Prime of my Life

05AB1E, 18 bytes

2101GтLDpÏN+DpÏ¥Q–

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Explanation

2101G               # for each N in [1 ... 2100]
     тL             # push the range [1 ... 100]
       DpÏ          # keep only numbers that are prime
          N+        # add N to each
            DpÏ     # keep only numbers that are prime
               ¥    # calculate delta's
                Q–  # print N if the resulting list is equal to input

Jelly, 19 bytes

97ÆR+⁸ÆPÐfI⁼
⁽¥cçÐf

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How it works

⁽¥cçÐf        Main link. Argument: A (array of integers)

⁽¥c           Yield 2100.
   çÐf        Apply the helper link to each n in [1, ..., 2100], with right
              argument A. Yield all values of n that return 1.


97ÆR+⁸ÆPÐfI⁼  Helper link. Left argument: n. Right argument: A

97ÆR          Prime range; yield all prime numbers in [1, ..., 97].
    +⁸        Add n to each of these primes.
      ÆPDf    Filter by primality, keeping only prime sums.
          I   Increments; compute all forward differences.
           ⁼  Return 1 if the result and A are equal, 0 if not.