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Show that the number of solution to x2 - y2 ≡ a (p) is p-1 if...

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Advance Math

Show that the number of solution to x2 -y2 ≡ a (p) is p-1 if p not divides a and 2p -1if p divides a, Use u = x+y and v= x-y

Then show that summation y=0 to p-1 ((y^2 +a)/p) = -1 ifp not divides a and

= p-1 if p divides a

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4.2 Ratings (569 Votes)
Given Put and Then1 becomes Since we are solving modulo p x and y can take values from 0 1 2 p1 Case 1 divides 2 implies is divisible by But divides and so divides That    See Answer
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