A mistake often made by first year calculus students is in thinking that

(a+b)n = an + bn.  This mistake has acquired the nickname of the “Freshman’s Dream”.

 

A related result (that is true) known as the the “Freshman’s Dream Theorem” is that

 

(a+b)n = an + bn  (mod n) when n is a prime or a Carmichael number.

 

Source: Pratibha Ghatage and Brian Scott, When is (a+b)n = an + bn  (mod n)?, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), p. 322.

 

Also see: http://en.wikipedia.org/wiki/Carmichael_number

 

If there was a calculus “Professor’s Dream” it would likely be that all students carefully read (and then read again several more times) the following pages before they reach calculus:

 

Paul's Online Math Notes: Common Math Errors

http://tutorial.math.lamar.edu/extras/commonerrors/commonmatherrors.aspx

 

The Most Common Errors in Undergraduate Mathematics

http://www.math.vanderbilt.edu/~schectex/commerrs/