Find prime gaps

Jelly, 10, 9, 8 10 bytes

Æn_$:ð1#»2

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Two bytes saved thanks to @Dennis! (and then added back again due to edge-cases)

Explanation:

Æn          #   The next prime after 'P'
  _$        #   Minus 'P'
    :       #   Divided by 'N'
            #
            # This will give a falsy value unless the distance to the next prime is >= N
            #
     ð      # Treat all of that as a single dyad (fucntion with two arguments). 
            # We'll call it D(P, N)
            #
      1#    # Find the first 'P' where D(P, input()) is truthy
        »2  # Return the maximum of that result and 2

Mathematica, 30 bytes

2//.x_ /;NextPrime@x-x<#:>x+1&

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Mathematica, 35 bytes

(t=2;While[NextPrime@t-t<#,t++];t)&

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Mathematica, 77 bytes

Prime@Min@Position[s=Differences@Prime@Range[(r=#)^3+1],#&@@Select[s,#>=r&]]&

Haskell, 106 102 93 77 73 72 bytes

This generates an infinite list of primes first, then looks for the prime gaps. The prime list was taken from here. It can probably be shortened, but I haven't figured out how yet:)

Thanks to @BruceForte for -4 bytes and @Zgrab for -1 byte!

f n=[x|(y,x)<-zip=<<tail$[n|n<-[2..],all((>0).rem n)[2..n-1]],y-x>=n]!!0

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