(a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b) Determine if x2 -x+5 0 (mod 137) has any solutions.

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(a) Let p be an odd prime. Give an analogue of the quadratic formula
mod p, and prove it.
(b) Determine if x2 – x+5 = 0 (mod 137) has any solutions.
Transcribed Image Text:(a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b) Determine if x2 – x+5 = 0 (mod 137) has any solutions.
Expert Solution
Step 1

(a) Given that p is an odd prime. 

The quadratic formula mod p is given by:

ax2+bx+c0mod p

Here, p does not divide a.

The above equation can be rewritten as:

ax2+bx-cmod p

Multiplying both sides by 4a,

4a2x2+4abx-4acmod p

Now, adding b2 to both sides,

4a2x2+4abx+b2b2-4acmod p

Step 2

Factor the L.H.S,

2ax+b2b2-4acmod p

Thus, it has a solution if and only if b2-4ac is a quadratic residue or p divides b2-4ac.

If p divides b2-4ac, the R.H.S of the above equation will be 0 and it gives a single solution.

If it is a quadratic residue , it will have two solutions.

Hence proved.

(b) x2-x+50mod 137

Multiply both sides by 4,

4x2-4x-20mod 137

Adding -12=1 on both sides,

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