Numbers of the form 2ᵖ − 1 with p prime are often prime themselves — but not always. Which of these is NOT a prime number?

Work each one out: $2^2-1=3,2^3-1=7$ and $2^5-1=31$ are all prime. But $2^{11}-1=2047=23\times 89$, so it is composite even though its index 11 is prime — the pattern breaks here for the first time. Only trial division settles it: 2047 has to be tested against every prime up to √2047 ≈ 45, and it survives 2, 3, 5, 7, 11, 13, 17 and 19 before 23 finally divides it exactly.