Number theroy: Pythagorean Triple
- Thread starter cathwelch
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Hello, cathwelch!
\(\displaystyle \text{We know that: }\;a^2+b^2 \:=\:c^2\;\;[1]\)
\(\displaystyle \text{Since }a,b,c\text{ are consecutive integers: }\;\begin{Bmatrix}a &=& a \\ b &=& a+1 \\ c &=& a+2 \end{Bmatrix}\)
\(\displaystyle \text{Substitute into [1]: }\;a^2 + (a+1)^2 \:=\
a+2)^2\)
. . \(\displaystyle \text{which simplifies to: }\;a^2-2a-3 \:=\:0\)
. . \(\displaystyle \text{which factors: }\;(a-3)(a+1) \:=\:0\)
. . \(\displaystyle \text{and has roots: }\;a \;=\;3,\:-1\)
\(\displaystyle \text{Since }a,b,c\text{ are }positive\text{ integers, }a = 3 \text{ is the only solution.}\)
\(\displaystyle \text{Therefore, }(a,b,c) = (3,4,5)\text{ is the only Pythagorean triple with consecutive integers.}\)
\(\displaystyle \text{We know that: }\;a^2+b^2 \:=\:c^2\;\;[1]\)
\(\displaystyle \text{Since }a,b,c\text{ are consecutive integers: }\;\begin{Bmatrix}a &=& a \\ b &=& a+1 \\ c &=& a+2 \end{Bmatrix}\)
\(\displaystyle \text{Substitute into [1]: }\;a^2 + (a+1)^2 \:=\
a+2)^2\). . \(\displaystyle \text{which simplifies to: }\;a^2-2a-3 \:=\:0\)
. . \(\displaystyle \text{which factors: }\;(a-3)(a+1) \:=\:0\)
. . \(\displaystyle \text{and has roots: }\;a \;=\;3,\:-1\)
\(\displaystyle \text{Since }a,b,c\text{ are }positive\text{ integers, }a = 3 \text{ is the only solution.}\)
\(\displaystyle \text{Therefore, }(a,b,c) = (3,4,5)\text{ is the only Pythagorean triple with consecutive integers.}\)