面白いものを見つけた
　>>41より
https://abel.a.la9.jp/
東京理科大学理学部第一部数学科　教授　安部直人
https://abel.a.la9.jp/sub11.html
東京理科大学 数学科　教授　安部直人　(2013)07月06日
　素因数分解を習った中学生なら誰でもわかる３行の直接証明：
　「自然数　a,b につき、
　　aa と 2bb の素因数の個数は偶数と奇数
　　で異なるから aa≠2bb、よって √2≠a/b。」
(引用終り)

これ wikipedia Square root of 2
の Proof by infinite descent にそっくりそのままある
”Proof by unique factorization
As with the proof by infinite descent, we obtain
a^=2b^2.
Being the same quantity, each side has the same prime factorization by the fundamental theorem of arithmetic, and in particular, would have to have the factor 2 occur the same number of times.
However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.”
だ（強調するが”a contradiction”な）

英語圏では ”a contradiction”つまり 背理法認定ｗ（＾＾
どうすんのこれ？ｗｗ

(参考)（他の面白そうな証明も引用する）
https://en.wikipedia.org/wiki/Square_root_of_2#Proof_by_infinite_descent
Square root of 2
Proofs of irrationality
Proof by infinite descent
One proof of the number's irrationality is the following proof by infinite descent. It is also a proof of a negation by refutation: it proves the statement
"√2 is not rational" by assuming that it is rational and then deriving a falsehood.
1.Assume that √2 is a rational number, meaning that there exists a pair of integers whose ratio is exactly √2.
略

Proof using reciprocals
Assume by way of contradiction that
√2 were rational.

Proof by unique factorization
As with the proof by infinite descent, we obtain
a^=2b^2.
Being the same quantity, each side has the same prime factorization by the fundamental theorem of arithmetic, and in particular, would have to have the factor 2 occur the same number of times.
However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.

Application of the rational root theorem
The irrationality of
√2√2 also follows from the rational root theorem, which states that a rational root of a polynomial, if it exists, must be the quotient of a factor of the constant term and a factor of the leading coefficient. In the case of
p(x)=x^2-2, the only possible rational roots are
±1 and ±2.
As √2 is not equal to ±1 or ±2, it follows that √2 is irrational.
略