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Freshman's dream - mklabgr - 08-10-2026
Freshman's dream
Summary
The Freshman’s Dream is a generally false mathematical rule that incorrectly assumes powers distribute over addition, written as $(x+y)^n=x^n+y^n$. It is a common mistake made by beginning students, because the correct expansion usually contains additional terms, as shown by the binomial theorem; for example, $(x+y)^2=x^2+2xy+y^2$, not $x^2+y^2$.
However, the identity is genuinely valid in certain mathematical settings, particularly in commutative rings of prime characteristic $p$, where $(x+y)^p=x^p+y^p$ because the intermediate binomial coefficients vanish modulo $p$. The article also discusses special cases where the formula happens to hold, its connection to the Frobenius endomorphism, and the historical development of the term “Freshman’s Dream.”
ARTICLE