Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically: ∀ n ∈ Z , n 2 is even ⇒ n is even ∀ n ∈ Z , n 2 is even ⇒ n is even

Question

Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically:  ∀  n  ∈      Z    ,            n    2     is even   ⇒  n   is even     ∀  n  ∈      Z    ,            n    2     is even   ⇒  n   is even

GrEettarim3 · Accepted Answer

Ryland Houston · Accepted Answer

Let   n  =      m    2  . Let   m  =  ∏      p    i                  n        i             be the unique prime factorization of   m. Then       n    2    =  ∏      p    i          2              n        i             is the prime factorization of       n    2  . As       n    2   is even one of those primes is   2. Those are the primes that factor   m so   m is even.

Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically: forall n in Z, n^2 is even =>n is even

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