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

(

*x*+*y*)^{n}=*x*^{n}+*y*^{n},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} = *x*^{2} + 2*xy* + *y*^{2}. 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!

A paper : ” The Sophore’s Dream Function ” :

http://www.scribd.com/JJacquelin/documents

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Did some <sup> tags get lost?

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