Mind Blowing!

Firstly we need to find the general solution of $x$ and $y$ .

Let $\frac{1234}{973}$ be converted into continued fraction as follows,

$\frac{1234}{973}=1+\cfrac { 1 }{ 3+ } \cfrac { 1 }{ 1+ } \cfrac { 1 }{ 2+ } \cfrac { 1 }{ 1+ } \cfrac { 1 }{ 2+ } \cfrac { 1 }{ 11+ } \cfrac { 1 }{ 2 }$ ,

The convergent just preceding $\frac{1234}{973}$ is $\frac{591}{466}$ such that $973(591)-1234(466)=\pm{1}$ . By substitution we found that $973(591)-1234(466)=-1$ .

Now, $973(1190865)-1234(938990)=-2015$ . . . . . . . . . . . . . . .(1)
Adding the given Eq. and Eq. 1 we get, $973(x+1190865)=1234(y+938990)$ $\Rightarrow$ $\frac{x+1190865}{1234}=\frac{y+938990}{973}=t$ $\Rightarrow$ $x=1234t-55$ $\Rightarrow$ $\boxed{x_{min}=1179}$ . $y=973t-45$ $\Rightarrow$ $\boxed{y_{min}=928}$ .

$\therefore$ number of digits in $x^{y}$ = number of digits in $10^{y\log _{ 10 }{ x }}= \boxed{\boxed{\boxed{2851}}}$ .

This is not a solution.

Hint:

1)Solve linear diphontine equation.General form of solution of linear diphontine equation is $x=x'+(b/d)k$ and $y=y'-(a/d)k$ , where $x',y'$ are any solutions to equation and $a,b,d$ are coef of x ,coef of y and $gcd(a,b)$ respectively.

2) Solve inequalities $x>0$ and $y>0$ and find least integral solution in positive integers.

3)Now use base $10$ logarithms to find number of digits,In this part you will require given data by problem poser.