Sequence of 11 Primes by Trebling and Adding 16

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): $587$