5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic re...

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5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between

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Answer 1

Answer #1

Solution: Here 'p ' is an odd prime number such that $Pー!mod 4$ and there are (p-1) / 2 quadratic residues of p.

(p-1)/2 ai-a1 + a2 + a3 + a(p-1)/2

-(a1 +a(p-1)/2) +(a2a3)/2)+(a3+ap-5)/2) +[(p - 1)/2|/2 times

(al + a(pー1)/2) + (a2 + a(p-3/2) + (a3 + a(pー5)/2 ) (p- 1)/4 times

As $(a_{1}+a_{(p-1)/2}) =(a_{2}+a_{(p-3)/2})=(a_{3}+a_{(p-5)/2})+.....=p$ ( If a₁ is quadratic residue of p then p - a₁ = a_p-1/2 ) so,

(p-1)/2 aip+pp..(p 1)/4 times

-plp - 1)/4

We will see it with an example,

Let p = 13 and 1,3,4,9,10 and 12 are quadratic residues of p=13.

We can see that number of quadratic residues = p-1 /2 = 13-1 /2 = 6 residues (1,3,4,9,10 and 12 )

Now 1+12 = 3+10 = 4+9 = 13

i.e 1 + 3 + 4 +9 + 10 + 12 = (1+12)+(3+10)+(4+9) = 13+13+13

as p-1 / 4 = 13-1 / 4 = 3 so 3 pairs of 13.

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Answer 2

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