Euclid's proof.

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Assume,

a positive integer.

By definition it is equivalent to say,

.

Now the prime decomposition of has an even amount of prime factors because of the square. While does not. By uniquness this is impossible.

Pythagorus' Proof.

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Assume that where is a reduced fraction, meaning no common factors.

Then,

.

Note, the right hand side is divisible by because the left hand side. Meaning divides . But then itself is divisible by by properties of prime numbers. That is .

Subsitute,

But then the left hand side is divisble by , that is divides . But then divides . Hence and have common factors, contrary to assumption.

(This is also a similar approach to Fermat's principle of infinite descent.)

Eisenstein Proof.

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The polynomial fits the conditions of Eisenstein irreducibility criterion for . Thus, there are no solution in and hence none in .

Rational Roots Proof.

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The polynomial can only have zeros for . None of which work. Thus, it is irrational.