proof of congruence of Clausen and von Staudt
In the equation
substitute and add, obtaining,
Now by comparing coefficients of and then multiplying by , we obtain the result.
We will write this identity in the following form:
This follows by replacing by and then switching and in the theorem.
Let be prime and . Then is -integral, and if is even, then .
is -integral for . By induction is -integral for , and is -integral since for all primes . It follows that is -integral. To establish the congruence, we need to show that if , then
For , , since . For , we have
since is even. In fact, since for even, it suffices to check it for , which is obvious.
Let be prime. Then
Let be a primitive root modulo . Then
If , then , and . If , then .
We are now ready to prove the congruence.
Proof 4 (Proof of von Staudt-Claussen congruence)
is a -integer for all primes . For if is a prime, and , then is -integral and hence is as well, since the sum contributes no negative power of . Otherwise, and
which is clearly -integral. Since is -integral for all primes , it must be the case that . That is,
|Title||proof of congruence of Clausen and von Staudt|
|Date of creation||2013-03-22 15:33:58|
|Last modified on||2013-03-22 15:33:58|
|Last modified by||slachter (11430)|