1. Show that the number of solutions (x mod p, y mod p) to the equation...
22. Suppose that (ab, p)- 1 and that p> 2. Show that the number of solutions (x, y) of the congruence ax2 + by 1 (mod p) is -ab 22. Suppose that (ab, p)- 1 and that p> 2. Show that the number of solutions (x, y) of the congruence ax2 + by 1 (mod p) is -ab
Prove that there are no natural number solutions to the equation where x, y ≥ 2 ... (See Picture Below) Prove that there are no natural number solutions to the equation where X, Y > 2. x2 - y2 = 1.
1. (Complex Multiplication) Let E : y x3 y23 to this congruence mod p. So for example, #E(Z3) = 3 because we have the solutions (0, 0), (1,0) and (2,0) and no more. - x. Then we can reduce E mod p to get mod p for various primes p. We write #E(Z») for the number of solutions This particular equation has some miraculous explore here patterns we (a) Make a chart that lists p, #E(Zp), and #E(Z) - p...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
8. For an equation y',-y'-6y-0 show that yı + y2 and Cyı are also solutions for any constant C where yi - e3t and y2 e2
Consider the following differential equation to be solved using a power series. y" - y' = 0 Using the substitution y = į coxn, find an expression for Ck + 2 in terms of Ck + 1 for k = 0, 1, 2, .... k+2= + + + Find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the s 4.3. Try to explain any differences between the two forms of the...
Consider the differential equation, L[y] = y'' + p(t)y' + q(t)y = 0, (1) whose coefficients p and q are continuous on some open interval I. Choose some point t0 in I. Let y1 be the solution of equation (1) that also satisfies the initial conditions y(t0) = 1, y'(t0) = 0, and let y2 be the solution of equation (1) that satisfies the initial conditions y(t0) = 0, y'(t0) = 1. Then y1 and y2 form a fundamental set...
16. a.) Show that Y. and Y2 = In x are both solutions of the non-linear differential equation y' + (y') = 0 by substituting. b.) Is Y + Y also a solution, yes or no? Demonstrate by substituting, c.) Is 99, + 2Y2 also a solution, yes or no? Demonstrate by substituting.
Show that there are no solutions to the equation p+ q2 = y2 + y2 + t2 where p, q, r, s, t are primes. (Hint: Consider the remainder of the square of an odd integer when divided by 8, and then consider cases.]
One of the solutions to the following differential equation (1 – 2x – 2y + 2(1+x)y – 2y = 0 is known to be yı (x) = 1 +1 Find the second linearly independent solution y2 (2) using the method of Reduction of Order.