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I have this equation:

$3^x + 4^x = 5^x$, according to fermat theorem, $x <= 2$, so the answer is $2$.

But how can I get to the result through an elementary algebraic procedure?

I have tried many things, but I have not come up with any concrete results, arriving only at:

First try, $log(3^x+4^x) = 3log5$

Second try, $(\frac{3}{5})^x + (\frac{4}{5})^x = 1$

I have also tried to search several programs, such as symbolab, mathway, etc. But, they have not been able to solve it, the only program that has given me an answer is wolframAlpha, but I can not visualize it step by step, thanks in advance.

ESCM
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  • i'm not sure, but i don't think this is solvable using elementary algebra... – tc216 Jul 03 '18 at 22:28
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    Observe that $\left(\frac{3}{5}\right)^x + \left(\frac{4}{5}\right)^x$ is a decreasing function of $x$. So there must be only one solution, which luckily turned out to be an integer, i.e., $x=2$. – Math Lover Jul 03 '18 at 22:36
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    See also 1, 2, 3. – dxiv Jul 03 '18 at 22:40
  • I don't want to prove fermat theorem, i want to get the $x$. I don't think that is a duplicate – ESCM Jul 03 '18 at 22:42
  • The Fermat theorem you mean refers only to integer exponents. So: how would you solve $$ 98^x + 99^x = 100^x ; $$ with the only restriction being that $x$ be a real number? – Will Jagy Jul 03 '18 at 23:02
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    oh, well. $x$ is between 47 and 48. – Will Jagy Jul 03 '18 at 23:09
  • @Will Jagy Newton's method. – Phil H Jul 03 '18 at 23:21
  • How you know that $x$ is between $[47, 48]$ ? – ESCM Jul 03 '18 at 23:25
  • BETTER: solve $$8^x + 9^x = 10^x ; .$$ Well, $8^4 + 9^4 = 4096+6561= 10657 > 10000 = 10^4.$ Next, $8^5 + 9^5 = 32768 +59049 = 91817 < 100000 = 10^5.$ Therefore $x$ is between $4$ and $5.$ My calculator says $x \approx 4.424392331.$ Also $8^{4.424392331} \approx 9899.754222 ; , $ then $9^{4.424392331} \approx 16670.29329 ; , $ sum of the two $26570.04751 ; .$ Then it says $10^{4.424392331} \approx 26570.04751 ; . $ So this is probably a pretty good numerical solution. – Will Jagy Jul 04 '18 at 20:58
  • Put the other way, my calculator says $(0.8)^{4.424392331} \approx 0.372590761$ and $(0.9)^{4.424392331} \approx 0.627409239 ; .$ It says the sum is $1.000000000$ – Will Jagy Jul 04 '18 at 21:21

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