"Forget I exist"
Jul 2009
Dumbassville
20C0_{16} Posts

Quote:
Originally Posted by allasc
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^n1, m=(2^n1)*(n1)n+2, x=m*(2^n1), 2^(x1)==(a+1)(mod x), m^(x1)==(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=2^31=7
m=(2^31)*(31)3+2=13
x=m*(2^n1)=13*7=91
2^(x1)==(a+1)(mod x);2^90==(63+1)(mod 91), a=63
m^(x1)==(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 ^ n1
m = (2 ^ n1) * (n1)n +2,
x = m * (2 ^ n1)
2 ^ (x1) == (a +1) (mod x),
m ^ (x1) == (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 ^ 31 = 7
m = (2 ^ 31) * (31) 3 +2 = 13
x = m * (2 ^ n1) = 13 * 7 = 91
2 ^ (x1) == (a +1) (mod x); 2 ^ 90 == (63 +1) (mod 91), a = 63
m ^ (x1) == (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 ^ 32171)
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.
