I think you forgot to mention that must be an odd integer...

----, and let be two odd primes.Suppose Euler's version holds

() Assume first that and put , then:i

Since , we have that , and this

is just Legendre's version for (why?) .

() Now assume and do exactly as above. This time though we getii

that , and since , we

have that , and again this is Legendre's version (why?)

----, and let , for odd primes , as:Suppose now Legendre's version holds

1) ;

2) Since also , and thus:

...now try to complete the

proof (two steps more). Where did we use that a is an odd integer?

Tonio