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.

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- Mar 13th 2013, 04:35 PMShadowKnight8702solving 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 PMILikeSerenaRe: solving for x in 78^x =x (mod 10^12)
- Mar 13th 2013, 04:59 PMShadowKnight8702Re: 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 PMILikeSerenaRe: solving for x in 78^x =x (mod 10^12)
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 $\displaystyle x \equiv 3 \pmod{10}$, you get:

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

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

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

$\displaystyle 2 \equiv 3 \pmod{10}$

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