My problem is solved: I had a problem with a certificate being deactivated. I activated it and I can now connect successfully. I'm sorry that I don't know which cert it was, I had many that were turned off and I turned them on in groups. And these things have a language all their own!

I looked for the cert in my phones listings and didn't see it!
For the samsung S4 it was found:
phone settings
security
trusted credentials
system ... gives you a list of certs your have on your phone.
Likely that most android phones would be similar.
At the risk of being embarrassed I'll tell you how this happened to me.
Half year ago I was in an airport, late at night and I was going to sleep there. So I signed on to free wifi. After that, my phone started acting odd ... so I freaked out. I added AVG security, etc. Somewhere along the line I looked at the security settings, found these certificates and the names are ... well they frightened me (thought I was hacked). For instance, there are numerous from other countries, including China!
At that point I turned off any cert that didn't look right to me. Poor decision as apparently we need certs from "america online" .... etc !
Let me summarize for anyone who finds this (and likely is having a connection problem to the lds server).
If you can log into lds.org without your browser warning you, it's likely that you don't have the same problem I had.
If you do get this warning you have messed with your certs, been pranked, handed your phone to one to many toddlers, etc

Lastly, I'm no expert on this stuff.