1. ## Pythagorean triples

I have been trying to figure out this problem for hours with no luck.

There exist infinitely many primitive Pythagorean triples in which one of the legs is 8 (which means the difference between the other, longer, leg and hypotenuse is 8). Give the proof for this theorem and the three first triples, those with the shortest hypotenuse.

2. ## Re: Pythagorean triples

Originally Posted by Mischa
I have been trying to figure out this problem for hours with no luck.

There exist infinitely many primitive Pythagorean triples in which one of the legs is 8 (which means the difference between the other, longer, leg and hypotenuse is 8). Give the proof for this theorem and the three first triples, those with the shortest hypotenuse.

If x^2+y^2=z^2

2|x and gcd(x,y,z)=1

then:

x=2st, y=s^2-t^2, z=s^2+t^2

3. ## Re: Pythagorean triples

Originally Posted by Mischa
I have been trying to figure out this problem for hours with no luck.

There exist infinitely many primitive Pythagorean triples in which one of the legs is 8 (which means the difference between the other, longer, leg and hypotenuse is 8).

Which means, I'd say, that the difference between the squares of the hypotenuse and the other leg is $8^2=64$

Tonio

Give the proof for this theorem and the three first triples, those with the shortest hypotenuse.
.

4. ## Re: Pythagorean triples

Originally Posted by Also sprach Zarathustra
If x^2+y^2=z^2

2|x and gcd(x,y,z)=1

then:

x=2st, y=s^2-t^2, z=s^2+t^2

How does this prove what the OP asked?

Tonio

5. ## Re: Pythagorean triples

Originally Posted by tonio
.

Suppose $a^2+8^2=b^2 \iff 64=(b+a)(b-a)\Longrightarrow a+b<64$ , and as we're talking about natural numbers there can only be

a finite number of such triples, so: what exactly did you try to prove?

Tonio

6. ## Re: Pythagorean triples

naturally i was curious, just how many such triples could we find? it was clear to me that b+a, being the larger of the 2 factors of 64, had to be 16 or 32.

b+a = 16 --> a = 6, b = 10. this is not primitive.

b+a = 32 --> a = 15, b = 17.

so it would appear there is but one such primitive triple: (8,15,17).

since the original question asks for the first 3 such triples, i suspect we are not answering the same question the OP is trying to ask.

7. ## Re: Pythagorean triples

x=2st, y=s^2-t^2, z=s^2+t^2

Is there other triples than 8,15,17? I could not find other with this theorem.

8. ## Re: Pythagorean triples

Originally Posted by Mischa
x=2st, y=s^2-t^2, z=s^2+t^2

Is there other triples than 8,15,17? I could not find other with this theorem.

DIdn't you read/understand what I and Deveno wrote? And what "theorem" are you talking about?

Tonio