Modular equation
in [Math]
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From: joe.doubtful on 5 Apr 2010 16:12 Hi, consider the modular equation s^4 = -4 mod q, where q=3 mod 4. Does anyone knows how to prove that it has no solutions? Thanks in advance
From: joe.doubtful on 5 Apr 2010 16:38 On 5 Apr, 22:12, "joe.doubtful" <joe.doubt...(a)yahoo.it> wrote: > Hi, consider the modular equation s^4 = -4 mod q, where q=3 mod 4. > Does anyone knows how to prove that it has no solutions? > Thanks in advance I forgot: q is an odd prime.
From: bert on 5 Apr 2010 17:00 On 5 Apr, 21:38, "joe.doubtful" <joe.doubt...(a)yahoo.it> wrote: > On 5 Apr, 22:12, "joe.doubtful" <joe.doubt...(a)yahoo.it> wrote: > > > Hi, consider the modular equation s^4 = -4 mod q, where q=3 mod 4. > > Does anyone knows how to prove that it has no solutions? > > Thanks in advance > > I forgot: q is an odd prime. Let t = s^2 / 2, then t^2 = -1 mod q, so what you want is the existing and well-known proof that -1 is a quadratic residue of only those primes of the form 4n+1, not of primes of the form 4n+3. --
From: joe.doubtful on 5 Apr 2010 19:14 On 5 Apr, 23:00, bert <bert.hutchi...(a)btinternet.com> wrote: > On 5 Apr, 21:38, "joe.doubtful" <joe.doubt...(a)yahoo.it> wrote: > > > On 5 Apr, 22:12, "joe.doubtful" <joe.doubt...(a)yahoo.it> wrote: > > > > Hi, consider the modular equation s^4 = -4 mod q, where q=3 mod 4. > > > Does anyone knows how to prove that it has no solutions? > > > Thanks in advance > > > I forgot: q is an odd prime. > > Let t = s^2 / 2, then t^2 = -1 mod q, > so what you want is the existing and > well-known proof that -1 is a quadratic > residue of only those primes of the form > 4n+1, not of primes of the form 4n+3. > -- Interesting, I didn't know that result. A last question: how do you know that s is even?
From: Rob Johnson on 5 Apr 2010 20:19
In article <9f85ba15-8ba5-425e-8d3f-8f22a5f03e20(a)11g2000yqr.googlegroups.com>, "joe.doubtful" <joe.doubtful(a)yahoo.it> wrote: >On 5 Apr, 23:00, bert <bert.hutchi...(a)btinternet.com> wrote: >> On 5 Apr, 21:38, "joe.doubtful" <joe.doubt...(a)yahoo.it> wrote: >> >> > On 5 Apr, 22:12, "joe.doubtful" <joe.doubt...(a)yahoo.it> wrote: >> >> > > Hi, consider the modular equation s^4 = -4 mod q, where q=3 mod 4. >> > > Does anyone knows how to prove that it has no solutions? >> > > Thanks in advance >> >> > I forgot: q is an odd prime. >> >> Let t = s^2 / 2, then t^2 = -1 mod q, >> so what you want is the existing and >> well-known proof that -1 is a quadratic >> residue of only those primes of the form >> 4n+1, not of primes of the form 4n+3. >> -- > >Interesting, I didn't know that result. A last question: how do you >know that s is even? It doesn't matter whether s is even or not. If q is odd, 2 has an inverse mod q, so s^2/2 is well-defined mod q. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font |