Question
In: Advanced Math
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.
Solutions
Colby Messinger answered 1 year agoRelated Solutions
If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1
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.
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.
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.
prove that there exist infinitely many primitive Pythagorean triples
prove that there exist infinitely many primitive Pythagorean
triples
Use spherical coordinates to evaluate the triple integral ∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where...
Use spherical coordinates to evaluate the triple integral
∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where E is the region
bounded by the spheres x^2 + y^2 + z^2 = 4 and x^2 + y^2 + z^2 =
9
The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...
The curried version of let f (x,y,z) = (x,(y,z)) is
let f (x,(y,z)) = (x,(y,z))
Just f (because f is already curried)
let f x y z = (x,(y,z))
let f x y z = x (y z)
Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}....
Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )
If X, Y and Z are three arbitrary vectors, prove these identities: a. (X×Y).Z = X.(Y×Z)...
If X, Y and Z are three arbitrary vectors, prove these
identities:
a. (X×Y).Z = X.(Y×Z)
b. X×(Y×Z) = (X.Z)Y – (X.Y)Z
c. X.(Y×Z) = -Y.(X×Z)
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