The next set of exercises unpacks and verifies the details of Don Zagier\'s one

Question

The next set of exercises unpacks and verifies the details of Don Zagier's one sentence proof that every prime p = 1 (mod 4) is a sum of two squares. Let T be a set. An involution psi on T is any function psi: T rightarrow T such that psi is its own inverse, i.e. psi^2 (x) = psi (psi (x) = x for all x element T. Let p be a prime such that p 1 mod 4. Define S: = {(x, y, z) element N^3| x^2 + 4yz = p}. Define the function phi by phi: S rightarrow S (x, y, z) phi (x, y, z): = {(x + 2z, z, y - x - z) if x

Explanation / Answer

in mathematics an involution is a function that is its own inverse for all x in the domain f, hence in the given question the function is an involutory function on the finite set T with a odd number of fixed point, since it is define only for odd number that's why if in the mapped element there is only odd number so the cardinality of T will be the odd only. if there is n element in T and all are odd element the then the cardinality of T will be n

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