Fermat Number

A Fermat number is built by taking 2 to a power that is itself a power of 2, then adding 1: F_n = 2^{2^n} + 1. This creates a very rapidly growing sequence: each step doubles the exponent, so the numbers explode in size even for small n. They matter in number theory because Fermat numbers have special algebraic properties, including the fact that any two different Fermat numbers are relatively prime, which comes from the identity that the product of earlier Fermat numbers is one less than the next power of 2. This structure also explains why they were historically studied in connection with constructible polygons and prime numbers.

Use this formula when you need the n-th Fermat number itself, especially in problems about divisibility, primality, or pairwise coprimality. It also comes up in algebraic number theory and classical geometry questions involving constructible polygons.

• Confusing 2^{2^n} with (2^2)^n; the exponent is a tower, not a product of exponents
• Forgetting the +1 at the end, which changes the value completely
• Evaluating powers in the wrong order and computing 2^{2^n} as 2^{2}^n instead of 2^{(2^n)}