP163.SPMP163
subject:Mersenne numbers, Mp163
Quote: D.Calc.Factors.FermatMers.Mersenne.CARMP163.SPMP163
We look at the usual long multiplication of, for
example, 123*48.
123
x 48
-------
984
+492
-------
5904.
However, I wish to concentrate just on the tens and
units figures of the resulting product. This is called
'multiplying modulo 100.'
That is, any whole number of 100s can be thrown away in
the process.
Then, (100+23)*48 == 48*100+23*48 (is congruent to)
== (20+3)*(40+ == 2*4*100+3*40+8*20+3*8
== 120+160+24
== 104 == 04 modulo 100.
In calculating the remainder of product (integers w*z)
after dividing by positive integer, t, it can often be
made easier by taking away convenient multiples of the
modulus, t, from w or z respectively before executing
the multiplication.
Let w = at+b, say, and z = ct+d, say.
Then, wz = (at+b)(ct+d)
= at(ct+d) +b(ct+d)
= t(act +ad +bc) +bd
== bd modulo t.
Bd should usually be less than w*z, but it may require
to be reduced further.
I photographed car number plate 'MP163' at Hanson St,
Newtown, Wellington on today, 26-5-2002. This number
plate could be shorthand for Mersenne numbers,
2^prime-1.
(Figure 2 raised to a prime index, subtract 1.)
The world's largest known prime number in 2001 often is
2^13.4million-1. (Greater than 13 million 2s
multiplied together.) In December 2003 is 2^20996011-1.
(Dominion Post, NZ Herald, GIMPS great
internet mersenne prime search.)
By the way, the 40th known Mersenne prime exponent is
a twin prime, 10x(23x63)^2 +/-1. And a previous record
holder was virtually a palindrome, Table Mountain, increasing
decreasing.
Exponent (1*2*34*44432+1)= 3021377. [sic.]
_____
/ \
A theorem of mathematics number theory says if
(a is a positive integer )..
a^prime-1 has factor/s they must be of the form
2*k*p+1.
However, the Penguin Dictionary of Curious and
Interesting Numbers (1997), entry 28, table of perfect
numbers, does not give M_163 as a Mersenne prime.
Therefore, I have written myprogram 'powabc1' in BBC
interpreted Basic64, which tests for just such factors.
(On Acorn A5000 computer, UK 1990, RISC OS-reduced instruction
set computer, 4 Megabytes RAM randomaccess memory.)
In this case, several programs found 150287 and 704161,
etc. divide M_163.
Or 2^163 == 1 modulo 150287.
Have I found a factor of M_163?
The following, I believe, shows that I have.
Check. 2^163= 2^3*(2^20)^8 (is congruent to)
== 8*1048576^8
2^20==1048576-6*150287
== 146854 mod 150287
== -3433.
Squaring 2^40= 11785489
== 63103
Squaring 2^80==3,981,988,609
==134544== -15743
Squaring 2^160==247,842,049
==18786
2^163==8*18786
==150288==1 mod 150287.
Q.e.d.
By the way, my methods found (the least prime) factor
1580,187,223 of (10^9)^(10^9)+3. (sci.math sensation 2001.)
This is, digit 1 followed by 9 billion zeroes. Surpassed
only by Graham's number. I suspect gigaplex =
10^billion. (There is a typo error in Penguin Dictionary
of Curious and Interesting Numbers, 1997.)
yours sincerely,
/ Donald S. McDonald (Wellington, New Zealand) |
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