3. (Consecutive squares modulo p) We can use quadratic reciprocity to answer the question: "How many residue class modulo p are there such that both x and x+1 are squares?" (a) Show the identity: E)-...

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Extracted text: 3. (Consecutive squares modulo p) We can use quadratic reciprocity to answer the question: "How many residue class modulo p are there such that both x and x+1 are squares?" (a) Show the identity: E)- = 0 x=0 and that if x #0 (mod p) () | 1, if y? = x (mod p) is solvable if y? = x (mod p) has no solutions 1 2 (b) Consider the summation р-2 (금) x + 1 1+ N = 1+ = 1+ + 1+ 2 x=1 Justify why N is the number of consecutive pairs of square modulo р. (c) Prove that p+2+ 4