All prime Pythagorean triple

May 2015
516
32
Arlington, VA
Does there exist a Pythagorean triple consisting only of prime numbers?
 
Nov 2010
174
35
Berkeley, CA
Hint: At least one term of a Pythagorean triple must be even.
 
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May 2016
1,310
551
USA
Hint: At least one term of a Pythagorean triple must be even.
$k,\ m,\ n \in \mathbb N^+.$

$(2k- 1)^2 + (2m - 1)^2 = (2n - 1)^2 \implies$

$4k^2 - 4k + 1 + 4m^2 - 4m + 1 = 4n^2 - 4n + 1 \implies$

$1 + 1 - 1 = 1 = 4(n^2 - n - k^2 - m^2 + k + m) \implies$

$\dfrac{1}{4} = n^2 - n - k^2 - m^2 + k + m \implies$

$k \not \in \mathbb N^+ \text { or } m \not \in \mathbb N^+ \text { or } n \not \in \mathbb N^+.$
 
Nov 2010
174
35
Berkeley, CA
$k,\ m,\ n \in \mathbb N^+.$

$(2k- 1)^2 + (2m - 1)^2 = (2n - 1)^2 \implies$

$4k^2 - 4k + 1 + 4m^2 - 4m + 1 = 4n^2 - 4n + 1 \implies$

$1 + 1 - 1 = 1 = 4(n^2 - n - k^2 - m^2 + k + m) \implies$

$\dfrac{1}{4} = n^2 - n - k^2 - m^2 + k + m \implies$

$k \not \in \mathbb N^+ \text { or } m \not \in \mathbb N^+ \text { or } n \not \in \mathbb N^+.$
This argument appears to be correct. However, more simply, if {a, b, c} is a Pythagorean triple all of whose terms are odd, then $a^2+b^2 = c^2$ implies that odd + odd = odd, an obvious contradiction.