How Old Am I?

On my last birthday, my friend said to me:

"In 15 years' time, your age will be the square of your age 15 years ago!"
Can you work out how old I am?
 

This got me thinking...
 
Was there ever a time in my life when I had other birthdays that were special in this way?
 
Could I have said:
"In 3 years' time, my age will be the square of my age 3 years ago"
or:
"In 4 years' time, my age will be the square of my age 4 years ago"
or:
"In 5 years' time, my age will be the square of my age 5 years ago"
or:
"In 6 years' time, my age will be the square of my age 6 years ago"
or...?
 
Can you make any generalisations about which birthdays are special in this way?
Can you prove your findings?

Click here for a poster of this problem.