High Quality Content by WIKIPEDIA articles! The freshman's dream is a name sometimes given to the error (x + y)n = xn + yn, where n is a real number (usually a positive integer greater than 1). Beginning students commonly make this error in computing the exponential of a sum of real numbers.[1][2]. When n = 2, it is easy to see why this is incorrect: (x + y)2 can be correctly computed by the FOIL rule as x2 + 2xy + y2. For larger positive integer values of n, the correct result is given by the binomial theorem. The name freshman's dream also sometimes refers to the theorem that says that for a prime number p, if x and y are members of a commutative ring of characteristic p, then (x + y)p = xp + yp. In this case, the "mistake" actually gives the correct result, due to p dividing all the binomial coefficients save the first and the last. This theorem demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring.

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