---- Suppose Euler's version holds, and let be two odd primes.
(i) Assume first that and put , then:
Since , we have that , and this
is just Legendre's version for (why?) .
(ii) Now assume and do exactly as above. This time though we get
that , and since , we
have that , and again this is Legendre's version (why?)
---- Suppose now Legendre's version holds , and let , for odd primes , as:
2) Since also , and thus:
...now try to complete the
proof (two steps more). Where did we use that a is an odd integer?