I Euclid's Formula as a test for sufficiency

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If any one side of a triangle cannot be derived from Euclid’s formula for pythagorean triples, is this sufficient to prove that a right triangle with integer sides is impossible?

For example, lets take the leg expressed by k2mn in Euclid's formula,, where k,m,n, are integers. If one of the sides of a triangle is expressed by prime and non-integer factors that do not conform to k2mn, is this sufficient to prove that the other remaining sides will never form a right triangle with integer sides?
 
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is this sufficient to prove that a right triangle with integer sides is impossible?
They exist, you can't prove that they are impossible.

If none of the squares of a side is the sum of squares of the other two sides then this particular triangle won't be a right triangle.
If one of the sides of a triangle is expressed by prime and non-integer factors that do not conform to k2mn, is this sufficient to prove that the other remaining sides will never form a right triangle with integer sides?
How do you know the side is not one of the other two sides in Euclid's formula? 3 and 5 cannot be written as 2mn, but clearly both can be in such a triangle (3,4,5).
 
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They exist, you can't prove that they are impossible.
Maybe I don't understand what you are implying. Are you saying that right triangles with integer sides are possible that do not fall under the net of Euclid's formula?
 
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Are you saying that right triangles with integer sides are possible that do not fall under the net of Euclid's formula?
No, the formula covers all, but your first post had a different question.
 
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Ok. Let me restate my inquiry. Suppose we have two line segments and it is given that they each touch at one end, and that they are perpendicular to each. Suppose one line segment can be expressed by 2mnk, where m > n and m,n, and k are integers. Suppose the second line segment can be expressed by m^sq - n^sq. Does it follow by citing Euclid's formula that a line segment connecting the open ends of the two segmens must be equal to m^2 + n^2?

And if this is true, can we assert the following. Suppose the first line segment equals 2mnk as before, but the second line segment does not equal to m^2 - n^2, but some other integer value not equal to m^2 - n^2. Does it follow from Euclid's formula that a line segment connecting the open ends of these two segments can never be an integer?
 
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sq=2?
From Pythagoras we know the third line will have a squared length that is the sum of the other two squared lengths. ##(2mnk)^2 + (k(m^2-n^2))^2 = k^2 (m^2+n^2)^2## which is a square. You missed the k for two lengths.
And if this is true, can we assert the following. Suppose the first line segment equals 2mnk as before, but the second line segment does not equal to m^2 - n^2, but some other integer value not equal to m^2 - n^2. Does it follow from Euclid's formula that a line segment connecting the open ends of these two segments can never be an integer?
That depends on how you define m and n. For a given first line you can find m,n such that the second line doesn't have length k(m^2 - n^2) even if it is part of a triple with integer side lengths.
As an example, consider the triangle 3,4,5. Clearly 4 can be expressed as 2mnk where k=2, m=1, n=1. The the second side (3) is not equal to k(m^2 - n^2), but you still have a triangle with integer side lengths. You (well, I did) just made a poor choice of m,n,k. If there is no combination of k,n,m such that the two sides can be expressed in this way then the third side length is not an integer.
 
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Suppose the second line segment can be expressed by m^sq - n^sq.
Do you mean ##m^2 - n^2##? If so, what you wrote is an unusual and very confusing way to write this.
 

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