Gilbreath's Conjecture

Mathematica, 66 bytes

(For[z=1,Last@Nest[Abs@*Differences,Array[Prime,z+#],#]<3,z++];z)&

Pure function taking a positive integer as argument and returning a 1-indexed integer. Nest[Abs@*Differences,Array[Prime,z+#],#] computes the #th iterated absolute difference list of the list of the first z+# primes. For[z=1,Last@...<3,z++] loops this computation until the last element of the resulting list is at least 3, and then z is output. (Note that the correctness of the algorithm assumes Gilbreath's conjecture!)

Pyth, 32 bytes

-l.W>3h.WtHaVZtZ>QH+ZfP_TheZ.fP_

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Uses 2-indexing.

MATL, 18 bytes

`@:YqG:"d|]3<A}@G-

Input and output are 1-based. It takes less than 40 seconds in TIO for each of the test cases.

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Explanation

This keeps trying longer initial sequences of primes until the iterated absolute consecutive differences give at least one value exceeding 2.

`        % Do... while loop
  @:Yq   %   Array of first k primes, where k is iteration index
  G:"    %   Do this as many times as the input
    d|   %     Absolute value of consecutive differences
  ]      %   End
  3<A    %   Are they all less than 3? This is the loop condition
}        % Finally (execute before exiting loop)
  @G-    %   Push last iteration index minus input. This is the output
         % End (implicit). Continue with next iteration if top of stack is true
         % Display (implicit)