Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
How many ways can 1000 be written as the sum of two squares?
List the unique solutions to 1000 = a^2 + b^2, where a and b are positive integers. Prove that that's all of them.
Hint: Some of the questions I've asked recently.
The first two answers are both incorrect. In fact, the first one doesn't even make any sense.
So is that all of them, BH? If so, can you prove it?
Moshi,
That's absolutely the right idea, but you got a bit sloppy.
Wang,
That definitely works, but you're only getting best answer if nobody actually writes out a more complete proof than "Well, I did it by trial and error, and trust me -- this is the answer." :)
5 Answers
- wangsaclLv 41 decade agoFavorite Answer
Using the Pigeonhole Principle, we get that one of a^2 and b^2 should be less than 500.
wlog, a<b.
So, a^2 < 500, a < 22.36 Since a is positive integer, a <=22.
Using the trial and error for interval between 1<=a<=22,
We get (a,b) = (10,30), (18,26)
Since we could reverse them,
(a,b) = (10,30), (18,26), (26,18), (30,10)
This is all of them, because we use trial and error.
- 1 decade ago
Write
a^2+b^2=(a+bi)(a-bi)
and
1000 = 2^3*5^3 = (1+i)^3(2+i)^3(1-i)^3(2-i)^3
By unique prime factorization Z[i] and because a+bi and a-bi are conjugates, the only possibilities for a+bi are(up to multiplication by units which doesn't change anything)
(1+i)^3(2+i)^3
(1+i)^3(2+i)^2(2-i)
(1+i)^2(1-i)(2+i)^3
(1+i)^2(1-i)(2+i)^2(2-i)
when taken account of repeats. Thus, we have 4 different solutions.(given that (a,b) is counted as an ordered pair)
- How do you think about the answers? You can sign in to vote the answer.
