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There's an easier proof: p + 1 must be even except where p is 2. Even implies one factor is 2. If one factor is 2, both must be two. This uniquely describes 4. 2 + 1 = 3, so p is not 2. This means p is uniquely 3.
I don't quite follow "If one factor is 2, both must be two." There are plenty of even squares which are not equal to 4. (For example 16, 36, ...)
There's an easier proof: p + 1 must be even except where p is 2. Even implies one factor is 2. If one factor is 2, both must be two. This uniquely describes 4. 2 + 1 = 3, so p is not 2. This means p is uniquely 3.
I don't quite follow "If one factor is 2, both must be two." There are plenty of even squares which are not equal to 4. (For example 16, 36, ...)