Freshman's dream

[1][2] When n = 2, it is easy to see why this is incorrect: (x + y)2 can be correctly computed as x2 + 2xy + y2 using distributivity (commonly known by students in the United States as the FOIL method).

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 more exotic type of arithmetic, the "mistake" actually gives the correct result, since p divides all the binomial coefficients apart from the first and the last, making all the intermediate terms equal to zero.

We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation.

The demand that the characteristic p be a prime number is central to the truth of the freshman's dream.

A related theorem states that if p is prime then (x + 1)p ≡ xp + 1 in the polynomial ring

[4] In 1938, Harold Willard Gleason published a poem titled «"Dark and Bloody Ground---" (The Freshman's Dream)» in The New York Sun on September 6, which was subsequently reprinted in various other newspapers and magazines.

It consists of 2 stanzas, each containing 8 lines with alternating indentation; it has an ABCB rhyming scheme.

Words and phrases that hint that it might be related to this concept include: "Algebra", "Wild corollaries twine", "surds", "of plus and minus sign", "binomial", "quadratic", "parenthesis", "exponents", "in terms of x and y", "remove the brackets, radicals, and do so with discretion", and "factor cubes".

In a 1940 article on modular fields, Saunders Mac Lane quotes Stephen Kleene's remark that a knowledge of (a + b)2 = a2 + b2 in a field of characteristic 2 would corrupt freshman students of algebra.

This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic.

[6] Since then, authors of undergraduate algebra texts took note of the common error.

The first actual attestation of the phrase "freshman's dream" seems to be in Hungerford's graduate algebra textbook (1974), where he states that the name is "due to" Vincent O.

[8] The term "freshman's dream" itself, in non-mathematical contexts, is recorded since the 19th century.