How many miles would you have to ride on a bike, to make it equivalent to running mile?

I am not quite sure but about 3 miles for every 1. This assumes you aren't like a professional in either of these sports. It really depends on how fast you bike/run.

At first this struck me as similar to asking a pound of iron weighs more than a pound of feathers. A mile is a mile whether you run it or pedal it.
However, I suspect what you really want to know how to equate the same physical effects from doing 2 different activities.
There are several opinions floating around but generally most people think a ratio of 3 or 4 bike miles to 1 running mile is in the ball park.
The question has a highly personal content as the result from one to another depends on conditioning and basically no physical impairments which might limit one of the exercises.
A way to determine your personal ratio would be to time a mile run at a consistent heart rate (after warming up). Then bike at the same heart rate for the same time and compare how far you have traveled. You would want to to do this exercise on relatively flat terrain, as maintaining a consistent heart rate ascending/descending hills would skew the results. Optimally, using a track would be best.

Good question.
I can just about run 1 mile ( but no more )
But can cycle 120 miles in one day. ( Including many hills )
So to say its the same as 4 miles seems wrong to me as there is no way I could ever run 30 miles !

It will be very different for every person and will also vary widely based on type of bike, etc.

The best way to find the answer for you is to get a heart rate monitor (available at Best Buy or all over the internet for $30 and up - don't need anything fancy). Wear it while running a mile at a comfortable pace/exertion level and pay attention to your heart rate. Then ride your bike at the same heart rate for the same time. Actually, the HRM's instructions may have better ways to find your answer.

By your question it sounds as if you are interested in running/cycling as a workout. Here is a short article about HRM's that drastically changed the way I work out and I am making progress I never thought possible: http://www.duathlon.com/articles/1460