Sequence of 11 Primes by Trebling and Adding 16
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Theorem
The process of multiplication by $3$ and then adding $16$ produces a sequence of $11$ primes when starting from $587$:
- $587, 1777, 5347, 16 \, 057, 48 \, 187, 144 \, 577, 433 \, 747, 1 \, 301 \, 257, 3 \, 903 \, 787, 11 \, 711 \, 377, 35 \, 134 \, 147$
Proof
\(\ds \) | \(\) | \(\ds 587\) | is prime | |||||||||||
\(\ds 3 \times 587 + 16\) | \(=\) | \(\ds 1777\) | which is prime | |||||||||||
\(\ds 3 \times 1777 + 16\) | \(=\) | \(\ds 5347\) | which is prime | |||||||||||
\(\ds 3 \times 5347 + 16\) | \(=\) | \(\ds 16 \, 057\) | which is prime | |||||||||||
\(\ds 3 \times 16 \, 057 + 16\) | \(=\) | \(\ds 48 \, 187\) | which is prime | |||||||||||
\(\ds 3 \times 48 \, 187 + 16\) | \(=\) | \(\ds 144 \, 577\) | which is prime | |||||||||||
\(\ds 3 \times 144 \, 577 + 16\) | \(=\) | \(\ds 433 \, 747\) | which is prime | |||||||||||
\(\ds 3 \times 433 \, 747 + 16\) | \(=\) | \(\ds 1 \, 301 \, 257\) | which is prime | |||||||||||
\(\ds 3 \times 1 \, 301 \, 257 + 16\) | \(=\) | \(\ds 3 \, 903 \, 787\) | which is prime | |||||||||||
\(\ds 3 \times 3 \, 903 \, 787 + 16\) | \(=\) | \(\ds 11 \, 711 \, 377\) | which is prime | |||||||||||
\(\ds 3 \times 11 \, 711 \, 377 + 16\) | \(=\) | \(\ds 35 \, 134 \, 147\) | which is prime | |||||||||||
\(\ds 3 \times 35 \, 134 \, 147 + 16\) | \(=\) | \(\ds 105 \, 402 \, 457\) | which is not prime : $105 \, 402 \, 457 = 67 \times 137 \times 11 \, 483$ |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $587$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $587$