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Prove if the following quadratic congruence has a solution or not (do not find an answer):

x² ≡ 7 (mod 107)

Question:
Solve the following Diophantine equation:

x² - 19*y² = 1

After getting the first solution, what is another solution?

All solutions should be integers.

Answer:
Use continued fractions to estimate √(19).

√(19) ~= 4.3589 = 4 + .3589
1/.3589 = 2 + .78629
1/.78629 = 1 + .2718
1/.2718 = 3 + .67918
1/.67918 = 1 + .47236
1/.47236 = 2 + .11703
1/.11703 = 8 + .54482 //Stop here on continued fraction

Continued fraction reduces to:

170/39

(170)² - 19 * (39)² = 1

In order to find another solution, square the equation:
(x - y * √(19)) * (x + y * √(19)) =
x2 = x² + 19 * y²
y2 = 2*x*y

x2 = 57799
y2 = 13260