solving for x in 78^x =x (mod 10^12)

• Mar 13th 2013, 04:35 PM
solving for x in 78^x =x (mod 10^12)
Hi Guys,

Any idea how to solve the equation 78^x = x (mod 10^12) ? Assume that x is a 12 digits number
I've looked over every congruence and can't seem to find anything useful.
• Mar 13th 2013, 04:50 PM
ILikeSerena
Re: solving for x in 78^x =x (mod 10^12)
Quote:

Originally Posted by ShadowKnight8702
Hi Guys,

Any idea how to solve the equation 78^x = x (mod 10^12) ?
I've looked over every congruence and can't seem to find anything useful.

Let's see... you're looking for a number x such that the last 12 digits of 78^x is the same as the last 12 digits of x.
Perhaps we should start with just the last digit.
Can you solve 78^x = x (mod 10)?
• Mar 13th 2013, 04:59 PM
Re: solving for x in 78^x =x (mod 10^12)
Im not quite sure how that is solved. But one piece of info i forgot to mention is that x is a 12 digits number.
• Mar 13th 2013, 05:04 PM
ILikeSerena
Re: solving for x in 78^x =x (mod 10^12)
Quote:

Originally Posted by ShadowKnight8702
Im not quite sure how that is solved. But one piece of info i forgot to mention is that x is a 12 digits number.

The way to solve it, is to try each modulo class of x.
That's quite doable since there are only 10.
For instance for $x \equiv 3 \pmod{10}$, you get:

$78^3 \equiv 3 \pmod{10}$

$(78 \pmod{10})^3 \equiv 3 \pmod{10}$

$8^3 \equiv 3 \pmod{10}$

$2 \equiv 3 \pmod{10}$

This is a contradiction, so any number x that ends in 3 will not satisfy the equation.