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



The Most Common Errors in Undergraduate Mathematics