The offset part of the LIMIT clause refers to the Count of records found, not the value of the primary key. Given that entries in the key sequence may be missing, the records with 100 as it's primary key may only be the 20th record in the table while the 100th record has an id of 257, or whatever.
to achieve what I think you mean you need
select * from table_name WHERE id>=$primary_key ORDER BY id LIMIT 10;
I've thrown the order by in there to bullet proof it: it should default to that anyway but you never can tell.