Question

Prove: Let (a,b,c) be a primitive pythagorean triple. then we have the following

1. gcd(c-b, c+b) =1

2. c-b and c+b are squares

Answer #1

Prove: If (a,b,c) is a primitive Pythagorean triple, then either
a or b is divisible by 3.

prove that in any primitive pythagorean triple (a,b,c), abc is a
multiple of 30

In class we proved that if (x, y, z) is a primitive Pythagorean
triple, then (switching x and y if necessary) it must be that (x,
y, z) = (m2 − n 2 , 2mn, m2 + n 2 ) for some positive integers m
and n satisfying m > n, gcd(m, n) = 1, and either m or n is
even. In this question you will prove that the converse is true: if
m and n are integers satisfying...

Let a, b, c be natural numbers. We say that (a, b, c) is a
Pythagorean triple, if a2 + b2 = c2 . For example, (3, 4, 5) is a
Pythagorean triple. For the next exercises, assume that (a, b, c)
is a Pythagorean triple.
(c) Prove that 4|ab Hint: use the previous result, and a proof
by con- tradiction.
(d) Prove that 3|ab. Hint: use a proof by contradiction.
(e) Prove that 12 |ab. Hint : Use the...

The table of exercise 1 suggests that one of the numbers in any
primitive Pythagorean triple is divisible by 4, one is divisible by
3 and one is divisible by 5. Please Prove this

(a) Prove that if y = 4k for k ≥ 1, then there exists a
primitive Pythagorean triple (x, y, z) containing y.
(b) Prove that if x = 2k+1 is any odd positive integer greater
than 1, then there exists a primitive Pythagorean triple (x, y, z)
containing x.
(c) Find primitive Pythagorean triples (x, y, z) for each of z =
25, 65, 85. Then show that there is no primitive Pythagorean triple
(x, y, z) with z...

4. Let a, b, c be integers.
(a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) =
1. (Hint: use the GCD characterization theorem.)
(b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) =
1. (Hint: you can use the GCD characterization theorem again but
you may need to multiply equations.)
(c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if
and...

Prove: If (a, a+1, a+2) is a Pythagorean triple, then a= 3

Let b be a primitive root for the odd prime p. Prove that b^k is
a primitive root for p if and only if gcd(k, p − 1) = 1.

Prove that if gcd(a,b)=1 and c|(a+b), then
gcd(a,c)=gcd(b,c)=1.

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