I was born in 1956 in northern ireland,my mother had me before she married.I have asked her on a number of occasions my fathers name and she keeps telling me that she can't remember,I have been trying for years to find him,all I have is a picture of him.

Do you have your birth certificate? It might also be worth asking your mother's friends and relatives who were around at the time. One of them may remember the name of the man she was with.

I find it hard to believe she has a photograph (which presumably means it wasn't a drunken one-night stand with no way to contact him again) but can't remember his name. It's more likely that she does not want to tell you about him for some reason.

Northern Ireland, even the cities like Belfast, is a small community. Chances are someone will remember him.