Experimental mathematics: conjectures and counterexamples

The Fermat numbers are $F_n = 2^{2^n} + 1$. What is the smallest $n$ for which $F_n$ is not prime?

Let $\varphi = (1 + \sqrt5)/2$. Which vector is an integer relation for $(\varphi^2, \varphi, 1)$?

Which of these are honest, adequately specified reports of a counterexample search for Goldbach's conjecture? (Choose all that apply.)

At 100 digits, PSLQ finds a relation with coefficients of size below $100$ among six constants, with residual $10^{-95}$; at 200 digits the same relation holds to $10^{-195}$. What may you claim?

Among the primes $p \le 50$, let $T$ be the number with $p \equiv 3 \pmod 4$ and $O$ the number with $p \equiv 1 \pmod 4$. Compute $T - O$.

In the proof of the BBP formula, which step needs an analytic justification rather than algebra alone?

With fermat n = 2 ^ (2 ^ n) + 1, which Lean statement says that Fermat's conjecture is false?

Which conjecture would be refuted by a finite counterexample that a computer can check, whatever form the counterexample takes?

Write first_counterexample(pred, start, bound) that checks pred(n) for every integer n with start <= n <= bound in increasing order and returns the first n for which pred(n) is false, or None if there is none (an exhaustive search up to bound). Then use it to find the first prime p for which $2^p - 1$ is not prime. The helper is_prime is provided.