"Suppose p is odd so that 2 is not a factor of p."
"Then p can be written as 2n+1."
=> p^2 = (2n+1)^2 = 4n^2 + 4n + 1
"Now "(4 n^2 + 4n + 1)" mod 2 = 1,"
"so "p^2" is odd."
p^2 = 2 m^2 " is impossible as such as "2 m^2" is even."
"Hence our assumption that p is odd is false, so p must be even."
"One can also work through the prime factorization that is"
"unique :"
p^2 " contains 2 in its prime factorization."
"Hence also "p" contains 2 in its prime factorization as a square"
"of a number has the same prime factorization but with the"
"exponents doubled."