Freshman’s dream and sophomore’s dream

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

(x + y)n = xn + yn,

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 = xp + yp.

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

sophomore's dream

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!

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