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

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Theorem

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:

\(\ds 6 \times 1 - 1\) \(=\) \(\ds 5\) which is prime
\(\ds 6 \times 1 + 1\) \(=\) \(\ds 7\) which is prime


\(\ds 6 \times 2 - 1\) \(=\) \(\ds 11\) which is prime
\(\ds 6 \times 2 + 1\) \(=\) \(\ds 13\) which is prime


\(\ds 6 \times 3 - 1\) \(=\) \(\ds 17\) which is prime
\(\ds 6 \times 3 + 1\) \(=\) \(\ds 19\) which is prime


\(\ds 6 \times 4 - 1\) \(=\) \(\ds 23\) which is prime
\(\ds 6 \times 4 + 1\) \(=\) \(\ds 25 = 5^2\) and so is composite


\(\ds 6 \times 5 - 1\) \(=\) \(\ds 29\) which is prime
\(\ds 6 \times 5 + 1\) \(=\) \(\ds 31\) which is prime


\(\ds 6 \times 6 - 1\) \(=\) \(\ds 35 = 5 \times 7\) and so is composite
\(\ds 6 \times 5 + 1\) \(=\) \(\ds 37\) which is prime


\(\ds 6 \times 7 - 1\) \(=\) \(\ds 41\) which is prime
\(\ds 6 \times 7 + 1\) \(=\) \(\ds 43\) which is prime


\(\ds 6 \times 8 - 1\) \(=\) \(\ds 47\) which is prime
\(\ds 6 \times 8 + 1\) \(=\) \(\ds 49 = 7^2\) and so is composite


\(\ds 6 \times 9 - 1\) \(=\) \(\ds 53\) which is prime
\(\ds 6 \times 9 + 1\) \(=\) \(\ds 55 = 5 \times 11\) and so is composite


\(\ds 6 \times 10 - 1\) \(=\) \(\ds 59\) which is prime
\(\ds 6 \times 10 + 1\) \(=\) \(\ds 61\) which is prime


\(\ds 6 \times 11 - 1\) \(=\) \(\ds 65 = 5 \times 13\) and so is composite
\(\ds 6 \times 11 + 1\) \(=\) \(\ds 67\) which is prime


\(\ds 6 \times 12 - 1\) \(=\) \(\ds 71\) which is prime
\(\ds 6 \times 12 + 1\) \(=\) \(\ds 73\) which is prime


\(\ds 6 \times 13 - 1\) \(=\) \(\ds 77 = 7 \times 11\) and so is composite
\(\ds 6 \times 13 + 1\) \(=\) \(\ds 79\) which is prime


\(\ds 6 \times 14 - 1\) \(=\) \(\ds 83\) which is prime
\(\ds 6 \times 14 + 1\) \(=\) \(\ds 85 = 5 \times 17\) and so is composite


\(\ds 6 \times 15 - 1\) \(=\) \(\ds 89\) which is prime
\(\ds 6 \times 15 + 1\) \(=\) \(\ds 91 = 7 \times 13\) and so is composite


\(\ds 6 \times 16 - 1\) \(=\) \(\ds 95 = 5 \times 19\) and so is composite
\(\ds 6 \times 16 + 1\) \(=\) \(\ds 97\) which is prime


\(\ds 6 \times 17 - 1\) \(=\) \(\ds 101\) which is prime
\(\ds 6 \times 17 + 1\) \(=\) \(\ds 103\) which is prime


\(\ds 6 \times 18 - 1\) \(=\) \(\ds 107\) which is prime
\(\ds 6 \times 18 + 1\) \(=\) \(\ds 109\) which is prime


\(\ds 6 \times 19 - 1\) \(=\) \(\ds 113\) which is prime
\(\ds 6 \times 19 + 1\) \(=\) \(\ds 115 = 5 \times 23\) and so is composite


\(\ds 6 \times 20 - 1\) \(=\) \(\ds 119 = 7 \times 17\) and so is composite
\(\ds 6 \times 20 + 1\) \(=\) \(\ds 121 = 11^2\) and so is composite

$\blacksquare$


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