Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer t⩾3 is said to be exceptional if f(x)=xt is APN (Almost Perfect Nonlinear) over Fn2 for infinitely many values of n. Equivalently, t is exceptional if the binary cyclic code of length n2−1 with two zeros ω,ωt has minimum distance 5 for infinitely many values of n. The conjecture we prove states that every exceptional number has the form i2+1 or i4−i2+1.