Question

In: Advanced Math

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

Solutions

Expert Solution

If   is a primitive Pythagorean triple the and  

Since every integer is either even or odd so three case may arise both are even or both are odd or one of is even and another is odd .

Case 1 : If both x,y is even .

Then , for some

a contradiction to .

Case 2 : If both x , y are odd .

for some

is of the form

2 divides but 4 does not divided .

As 2 divides

a contradiction to 4 does not divides .

So both cannot be even and also both cannot be odd .

one of is odd and another is even .

Without loss of generality assume that is odd and is even .

   for some

,[ where s=k2+m2+1 ]

Hence is of the form .

Hence

Colby Messinger answered 3 years ago
Next > < Previous

Related Solutions

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 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.
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.
prove that there exist infinitely many primitive Pythagorean triples
prove that there exist infinitely many primitive Pythagorean triples
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)
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...
Let x, y ∈ Z. Prove that x ≡ y + 1 (mod 2) if and...
Let x, y ∈ Z. Prove that x ≡ y + 1 (mod 2) if and only if x ≡ y + 1 (mod 4) or x ≡ y + 3 (mod 4)
Use two different ways to prove X Y + Z = (X + Z)(Y + Z)....
Use two different ways to prove X Y + Z = (X + Z)(Y + Z). a) Use pure algebraic way b) k-maps
Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be...
Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one to one and onto functions. Then g ◦ f : X → Z is one to one and onto; and (g ◦ f)−1 = f−1 ◦ g−1 ).
Find ??, ?? and ?? of F(x, y, z) = tan(x+y) + tan(y+z) – 1
Find ??, ?? and ?? of F(x, y, z) = tan(x+y) + tan(y+z) – 1
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT