Freshman’s dream and sophomore’s dream

For those who haven’t heard of this yet, the freshman’s dream is given to the (common) error:

where n is usually a positive integer greater than 1 (can be real too). You’d be surprised how many university students make this mistake! Simplying looking at n=2 shows why it doesn’t work in general: (x + y)² = x² + 2xy + y². However, there is a theorem referred to as the “Freshman’s Dream” which says if p is a prime number, and x,y are members of a commutative ring of characteristic p, then (x + y)^p = x^p + y^p.

The sophomore’s dream is used for the following identity:

This formula was discovered in 1697 by J. Bernoulli. The sophomore’s dream seems too good to be true (like the freshman’s dream), but is in fact true!