wonderings

Thread Rating:
  • 6 Vote(s) - 2.5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
wonderings
#38
RE: wonderings
moreover we may exactly classify all numbers that can be written as a sum of two squares

lemma: Let p = 3 mod 4 and prime

if a^2+b^2 = 0 mod p then either a,b = 0 mod p (this is one solution) or we can divide by b (b is coprime to p so we can do this):

(a/b)^2=-1 mod p. So (a/b)^4=1 mod p, and thus the order of a/b in Z/pZ is 4. but wait, the cardinality of Z/pZ is p-1. and this is 2 mod 4. but by Lagrange's Theorem all element's order divide |Z/pZ| = p-1 so this is impossible!

therefore a,b = 0 mod p is the only solution.

(yeah ok we didn't really need group theory there, quadratic reciprocity does the trick too.)

anyway the punchline is that numbers with an odd number of any 3 mod 4 prime factor cannot be written as the sum of two squares. some minor details left out but you get the idea

(this is a necessary, and as it turns out, sufficient condition - what remains is to be able to show that all primes 1 mod 4 can be expressed as the sum of two squares, and you can use quadratic reciprocity)
Quote


Messages In This Thread
wonderings - by a52 - 11-27-2016, 09:11 PM
RE: wonderings - by a52 - 11-27-2016, 09:15 PM
RE: wonderings - by a52 - 11-27-2016, 09:17 PM
RE: wonderings - by a52 - 11-27-2016, 09:27 PM
RE: wonderings - by a52 - 11-27-2016, 09:45 PM
RE: wonderings - by a52 - 11-27-2016, 09:59 PM
RE: wonderings - by OTTO - 11-28-2016, 06:39 AM
RE: wonderings - by a52 - 11-28-2016, 06:59 AM
RE: wonderings - by Dragon Fogel - 11-27-2016, 10:07 PM
RE: wonderings - by Our Lady of Lampreys - 12-20-2016, 03:34 AM
RE: wonderings - by a52 - 11-27-2016, 10:10 PM
RE: wonderings - by a52 - 11-27-2016, 10:14 PM
RE: wonderings - by a52 - 12-18-2016, 08:30 PM
RE: wonderings - by Dragon Fogel - 12-20-2016, 04:05 AM
RE: wonderings - by a52 - 12-20-2016, 07:18 AM
RE: wonderings - by Our Lady of Lampreys - 12-20-2016, 07:42 PM
RE: wonderings - by a52 - 12-24-2016, 07:09 PM
RE: wonderings - by Kaynato - 12-24-2016, 07:21 PM
RE: wonderings - by a52 - 12-24-2016, 08:31 PM
RE: wonderings - by Kaynato - 12-24-2016, 10:42 PM
RE: wonderings - by a52 - 01-08-2017, 05:11 PM
RE: wonderings - by Sleepy - 01-10-2017, 09:37 PM
RE: wonderings - by ICan'tGiveCredit - 01-10-2017, 10:16 PM
RE: wonderings - by a52 - 01-13-2017, 05:34 AM
RE: wonderings - by a52 - 02-03-2017, 03:15 AM
RE: wonderings - by ☆ C.H.W.O.K.A ☆ - 02-06-2017, 09:10 PM
RE: wonderings - by a52 - 02-03-2017, 03:16 AM
RE: wonderings - by a52 - 02-03-2017, 03:38 AM
RE: wonderings - by a52 - 02-03-2017, 04:01 AM
RE: wonderings - by a52 - 03-16-2017, 01:44 AM
RE: wonderings - by a52 - 03-16-2017, 04:40 AM
RE: wonderings - by a52 - 03-16-2017, 01:47 AM
RE: wonderings - by a52 - 03-16-2017, 01:53 AM
RE: wonderings - by Kíeros - 03-16-2017, 03:16 AM
RE: wonderings - by a52 - 04-06-2017, 04:52 PM
RE: wonderings - by a52 - 05-27-2017, 02:18 AM
RE: wonderings - by qwerx3 - 05-27-2017, 08:26 AM
RE: wonderings - by qwerx3 - 05-27-2017, 08:38 AM
RE: wonderings - by a52 - 06-04-2017, 05:48 AM
RE: wonderings - by ICan'tGiveCredit - 06-04-2017, 02:23 PM
RE: wonderings - by a52 - 06-09-2017, 01:48 AM
RE: wonderings - by a52 - 08-09-2017, 08:16 PM
RE: wonderings - by ICan'tGiveCredit - 08-09-2017, 09:03 PM
RE: wonderings - by a52 - 08-10-2017, 03:03 AM
RE: wonderings - by a52 - 08-14-2017, 10:06 PM
RE: wonderings - by a52 - 08-14-2017, 10:30 PM
RE: wonderings - by wyatt - 08-21-2017, 03:46 PM
RE: wonderings - by Reyweld - 08-21-2017, 04:08 PM
RE: wonderings - by a52 - 09-23-2017, 08:41 PM
RE: wonderings - by a52 - 12-11-2017, 01:17 AM
RE: wonderings - by Our Lady of Lampreys - 12-12-2017, 07:04 PM
RE: wonderings - by Myeth - 12-12-2017, 10:30 PM
RE: wonderings - by Reyweld - 12-12-2017, 10:54 PM
RE: wonderings - by Myeth - 12-13-2017, 12:22 AM
RE: wonderings - by a52 - 02-23-2018, 08:56 AM
RE: wonderings - by a52 - 04-23-2018, 06:23 PM
RE: wonderings - by a52 - 05-04-2018, 08:07 AM
RE: wonderings - by a52 - 05-04-2018, 08:14 AM
RE: wonderings - by Myeth - 05-04-2018, 08:31 AM
RE: wonderings - by a52 - 05-05-2018, 02:19 AM
RE: wonderings - by ICan'tGiveCredit - 05-05-2018, 01:42 PM
RE: wonderings - by a52 - 10-22-2018, 09:08 PM
RE: wonderings - by a52 - 10-22-2018, 09:16 PM
RE: wonderings - by a52 - 05-21-2019, 03:07 AM