Describe all the integer solutions of the equation x2 + y2 = 32. Positive integer solutions...
2: (a) Find all solutions (x, y) = Z2 to Pell's Equation x2 – 29 y2 = 1. (b) Find all solutions (x, y) € Z to the Pell-like equation x2 - 21 y2 = 4.
(3) Show that {(2,3) € R? : x2 + y2 = 1,2 + –1} = {(*):tER}. (ii) Show that {(x, y) € Q2 : x2 + y2 = 1, x + -1} = {(1742, 1942): ted} (iii) Show that {(x, y, z) € ZP : x2 + y2 = z2} = {(m? – n2, 2mn, m? +n2): m, n € Z} (Hints: For (i) consider the equation of the line joining (-1,0) and (x, y) with slope t; For (ii)...
Determine all the integer solutions to the equation X1 + X2 + X3 + X4-7 where xj 2 0 for all i - 1,2,3,4
Determine the number of integer solutions of x1 + x2 + x3 + x4-32, where a) xi 2 0, 1 3is4 b) x1, x2 2 2, x3, X4 2 1
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.]
Use the elimination method to find all solutions of the system: x2 + y2 = 8 1 x2 - y2 = 1 The four solutions of the system are: (the one with x < 0, y < 0 is) x = (the one with x < 0, y > 0 is) X = y = (the one with x > 0, y < 0 is) x = y = (the one with x > 0, y > O is) c...
)Consider the non-negative integer solutions to x1 + x2+ x3 + x4 + x5 = 2020. (A) How many solutions does Equation (1) have satisfying 0 ≤ x1 ≤ 100? Explain. (B) Remember to explain your work. How many solutions does Equation (1) have satisfying 0 ≤x1 ≤ 100, 1 ≤x2 ≤ 150, 10 ≤x3 ≤ 220?
4. Show that the equation x + y + 32 - 2xyz has no nontrivial integer solutions.
Write the equation with rectangular coordinates. r2 = 5 cos20 a. (x2 + y2)2 = 5(x2 + y2) ob. (x2 + y2)2 = 6(x2 –y?) Oc. (x² + y2)2 = 6(x2+y?) O d. (x+y?)? = 6(x2-y?) O e. (x2+x2)? = 5(x2-y2)
Write the equation in spherical coordinates. (a) x2 + y2 + Z2 = 64 (b) x2 - y2 - 2 = 1 | eʼsin?(p)cos? (0) – eʼsin(@)sin?(0) – e cos?(p) = 1