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We prove a conjecture that classifies exceptional numbers. This conjecturearises in two different ways, from cryptography and from coding theory. An oddinteger $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (AlmostPerfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$.Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$with two zeros $\omega, \omega^t$ has minimum distance 5 for infinitely manyvalues of $n$. The conjecture we prove states that every exceptional number hasthe form $2^i+1$ or $4^i-2^i+1$.

Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions