The freshman's dream is the purported identity
 |
for a positive integer 
. It is not an identity over the
real numbers in general. For example, 
.
However, let 
be a commutative ring with a multiplicative
identity and ring characteristic 
, where 
is a prime number. Then
 |
for all 
.
In the binomial theorem, the intermediate binomial coefficients 
for 
are divisible
by 
and hence vanish in 
(O'Sullivan 2006, p. 19). Iteration gives 
for every
nonnegative integer 
. Together with 
, the identity shows
that the Frobenius map 
is a ring homomorphism
from 
to itself.
The similarly named sophomore's dream expresses definite integrals of 
and 
as infinite series.
