let p = 1031, Find the number of solutions to the equation x^2
-2 y^2=1 (mod...
let p = 1031, Find the number of solutions to the equation x^2
-2 y^2=1 (mod p), i.e., the number of elements (x,y),
x,y=0,1,...,p-1, which satisfy x^2 - 2 y^2=1 (mod p)
a. consider the plane with equation -x+y-z=2, and let p be the
point (3,2,1)in R^3. find the distance from P to the plane.
b. let P be the plane with normal vector n (1,-3,2) which passes
through the point(1,1,1). find the point in the plane which is
closest to (2,2,3)
a)
Let y be the solution of the equation y ′ =
(y/x)+1+(y^2/x^2) satisfying the condition y ( 1 ) = 0.
Find the value of the function f ( x ) = (y ( x ))/x
at x = e^(pi/4) .
b)
Let y be the solution of the equation y ′ = (y/x) −
(y^2/x^2) satisfying the condition y ( 1 ) = 1. Find the
value of the function f ( x ) = x/(y(x))
at x = e .
c)
Let y be the solution...
Number Theory:
Let p be an odd number. Recall that a primitive root, mod p, is
an integer g such that gp-1 = 1 mod p, and no smaller
power of g is congruent to 1 mod p. Some results in this chapter
can be proved via the existence of a primitive root(Theorem
6.26)
(c) Given a primitive root g, and an integer a such that a is
not congruent to 0 mod p, prove that a is a square...
For p = 5 , find a number x such that x^2 is congurent to -1 mod
p. here x is denoted by sqrt(-1). Determine if sqrt(-1) exists mod
p for p = 7, 11, 13 , 17, 23, 29.
1)Find the power series solution for the equation y'' − y =
x
2)Find the power series solution for the equation y'' + (sinx)y
= x; y(0) = 0; y'(0) = 1
Provide the recurrence relation for the coefficients and derive
at least 3 non-zero terms of the solution.
State the linearity of the equation: x (y^2 + 2z) p − y (x^2 +
2z) q = (x^2 −y^2) z. Find the general solution, and then the
particular solution that passes through the line: x + y = 0, z =
2.
Linearity is the category to which it belongs among: linear,
quasi-linear or non-linear.