Question

- may exist pythagora's triplet formed by primes ?
,,pythagora's triplet same with 3-4-5
5^2 =3^2 +4^2

- or allways need contain,need being one even between ?

Answer 1

Here's an interesting rule:

for a Pythagorean triad a^2 + b^2 = c^2 where a < b < c
you will find that b+c = a^2 if a is odd and 2(b+c) = a^2 if a is even. This can be a very interesting way to generate new triads.

3, 4, 5 --> 4+5 = 3^2
5, 12, 13 --> 12 + 13 = 5^2
7, 24, 25 --> 24 + 25 = 7^2
8, 15, 17 --> 2(15+ 17) = 8^2
9, 40, 41 --> 40 + 41 = 9^2

So the odd starting triads also have the property that c = b+1
Thus 2b + 1 = a^2.
Now having established a (and thus a^2 is odd, this implies b must then be even), so your triad must have one even number. Similarly given c = a^2 - b = ODD - EVEN, you'll have two odd numbers and one even in your triad.

If you start off with even triad, then 2(b+c) = a^2 and c = b+2
so 2(2b+2) = a^2 = 4(b+1) hence b must be odd and also c.

So it holds all Pythagorean triads have two odd and one even component.

Answer 2

@jhonyy9
unfortunately, there is always an even number >2, so cannot have a triad of all primes.
Why always an even number?
We know that odd^2=odd, even^2=even
So if the triad is a^2+b^2=c^2, then
odd^2+odd^2=odd+odd=even=even^2.
and of course
odd^2+even^2=odd+even=odd=odd^2.
so there, no all prime triads!

Answer 3

all triples are of the form \[m^2-n^2, 2mn, m^2+n^2\]
primitive if \(gcf(m,n)=1\) and opposite parity
so in fact not only is one even, it is divisible by 4

Answer 4

in fact, if \(a,b,c\) is a primitive triple, one is divisible by 3, one is divisible by 4 and one is divisible by 5
not necessarily distinct ones

Answer 5

@misty1212
Beautiful observation, and analysis. Thank you!

so @jhonyy9 referring to previous post
the closest as far as primes go, the best we can do is two primes out of three!
"not necessarily distinct" means 3 and 4 can be divisors of the same member of the triad, as in 5, 12, 13.

Answer 6

"not necessarily distinct" means, for example, 3 and 4 can be divisors of the same member of the triad, like 12 in (5, 12, 13), or
3 and 5 can be divisors of 15 in (8,15,17), while 4 is a divisor of 8.