For odd, .
For even, take , then .
Evaluate the Jacobi symbol ((n−1)(n+1)/n) for any odd natural number n.
Trying out some numbers, I THINK it alternates between 1 and -1, but how can we PROVE it formally?
Any help is appreciated!
[also under discussion in math link forum]
1 has no prime factorization, so the product is empty. Is it conventional to define the "empty" product to be equal to +1??
Also, is it true that, by definition, (a/1)=1 for any integer a?
Can someone clarify this? Thank you!
Example (which is not linked to your question, but just to show you) :
Why do we define 1 as not being prime ?
If we defined 1 as being prime, we would have to reformulate the Fundamental Theorem of Arithmetic in a rather heavy way, stating that "apart from adding 1's, the prime decomposition of a number is unique without taking into account the order" ...
And since there is no particular reason to define 1 as a prime, we just prefer to say it isn't one