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 |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $120$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $120$