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.