Breadcrumb 5B: We want to show that there is some \(L\) such that \( p \mid 2^{ 2^n + Lp} -2^n = 2^n ( 2^{ Lp + 2^n - n } -1) \).

Breadcrumb 6B: Take \( L = n - 2^n\). If we can apply Fermat's Little Theorem, then we are done, since \( 2^{Lp} \equiv 2^{L} \equiv 2^{n - 2^n} \pmod{p}\).

**Exercise 9:** Can we apply Fermat's Little Theorem directly? If no, why not?

Ponder this, then move on to the next note in this set.

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