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I have been mucking about with numbers for a while and would like to know if a Pythagorean triangle with the values of 1^2+2^2=3^2 is known?
The solution I have found is in Fermats Last Theroem x^n+y^n=z^n. It only works in whole numbers when n<3.
I have scanned and the smallest appears to be a 3, 4 & 5 triangle.
Does anybody know anything about an even smaller 1, 2 & 3 Pythagorean triangle?
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'Everything is impossible until it's not.' Cpt. JL Picard
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The solution I have found is in Fermats Last Theroem x^n+y^n=z^n. It only works in whole numbers when n<3.
FLT Proves the only whole number solutions to x^n+y^n=z^n is when n<3. The only equations that work in whole numbers are the 5 below.
x^-2+y^-2=z^-2
x^-1+y^-1=z^-1
x^0+y^0=z^0
x^1+y^1=z^1
x^2+y^2=z^2
123
456
789
3^2 = 3*3 = 9 Squares.
3^1=9 Triangles.
1
234
56789
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'Everything is impossible until it's not.' Cpt. JL Picard
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x^2+y^2=z^2
As this is the case 1^2+2^2=3^2 must work.
Or did Fermat believe the 1,2,3 triangle had no square relationship, and name the equation x,y&z instead of a,b&c?
Any comments?
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'Everything is impossible until it's not.' Cpt. JL Picard
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1^2+2^2=1+4=5
3^2=9
5 does not equal 9.
You also can't have a 1,2,3 triangle at all since the 3 side is as long as both the other sides combined.
A 3,4,5 triangle is the smallest pythagorean triangle.
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1^2+2^2=1+4=5
3^2=9
5 does not equal 9.You also can't have a 1,2,3 triangle at all since the 3 side is as long as both the other sides combined.
A 3,4,5 triangle is the smallest pythagorean triangle.
Yes I know.
But when it is treated abstractly I can see a smaller 1,2&3 solution.
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'Everything is impossible until it's not.' Cpt. JL Picard
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If you divide the sides of the 345 triangle by the the same number for all the sides it will become a smaller triangle. There are infinate combinations!
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5 does not equal 9.
I have concluded in my studies that in the equation x^n+y^n=z^n, n has only 5 possible variants in whole numbers. I have organized them into tetrahedral shapes.
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'Everything is impossible until it's not.' Cpt. JL Picard
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If you divide the sides of the 345 triangle by the the same number for all the sides it will become a smaller triangle.
What happens when a line is so small that it's the size of the smallest atom and cannot be divided anymore without changing the measurement to a different sub-atomic scale, which will in turn have a smallest scale of line.
The 3^2 in the smallest 3,4,5 Pythagorean triangle is too big. I think at this scale Pythagorus' theroem would not work, unless a 1,2&3 triangle exists. A series of hypotenuses would become more apparent.
With the switch of thickness to FLT of x^n+y^n=z^n it has an altered concept.
The double equilateral triangle unfolds to a square.
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'Everything is impossible until it's not.' Cpt. JL Picard
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On the scale of things an atom is huge?
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On the scale of things an atom is huge?
The Hydogen atom is a unit that is small relative to us, but sets our scale of life.
Just as the speed of light sets the scale of vision.
In the experiments and math I am involved in, The ^n function is seen as an increase in thickness.
The function ^n in FLT of x^n+y^n = z^n there is a triangular solution ^1 as well as a square solution.
I understand the 3.4.5. Pythagorean Triangle square hypothosis differently. This whole number right angled triangle tells me that an equilateral triangle of 3 lines can open to reveal 4 triangles inside, closes again making 5 triangles. When ^2, double triangles make 5 triangles two thick (10) or 5 squares.
I understand the 1.2.3 triangle to represent a stellated tetrahedron, oscillating to a cube.
[url=http://www.stargrail.co.uk]Ant[/url]
'Everything is impossible until it's not.' Cpt. JL Picard
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