Continuing in the same vein here are two one sentence proofs.

**Theorem:** For we have .

**Proof:** The coefficient of in the left hand side of the given expression, i.e. in , is obtained by selecting exactly parenthesis out of the total available to yield the , (the remaining parenthesis yield the ), which can be done in ways following which the coefficient is exactly .

**Theorem:** is irrational.

**Proof:** If is in lowest terms then is in lower terms.

**Remark:** The above proof generalizes to the case when is a non-square positive integer. Indeed, if then assuming that is in lowest terms also implies that is in lower terms.

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