Re: [math-fun] The Hippasus Integers
[By the way, nifty use of the blackboard font ℚ!] I didn't know the discover of the irrationality of sqrt(2) was known by name! I'd like to see a reference for that, since I always heard the history was unclear. ----- The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ℚ(√−1) numbers are known as Gaussian integers. The ℚ(√−3) numbers are known as Eisenstein integers. The ℚ(√−7) numbers are known as Kleinian integers. It's bugged me for awhile that ℚ(√−2) wasn't named. Today I decided that the obvious name was ... The ℚ(√−2) numbers are now known as Hippasus integers. Hippasus proved √2 was irrational. He was then murdered. Seemed like an apt name to use. I calculated the nine types of Heegner primes and plotted them out. http://community.wolfram.com/groups/-/m/t/965609 <http://community.wolfram.com/groups/-/m/t/965609> Anyone agree / disagree about calling ℚ(√−2) the Hippasus integers? ----- I might be misinterpreting what is meant by "The ℚ(√[whatever]) numbers" are known as [whoever] integers." Usually ℚ(√[whatever]) is labeled an example of a "number field", which I think is any field of algebraic numbers with a finite index over ℚ. But to use the word "integers" implies that they are the roots of monic polynomials, with integer coefficients, that lie in the number field. For the number field ℚ(√-h) where h is a Heegner number the set of algebraic integers is as follows: Let O_h be the ring of algebraic integers of ℚ(√-h), Then: O_1 = Z[i] O_2 = Z[sqrt(-2)] O_h = Z[(1 + sqrt(h))/2] for h = -3, -7, -11, -19, -43, -67, -163. —Dan
Correction:
On Nov 18, 2016, at 10:30 AM, Dan Asimov <asimov@msri.org> wrote:
Let O_h be the ring of algebraic integers of ℚ(√-h), Then:
O_1 = Z[i]
O_2 = Z[sqrt(-2)]
O_h = Z[(1 + sqrt(h))/2] for h = -3, -7, -11, -19, -43, -67, -163.
The last line should read: ----- O_h = Z[(1 + sqrt(h))/2] for -h = -3, -7, -11, -19, -43, -67, -163. ----- (and in that e-mail "discover" should be "discoverer". Argggh. —Dan
You were right the first time. If h=1 mod 4, the ring is Z[(1 + sqrt(h))/2]. Otherwise, the ring is Z[sqrt(h)]. It is understood that square factors are removed from h, so that h is squarefree. This is all derived in Harvey Cohn, "Advanced Number Theory", Dover, p. 45, a wonderful book. -- Gene From: Dan Asimov <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, November 18, 2016 11:33 AM Subject: Re: [math-fun] The Hippasus Integers Correction:
On Nov 18, 2016, at 10:30 AM, Dan Asimov <asimov@msri.org> wrote:
Let O_h be the ring of algebraic integers of ℚ(√-h), Then:
O_1 = Z[i]
O_2 = Z[sqrt(-2)]
O_h = Z[(1 + sqrt(h))/2] for h = -3, -7, -11, -19, -43, -67, -163.
The last line should read: ----- O_h = Z[(1 + sqrt(h))/2] for -h = -3, -7, -11, -19, -43, -67, -163. ----- (and in that e-mail "discover" should be "discoverer". Argggh. —Dan
Here's the primality method I came up with. HeegnerPrimeQ[{a_, b_}, heegner_] := Module[{norm}, If[Not[MemberQ[{1, 2, 3, 7, 11, 19, 43, 67, 163}, heegner]], "not heegner", Switch[heegner, 1, PrimeQ[{a, b}.{1, I}, GaussianIntegers -> True], 2, norm = Norm[{a, b}.{1, Sqrt[2] I}]; If[IntegerQ[norm], If[PrimeQ[norm] && MemberQ[{5, 7}, Mod[norm, 8]], True, False ], If[PrimeQ[norm^2] && MemberQ[{1, 2, 3}, Mod[norm^2, 8]], True, False ]], _, norm = Norm[{a, b}.{1, Sqrt[heegner] I}/2]; If[IntegerQ[norm], If[PrimeQ[norm] && JacobiSymbol[norm, heegner] == -1, True, False ], If[PrimeQ[norm^2] && Not[JacobiSymbol[norm^2, heegner] == -1], True, False ]]]]] --Ed Pegg Jr On Sat, Nov 19, 2016 at 2:15 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
You were right the first time. If h=1 mod 4, the ring is Z[(1 + sqrt(h))/2]. Otherwise, the ring is Z[sqrt(h)]. It is understood that square factors are removed from h, so that h is squarefree. This is all derived in Harvey Cohn, "Advanced Number Theory", Dover, p. 45, a wonderful book. -- Gene
From: Dan Asimov <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, November 18, 2016 11:33 AM Subject: Re: [math-fun] The Hippasus Integers
Correction:
On Nov 18, 2016, at 10:30 AM, Dan Asimov <asimov@msri.org> wrote:
Let O_h be the ring of algebraic integers of ℚ(√-h), Then:
O_1 = Z[i]
O_2 = Z[sqrt(-2)]
O_h = Z[(1 + sqrt(h))/2] for h = -3, -7, -11, -19, -43, -67, -163.
The last line should read:
----- O_h = Z[(1 + sqrt(h))/2] for -h = -3, -7, -11, -19, -43, -67, -163. -----
(and in that e-mail "discover" should be "discoverer". Argggh.
—Dan
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participants (3)
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Dan Asimov -
Ed Pegg Jr -
Eugene Salamin