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gaussints - Gaussian Integers NumThy Jonathan L.F King...

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Gaussian Integers : NumThy Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 20 October, 2011 (at 11:40 ) ( A natnum N is a SOTS , Sum Of Two Squares, if there are integers for which 2 + k 2 = N . If there exists such a pair with k , then N is coprime-SOTS . ( E.g, 25 has a non-coprime rep as 5 2 + 0 2 ; nonetheless, 25 is coprime-SOTS since 25 = 3 2 +4 2 . OTOHand, both 4 = 0 2 +2 2 and 40 = 2 2 +6 2 have these unique SOTS reps, so neither is coprime-SOTS. ) An odd integer L is 4Neg if L 4 1 and is 4Pos if L 4 1. Fermat’s Prime-SOTS Thm says: Oddprime p is SOTS iff p is 4 Pos . Mod N , a rono is a ( square ) Root Of Negative One; an integer I such that I 2 N 1. Use CRT for Chinese Remainder Thm . ) The ring. Let G := { b + c i | b, c Z } be the set of Gaussian integers , a subring of C . The norm of a gaussint B := b + c i is N ( B ) := B · B = b 2 + c 2 . To set notation, I will henceforth use B := b + c i , E := e + f i , and S := s + t i for gaussints. I will use β := N ( B ) , ε := N ( E ) , and σ := N ( S ) for their norms. A number in G or Z which is neither zero nor a unit will be called non-trivial . A gaussint S lying on the real or imaginary axis is said to be axial , i.e either s = 0 or t = 0. Lastly, use 0 for the complex number 0 + 0 i .
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