How can I find the exponent $n$ efficiently?
$begingroup$
Denote $$z=(2^{19}-1)cdot10^6+2^{18}-1$$ $$a=ord_2(z)$$ $$b=ord_{10}(z)$$
The object is to find a positive integer of the form $$n=ka+19$$ with positive integer $k$ such that $$m=f(n)=lceil(n-1)cdot log_2(10)rceil$$ is of the form $$m=lb+6$$
Motivation : An "ec-number" has the form $$ec(n)=(2^n-1)cdot 10^m+2^{n-1}-1$$ where $m$ is the number of decimal digits of $2^{n-1}$. I want to find an exponent $n>19$ , such that $$ec(19)mid ec(n)$$ If it helps, $z=ec(19)$ is a prime number.
number-theory elementary-number-theory prime-numbers
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
$endgroup$
add a comment |
$begingroup$
Denote $$z=(2^{19}-1)cdot10^6+2^{18}-1$$ $$a=ord_2(z)$$ $$b=ord_{10}(z)$$
The object is to find a positive integer of the form $$n=ka+19$$ with positive integer $k$ such that $$m=f(n)=lceil(n-1)cdot log_2(10)rceil$$ is of the form $$m=lb+6$$
Motivation : An "ec-number" has the form $$ec(n)=(2^n-1)cdot 10^m+2^{n-1}-1$$ where $m$ is the number of decimal digits of $2^{n-1}$. I want to find an exponent $n>19$ , such that $$ec(19)mid ec(n)$$ If it helps, $z=ec(19)$ is a prime number.
number-theory elementary-number-theory prime-numbers
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
$endgroup$
$begingroup$
math.stackexchange.com/questions/2635516/…
$endgroup$
– Peter
Dec 8 '18 at 12:06
add a comment |
$begingroup$
Denote $$z=(2^{19}-1)cdot10^6+2^{18}-1$$ $$a=ord_2(z)$$ $$b=ord_{10}(z)$$
The object is to find a positive integer of the form $$n=ka+19$$ with positive integer $k$ such that $$m=f(n)=lceil(n-1)cdot log_2(10)rceil$$ is of the form $$m=lb+6$$
Motivation : An "ec-number" has the form $$ec(n)=(2^n-1)cdot 10^m+2^{n-1}-1$$ where $m$ is the number of decimal digits of $2^{n-1}$. I want to find an exponent $n>19$ , such that $$ec(19)mid ec(n)$$ If it helps, $z=ec(19)$ is a prime number.
number-theory elementary-number-theory prime-numbers
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
$endgroup$
Denote $$z=(2^{19}-1)cdot10^6+2^{18}-1$$ $$a=ord_2(z)$$ $$b=ord_{10}(z)$$
The object is to find a positive integer of the form $$n=ka+19$$ with positive integer $k$ such that $$m=f(n)=lceil(n-1)cdot log_2(10)rceil$$ is of the form $$m=lb+6$$
Motivation : An "ec-number" has the form $$ec(n)=(2^n-1)cdot 10^m+2^{n-1}-1$$ where $m$ is the number of decimal digits of $2^{n-1}$. I want to find an exponent $n>19$ , such that $$ec(19)mid ec(n)$$ If it helps, $z=ec(19)$ is a prime number.
number-theory elementary-number-theory prime-numbers
number-theory elementary-number-theory prime-numbers
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
asked Dec 8 '18 at 12:01
PeterPeter
46.9k1039125
46.9k1039125
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math.stackexchange.com/questions/2635516/…
$endgroup$
– Peter
Dec 8 '18 at 12:06
add a comment |
$begingroup$
math.stackexchange.com/questions/2635516/…
$endgroup$
– Peter
Dec 8 '18 at 12:06
$begingroup$
math.stackexchange.com/questions/2635516/…
$endgroup$
– Peter
Dec 8 '18 at 12:06
$begingroup$
math.stackexchange.com/questions/2635516/…
$endgroup$
– Peter
Dec 8 '18 at 12:06
add a comment |
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math.stackexchange.com/questions/2635516/…
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– Peter
Dec 8 '18 at 12:06