Show that every Pythagorean triple (x, y, z) with x, y, z having no common factor...

Show that every Pythagorean triple (x, y, z) with x, y, z having no common factor d > 1 is of the form (r^2 - s^2, 2rs, r^2 + s^2) for positive integers r > s having no common factor > 1; that is

x = r^2 - s^2, y = 2rs, z = r^2 + s^2.

Expert Solution

Colby Messinger answered 1 year ago

Show that if (x,y,z) is a primitive Pythagorean triple, then X and Y cannot both be...

Show that if (x,y,z) is a primitive Pythagorean triple, then X and Y cannot both be even and cannot both be odd. Hint: for the odd case, assume that there exists a primitive Pythagorean triple with X and Y both odd. Then use the proposition "A perfect square always leaves a remainder r=0 or r=1 when divided by 4." to produce a contradiction.

If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1

Let x, y, z be a primitive Pythagorean triple with y even. Prove that x+y ≡...

Let x, y, z be a primitive Pythagorean triple with y even. Prove that x+y ≡ x−y ≡ ±1 mod 8.

A Pythagorean triplet is a set of positive integers (x, y, z) such that x2 +...

A Pythagorean triplet is a set of positive integers (x, y, z) such that x2 + y2 = z2. Write an interactive script that asks the user for three positive integers (x, y, z, in that order). If the three numbers form a Pythagorean triplet the script should a) display the message ‘The three numbers x, y, z, form a Pythagorean triplet’ b) plot the corresponding triangle using red lines connecting the triangle edges. Hint: place the x value on...

Suppose x,y ∈ R and assume that x < y. Show that for all z ∈...

Suppose x,y ∈ R and assume that x < y. Show that for all z ∈ (x,y), there exists α ∈ (0,1) so that αx+(1−α)y = z. Now, also prove that a set X ⊆ R is convex if and only if the set X satisfies the property that for all x,y ∈ X, with x < y, for all z ∈ (x,y), z ∈ X.

Question

Show that every Pythagorean triple (x, y, z) with x, y, z having no common factor...

Solutions

Expert Solution

Related Solutions

Show that if (x,y,z) is a primitive Pythagorean triple, then X and Y cannot both be...

If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1

Let x, y, z be a primitive Pythagorean triple with y even. Prove that x+y ≡...

A Pythagorean triplet is a set of positive integers (x, y, z) such that x2 +...

Suppose x,y ∈ R and assume that x < y. Show that for all z ∈...

Use spherical coordinates to evaluate the triple integral ∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}....

If X, Y and Z are three arbitrary vectors, prove these identities: a. (X×Y).Z = X.(Y×Z)...

Set up the triple integral of an arbitrary continuous function f(x, y, z) in spherical coordinates...