primes = Prime[Range[10^7]];
gaps = {Differences[primes], Most[primes]} // Transpose;
tmp = GatherBy[gaps, First][[All, 1]];
tmp = SortBy[tmp, First];
starts = Association[Rule @@@ tmp];
set = {Most[tmp[[All, 1]]], Abs@Differences[tmp[[All, 2]]]} // Transpose;
data = Table[{n, k} = SelectFirst[set, Last/*GreaterThan[10^i]];
   {10^i, n, starts[n], n + 2, starts[n + 2], k}, {i, 1, 7}];
StringTemplate["Earliest difference > `` between adjacent prime gap starting primes:
Gap `` starts at ``, gap `` starts at ``, difference is ``."]/*Print @@@ data;