Through DNA, I've traced my g.grandfather's ancestry to a Jewish man born Isaac Gomes da Costa in London in 1717, his father, Abraham, born in Portugal in 1681 (which fits family history). By researching trees of half a dozen 2nd, 3rd and 4th cousins who I share DNA with and with whom I've been in touch, I know each of these people is descended from a different child of Isaac. However, I've not been able to find out which of Isaac's children is an ancestor of my g.grandfather. I had been looking at men born about 25 years before my g.grandfather, but it seems his mother was only 16 when he was born, so now I'm going to look at men aged 16-18 to see if I can find out anything. I did have a DNA match on Ancestry with someone who was apparently descended from my g.grandfather's half sister, same mother but different father but before I could contact this person, she removed her DNA listing. Interestingly, the mother married a man with the same surname my g.grandfather used all his life, but he couldn't have been the father if he was only a half sibling who shared a mother. He isn't on census records with this married couple, although there is a child with his name with them but he is younger than my g.grandfather, I have the birth certificate of younger person, so they didn't just put a younger age on various census to cover up the fact that he was born before his mother married.
Unless I get a DNA match with another half sibling who shares a father with him, I am never going to find out who his father was. I live in hope.