Show that there are no solutions to the equation p+ q2 = y2 + y2 +...
Show that there are no solutions to the equation p2 + q2 = r2 + s2 + t2 where p, q, r, s, t are primes.
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
2) Prove that the square of any odd number leaves a remainder of one when divided by 8. (Hint: use cases expressing the odd number in the form 8k+r, where r<8)
differential equation a. Show that y = y2 + y2 is a solution of y" + P(x)y' + Q(x)y=T_(x) + 2(x) if y. and y2 are the solution of the following equations respectively; y" + P(x)g' + Q(x)y = Tz (X) and y" + P(x)y' + Q(x)y = T2(x)
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
Need the answer as soon as possible a. Show that y = y + y2 is a solution of y" + P(x)y' + Q(x)y = T (x) + T2(x) if y, and y, are the solution of the following equations respectively; y + P(x)y' + Q(x)y = Ti(x) and y" + P(x)y' + Q(x)y = T2(x) (CO2:P01 - 4 Marks) b. Determine the general solution of the given equation using method of undetermined coefficients y" +9y = 2 sin 3x...
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0
Q2. Consider the plane P C R3 given by the equation 2z-y+2z 7 and the point v2 (a) Show that the point p-5lies in P and calculate the distance between p and v (b) Find the point qE P that lies closest to v (c) What is the distance of v to P? (d) What is the angle between the vectors v - q and p -q? (e) Does the pythagoras theorem apply to the triangle formed by the points...
8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x,...
Hello, Can someone please show me two examples on how this proposition is being used? Please be legible. Thank you. 11.3.2 LAW OF QUADRATIC RECIPROCITY (Restatement) Let p and q be odd primes with p q. Then 1 q-1 = (-1) Not surnrisingly it has turned out that Phil 's ansuwer from the berin 11.3.2 LAW OF QUADRATIC RECIPROCITY (Restatement) Let p and q be odd primes with p q. Then 1 q-1 = (-1) Not surnrisingly it has turned...