Why does JH Conway say -1 is prime?
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From: Richard L. Peterson on 22 Jan 2010 14:01 > On Jan 22, 8:33 pm, "Richard L. Peterson" > <rl_p...(a)yahoo.com> wrote: > > I just read this in abook by William Stein. > > > > Thanks > > I suspect a misunderstanding has crept into the > discussion. -1 is not considered a prime. However > in quadratic sieving methods for large factoring > problems, it can be advantageous to include -1 as > a "small factor" of squares modulo the composite > to be factored. > > regards, chip That might be what it is. It turns out William Stein is referring to a book called The Sensual Quadratic Form by Conway, which I have no access to. Thanks.
From: FredJeffries on 23 Jan 2010 06:10 On Jan 22, 5:33 pm, "Richard L. Peterson" <rl_p...(a)yahoo.com> wrote: > I just read this in abook by William Stein. > > Thanks He prefers to call "the prime -1" what is usually referred to in algebraic number theory / valuation theory as "the infinite prime" or "prime at infinity" when including the usual archimedean absolute value in a discussion of p-adic valuations. http://en.wikipedia.org/wiki/Infinite_prime#Primes_and_places Go to Amazon's listing for "The Sensual (Quadratic) Form" http://www.amazon.com/Sensual-Quadratic-Carus-Mathematical-Monographs/dp/0883850303/ref=ntt_at_ep_dpi_7 do a Look Inside and search for "smile indulgently".
From: JEMebius on 23 Jan 2010 10:44 FredJeffries wrote: > On Jan 22, 5:33 pm, "Richard L. Peterson" <rl_p...(a)yahoo.com> wrote: >> I just read this in abook by William Stein. >> >> Thanks > > He prefers to call "the prime -1" what is usually referred to in > algebraic number theory / valuation theory as "the infinite prime" or > "prime at infinity" when including the usual archimedean absolute > value in a discussion of p-adic valuations. > > http://en.wikipedia.org/wiki/Infinite_prime#Primes_and_places > > Go to Amazon's listing for "The Sensual (Quadratic) Form" > > http://www.amazon.com/Sensual-Quadratic-Carus-Mathematical-Monographs/dp/0883850303/ref=ntt_at_ep_dpi_7 > > do a Look Inside and search for "smile indulgently". Original post by Richard L. Peterson ( news:sci.math 2010-01-23 02:33 CET ): -------------------------------------------------------------------------------- Why does JH Conway say -1 is prime? I just read this in a book by William Stein. Thanks -------------------------------------------------------------------------------- I guess because JH Conway wanted to overlook for a moment the unit-equivalence of elements which differ only by unit factors in the UFD in question. Just a guess: I do not know William Stein's book. Consider a positive number P in Z which is a prime number in N. In factorization questions it is common to speak about the prime factor P when the complete unit-equivalence class containing P is actually meant. It makes good sense to consider the negative number -P as a prime factor in Z too, if one wants to do so. Another illustration of factorization into primes: All possible factorizations of 5 in Z[i]: 5 = (2+i)(2-i) = (1+2i)(1-2i) = (-2-i)(-2+i) = (-1-2i)(1+2i) versus the common factorization 5 = (2+i)(2-i). Ciao: Johan E. Mebius
From: Inverse 19 mathematics on 23 Jan 2010 11:19 On Jan 23, 9:44 am, JEMebius <jemeb...(a)xs4all.nl> wrote: > FredJeffries wrote: > > On Jan 22, 5:33 pm, "Richard L. Peterson" <rl_p...(a)yahoo.com> wrote: > >> I just read this in abook by William Stein. > > >> Thanks > > > He prefers to call "the prime -1" what is usually referred to in > > algebraic number theory / valuation theory as "the infinite prime" or > > "prime at infinity" when including the usual archimedean absolute > > value in a discussion of p-adic valuations. > > >http://en.wikipedia.org/wiki/Infinite_prime#Primes_and_places > > > Go to Amazon's listing for "The Sensual (Quadratic) Form" > > >http://www.amazon.com/Sensual-Quadratic-Carus-Mathematical-Monographs... > > > do a Look Inside and search for "smile indulgently". > > Original post by Richard L. Peterson ( news:sci.math 2010-01-23 02:33 CET ): > > -------------------------------------------------------------------------------- > Why does JH Conway say -1 is prime? I just read this in a book by William Stein. > Thanks > -------------------------------------------------------------------------------- > > I guess because JH Conway wanted to overlook for a moment the unit-equivalence of elements > which differ only by unit factors in the UFD in question. > Just a guess: I do not know William Stein's book. > > Consider a positive number P in Z which is a prime number in N. > In factorization questions it is common to speak about the prime factor P when the > complete unit-equivalence class containing P is actually meant. > It makes good sense to consider the negative number -P as a prime factor in Z too, if one > wants to do so. > > Another illustration of factorization into primes: > > All possible factorizations of 5 in Z[i]: > 5 = (2+i)(2-i) = (1+2i)(1-2i) = (-2-i)(-2+i) = (-1-2i)(1+2i) > versus the common factorization > 5 = (2+i)(2-i). > > Ciao: Johan E. Mebius- Hide quoted text - > > - Show quoted text - -1 is prime 1 we have been saying this for a year and have now proven it with the Paragon of 19, because your zero is wrong, the correct zero is ofset by -1, that changes all mathematics. See our desription of simple 3 or 12 colmns of numbers , the verticle gap is -1 of the horizontal gap, that is obvious look here the prime line up at column 1, 3 , 1 , 3 , 1 , 1 and the -1 is obvious to a 3rd grader 1 * 2 3 4 5 * 6 7* 8 9 10 11* 12 ( 12-1=11 --- 13-1 =12-----1 zero) 13 * 14 15 16 17* 18 19* 20 21 22 23 * 24 so on see other post we posted for your for you math genuises This is infinite , figure it out genuises! , get rid of some of your theory and think -1 inverse zero,
From: Timothy Murphy on 23 Jan 2010 11:23 JEMebius wrote: > I guess because JH Conway wanted to overlook for a moment the > unit-equivalence of elements which differ only by unit factors in the UFD > in question. Just a guess: I do not know William Stein's book. I would guess slightly differently. Dirichlet's Units Theorem says that the group of units in a number field is F x Z^{e-1} where F is finite and e is the number of infinite places. So the last place could be assigned to F, and so to -1 as a representative of F. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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