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Old 2011-05-19, 14:23   #4
science_man_88
 
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"Forget I exist"
Jul 2009
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Quote:
Originally Posted by allasc View Post
Post of Russia
http://dxdy.ru/post447534.html#p447534

sequence in the OEIS
A190213

Numbers n such that a==0(mod k) and b==0(mod k), where k=2^n-1, m=(2^n-1)*(n-1)-n+2, x=m*(2^n-1), 2^(x-1)==(a+1)(mod x), m^(x-1)==(b+1)(mod x)

1, 3, 4, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217

EXAMPLE
n=3
k=2^3-1=7
m=(2^3-1)*(3-1)-3+2=13
x=m*(2^n-1)=13*7=91
2^(x-1)==(a+1)(mod x);2^90==(63+1)(mod 91), a=63
m^(x-1)==(b+1)(mod x);13^90==(77+1)(mod 91), b=77

test conditions
a==0(mod k), 63==0(mod 7)
b==0(mod k), 77==0(mod 7)
----------------------------------------

All odd numbers are of Mersenne exponents: primes n such that 2^n - 1 is prime A000043

4 is the only even number
why not 2? I do not know .....
Quote:
Proper application here
A190213

Numbers n such that a == 0 (mod k) and b == 0 (mod k),
where
k = 2 ^ n-1
m = (2 ^ n-1) * (n-1)-n +2,
x = m * (2 ^ n-1)
2 ^ (x-1) == (a +1) (mod x),
m ^ (x-1) == (b +1) (mod x)

1, 3, 4, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217

example
-------------------------------------------------- ---------------------------------
n = 3
k = 2 ^ 3-1 = 7
m = (2 ^ 3-1) * (3-1) -3 +2 = 13
x = m * (2 ^ n-1) = 13 * 7 = 91
2 ^ (x-1) == (a +1) (mod x); 2 ^ 90 == (63 +1) (mod 91), a = 63
m ^ (x-1) == (b +1) (mod x); 13 ^ 90 == (77 +1) (mod 91), b = 77

check the basic condition
a == 0 (mod k), 63 == 0 (mod 7)
b == 0 (mod k), 77 == 0 (mod 7)
-------------------------------------------------- ---------------------------------

all odd numbers are exponents Mersenne numbers A000043
test was only able to 3217, on the processor boils
but for me it is an achievement:) to find a test that does not make mistakes up to (2 ^ 3217-1)

4 - the only even number
interesting question is Why 4 instead of 2?? I don `t know ....

ready to hear any comments, maybe I invented the bicycle?
is what google turns it into it's basically the same for the math part though.
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