Timeline for Proof of "if $a^2 + b^2 = c^2$ then $abc$ is divisible by 60"
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 26, 2016 at 18:08 | comment | added | Qiaochu Yuan | @Poincare: you're right. My apologies. | |
| Jul 26, 2016 at 6:41 | comment | added | Poincare-Lelong | I dont see how just congruence modulo 4 would work. What if $b,c$ are odd (so $b^2,c^2$ congruent to 1 mod 4), but $a$ congruent to 2 mod 4. This will satisfy $a^2+b^2=c^2$ mod 4, but $abc$ is not divisible by 4. In this case I think one needs to go modulo 8. Since $b$ and $c$ are odd, $b^2,c^2$ are congruent to 1 mod 8, so $a^2$ is divisible by 8, and hence by 16. This forces a to be divisible by 4. Or am I missing something simple? | |
| Feb 28, 2010 at 8:46 | comment | added | Pete L. Clark | +1 for 3. especially. This is one of my favorite problems to assign to students who are learning about congruences. The conclusion is interesting, and once you realize you can just check this mod 3,4,5, it becomes a straightforward computation (which, as QY says, will teach you the significance of knowing the squares modulo N). Using the parameterization of all Pythagorean triples seems like overkill. | |
| Nov 6, 2009 at 1:26 | comment | added | Qiaochu Yuan | I've elaborated my answers; hopefully they're clearer now. | |
| Nov 6, 2009 at 1:25 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
Clarified the arguments.
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| Nov 6, 2009 at 1:24 | comment | added | Qiaochu Yuan | Sure, they're pretty easy to discover and have probably been rediscovered countless times. In addition to the nice geometric solution, there is a way to do it using what are called the Gaussian integers: en.wikipedia.org/wiki/Gaussian_integer | |
| Nov 6, 2009 at 1:02 | vote | accept | mailsuite | ||
| Nov 6, 2009 at 1:02 | comment | added | mailsuite | Thanks for reply. I'm trying to figure out your answers (2 and 3). Btw, do you think we can come up with the functions ourselves if we don't know that the Babylonians knew them already? | |
| Nov 6, 2009 at 0:49 | comment | added | Harald Hanche-Olsen | (A side comment: I much prefer the acronym WOLOG because WLOG could be taken to mean “with loss of generality”. There is presedence: Some people use “wo” as shorthand for “without”.) | |
| Nov 6, 2009 at 0:24 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |