Compare $A$ and $B$ with: $$A = \sqrt{2017} + \sqrt{2019} + \sqrt{2023}$$ $$B = \sqrt{2018} + \sqrt{2020} + \sqrt{2021}$$ I tried to prove $A^4 < B^4$ but it's too hard to do that.
2026-03-25 05:59:05.1774418345
Compare $A$ and $B$
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1
Well its enough to prove $$A < B$$ since taking power to a positive number is operation that does not change sign. Easy to prove seems $$A^2 < B^2$$ just by doing all of that products.
Quick guess without calculator would be also to use Taylor series $$\sqrt{2017 + x} \approx \sqrt{2017} + \frac{x}{2\sqrt{2017}} - \frac{1}{8}\frac{x^2}{2017^{3/2}} $$ so only third term makes difference and you effectively want to compare $x^2$ $$2^2+6^2 > 1^2 + 3^2 + 4^2$$ so that $$40 > 26$$ seems to hold.