Theorem: Smallest n such that 6 n + 1 and 6 n - 1 are both Composite

The smallest positive integer $n$ such that $6 n + 1$ and $6 n - 1$ are both composite is $20$.

Proof:
Running through the positive integers in turn:
{{begin-eqn}}
{{eqn | l = 6 \times 1 - 1
      | r = 5
      | c = which is prime
}}
{{eqn | l = 6 \times 1 + 1
      | r = 7
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 2 - 1
      | r = 11
      | c = which is prime
}}
{{eqn | l = 6 \times 2 + 1
      | r = 13
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 3 - 1
      | r = 17
      | c = which is prime
}}
{{eqn | l = 6 \times 3 + 1
      | r = 19
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 4 - 1
      | r = 23
      | c = which is prime
}}
{{eqn | l = 6 \times 4 + 1
      | r = 25 = 5^2
      | c = and so is composite
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 5 - 1
      | r = 29
      | c = which is prime
}}
{{eqn | l = 6 \times 5 + 1
      | r = 31
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 6 - 1
      | r = 35 = 5 \times 7
      | c = and so is composite
}}
{{eqn | l = 6 \times 5 + 1
      | r = 37
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 7 - 1
      | r = 41
      | c = which is prime
}}
{{eqn | l = 6 \times 7 + 1
      | r = 43
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 8 - 1
      | r = 47
      | c = which is prime
}}
{{eqn | l = 6 \times 8 + 1
      | r = 49 = 7^2
      | c = and so is composite
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 9 - 1
      | r = 53
      | c = which is prime
}}
{{eqn | l = 6 \times 9 + 1
      | r = 55 = 5 \times 11
      | c = and so is composite
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 10 - 1
      | r = 59
      | c = which is prime
}}
{{eqn | l = 6 \times 10 + 1
      | r = 61
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 11 - 1
      | r = 65 = 5 \times 13
      | c = and so is composite
}}
{{eqn | l = 6 \times 11 + 1
      | r = 67
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 12 - 1
      | r = 71
      | c = which is prime
}}
{{eqn | l = 6 \times 12 + 1
      | r = 73
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 13 - 1
      | r = 77 = 7 \times 11
      | c = and so is composite
}}
{{eqn | l = 6 \times 13 + 1
      | r = 79
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 14 - 1
      | r = 83
      | c = which is prime
}}
{{eqn | l = 6 \times 14 + 1
      | r = 85 = 5 \times 17
      | c = and so is composite
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 15 - 1
      | r = 89
      | c = which is prime
}}
{{eqn | l = 6 \times 15 + 1
      | r = 91 = 7 \times 13
      | c = and so is composite
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 16 - 1
      | r = 95 = 5 \times 19
      | c = and so is composite
}}
{{eqn | l = 6 \times 16 + 1
      | r = 97
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 17 - 1
      | r = 101
      | c = which is prime
}}
{{eqn | l = 6 \times 17 + 1
      | r = 103
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 18 - 1
      | r = 107
      | c = which is prime
}}
{{eqn | l = 6 \times 18 + 1
      | r = 109
      | c = which is prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 19 - 1
      | r = 113
      | c = which is prime
}}
{{eqn | l = 6 \times 19 + 1
      | r = 115 = 5 \times 23
      | c = and so is composite
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 6 \times 20 - 1
      | r = 119 = 7 \times 17
      | c = and so is composite
}}
{{eqn | l = 6 \times 20 + 1
      | r = 121 = 11^2
      | c = and so is composite
}}
{{end-eqn}}
{{qed}}