Timeline for How to find integer solution of this equation?
Current License: CC BY-SA 4.0
8 events
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| Jul 12, 2022 at 1:49 | vote | accept | minhthien_2016 | ||
| Jun 21, 2022 at 19:54 | comment | added | Daniel Lichtblau |
By the way, you don't really need Solve, just some algebra on paper (or in a notebook). Rewrite the equation as (x + y - 1) ((5 x + y)^3 - x y^3)==0. The constraints mean the first factor cannot vanish, so we have (5 x + y)^3 = x y^3. Take cube roots (these are real-valued) to get 5 x + y=x^(1/3)*y. Rearrange as y=5x/(x^(1/3)-1).
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| Jun 21, 2022 at 14:49 | history | edited | Akku14 | CC BY-SA 4.0 |
deleted 36 characters in body
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| Jun 21, 2022 at 0:02 | comment | added | Daniel Lichtblau |
From the equation y == (5 x)/(-1 + x^(1/3)) you know x is a cube of an integer, call it x=t^3. So the numerator is 5*t^3 and the denominator is t-1. This that has no factors in common with t^3 so it must divide 5. This means either t=2 or t=6, thus x is 2^3=8 or 6^3=216. Unless I missed something, this exhausts all cases.
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| Jun 20, 2022 at 20:08 | history | edited | Akku14 | CC BY-SA 4.0 |
added 214 characters in body
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| Jun 20, 2022 at 20:03 | history | undeleted | Akku14 | ||
| Jun 20, 2022 at 19:39 | history | deleted | Akku14 | via Vote | |
| Jun 20, 2022 at 11:18 | history | answered | Akku14 | CC BY-SA 4.0 |