Four square problem number theory The unit circle problem as

(Four square problem, number theory)

The unit circle problem asks: In the field Z_p how many incongruent solutions (ordered pairs) are there to

x^2 + y^2 congruent 1 (mod p)?

(a) Select three (4k + 1) primes p, and for each determine the number of solutions.

(b) Do the same for three (4k + 3) primes. Formulate conjectures.

Solution

the unit circle equation is given by x^2+y^2 = 1

in terms of the modulus its equivalent to

x^2 + y^2 = |p|

so the solution in terms of y will be

y = +-sqrt(|p| - x^2)

hence there would be indefinite number incongurent solutions

as x would vary x E (0 , |p| ]

a few of them are

(-x , sqrt(|p| - x^2))

(x , sqrt(|p| - x^2))

(-x , -sqrt(|p| -x^2))

and (x , -sqrt(|p| - x^2))

(a) when p = 4k+1

for k=1 , p=5 its a prime

so a few of the solution are:

(-x , sqrt(|5| - x^2))

(x , sqrt(|5| - x^2))

(-x , -sqrt(|5| -x^2))

and (x , -sqrt(|5| - x^2))            for this x E (0,5]

for k = 3 , p=13

a few of the solutions are:

(-x , sqrt(|13| - x^2))

(x , sqrt(|13| - x^2))

(-x , -sqrt(|13| -x^2))

and (x , -sqrt(|13| - x^2))               x E (0 , 13]

for k=4 , p=17

a few of the solutions are

(-x , sqrt(|17| - x^2))

(x , sqrt(|17| - x^2))

(-x , -sqrt(|17| -x^2))

and (x , -sqrt(|17| - x^2))    , x E (0 , 17]

(b) k=1 , p=7   , x E (0 , 7]

(-x , sqrt(|7| - x^2))

(x , sqrt(|7| - x^2))

(-x , -sqrt(|7| -x^2))

and (x , -sqrt(|7| - x^2))

k=2 , p=11   , x E (0, 11]

(-x , sqrt(|11| - x^2))

(x , sqrt(|11| - x^2))

(-x , -sqrt(|11| -x^2))

and (x , -sqrt(|11| - x^2))

k=4 ,    p = 19 , x E (0 19]

(-x , sqrt(|19| - x^2))

(x , sqrt(|19| - x^2))

(-x , -sqrt(|9| -x^2))

and (x , -sqrt(|19| - x^2))

so you could find the solution for these primes as well and they will be different form those found in part (a)