# Questions tagged [pythagorean-triples]

Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.

388 questions
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### Triple Pythagorean with $a^2+b^2=c^4$

It is well known that there exist integer solutions to the equation $a^2+b^2=c^2$. For example, an explicit formula for integer values of $a$ , $b$ , and $c$ is \begin{align}a&=2mn \\ b&=m^2-...
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### Odd Pythagorian triplets [closed]

How many Pythagorean triplets $\{a,b,c\}$ exist, where $a,b,c$ are all odd? As far as I know there are no such triplets. $\{3, 4, 5\}; \{5,12,13\} ; \{7,24,25\}$ and its multiples are examples. Is ...
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How can we obtain a set of nontrivial solutions of $$a^3+b^3=c^3+d^3,$$ for $a,b,c,d\in \mathbb{Z}$ where $(a,b)\neq (c,d)$ and $(a,b)\neq (d,c)$. Say in the range that $|a|,|b|,|c|,|d| \in [... 1answer 33 views ### Explain this n-modulus relation in Pythagorean triplets {a,a+n,c} I've found a seemingly consistent Pythagorean triplet relation for which I cannot find a proof or counterexample. Theorem: For all Pythagorean triplets {a, b, c} where$gcd(a,b)=1$, b-a is odd and 2 ... 1answer 43 views ### When can an odd integer$d$be represented as$d=a^2-2b^2$with coprime integers$a,b\ $? I found out that in a primitive pythagorean triple $$a^2+b^2=c^2$$ the difference$d=|a-b|$(which must be odd) can occur, if and only if we can write $$d=a^2-2b^2$$ with positive coprime integers$a,...
Why do the ratios of successive values of integers $a$ and $c$, where $a^{2}+(a+1)^{2}=c^{2}$, appear to converge to $$\frac{a_{n+1}}{a_{n}},\frac{c_{n+1}}{c_{n}}\rightarrow3+2\sqrt{2}$$ I rigorously ...