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edited Jul 31 2010 at 2:53 |
$\; v.u v \cdot u = v_1 u_1 + v_2 u_2 - v_3 u_3$. $\quad\quad v \mapsto v - 2 \dfrac{v.u}{u.u} dfrac{v \cdot u}{u \cdot u} u \quad\quad$ Reflectivity is clear: $\; u \mapsto -u$, and $\; v \mapsto v$ if $\; v\perp u, \;$ i.e. $v.u v\cdot u = 0$. $\quad\quad (x,y,z)\; \mapsto (x,y,z) - 2 \dfrac{(x,y,z).(1,1,1)}{(1,1,1).(1,1,1)} dfrac{(x,y,z)\cdot(1,1,1)}{(1,1,1)\cdot(1,1,1)} (1,1,1)$ $\quad\quad (-3,+4,5) \mapsto (-3,+4,5) - 2 \; (-3+4-5) \; (-3,+4,5) 1,1,1) = ( 5,12,13)$ $\quad\quad (-3,-4,5) \mapsto (-3,-4,5) - 2 \; (-3-4-5) \; (-3,-4,5) 1,1,1) = (21,20,29)$ $\quad\quad (+3,-4,5) \mapsto (+3,-4,5) - 2 \; (+3-4-5) \; (+3,-4,5) 1,1,1) = (15,8,17)$ There are also generalizations to different shape forms that were first used by L. Aubry in 1912 (Sphinx-Oedipe 7 (1912), 81-84) to give elementary proofs of the 3 & 4 square theorem , etc(seeAppendix 3.2 p. 292 of Weil's: Number Theory an Approach Through History). These results show that if anIn fact there is a function-field generalization by Cassels & Pfister analog that uses employs the Euclidean algorithm which was independently rediscovered by Cassels in place 1963. Namely, a polynomial is a sum of $n$ squares in $k(x)$ iff the same holds true in $k[x]$.Pfister immediately applied this to obtain a complete solution of the level problem for fields. These Shortly thereafter he generalized Cassels result to arbitrary quadratic forms, founding the modern algebraic theory of quadratic forms ("Pfister forms"). Aubry's results are, in fact, merely very special cases of general results of Wall, Vinberg, Scharlau et al. on reflective lattices, i.e. arithmetic groups of isometries generated by reflections in hyperplanes- so-called reflective lattices. However, Generally reflections generate the connections between these various strands orthogonal group of research appears to be little-known - if at allLorentzian quadratic forms in dim < 10.
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edited Jul 29 2010 at 18:03 |
One hidden A little-known chatoyant gem of elementary number theory is that the tree of Pythagorean triples clarify the points that you raiseabove. Below is a brief sketch extracted excerpted fromsome emails where I explained this sent to John Conway and R. K. Guy, after noticing that they mention this topic (too) briefly in their "Book of Numbers". Namely, on p. 172 they write: Below I explain briefly how to view this in terms of reflections and I mention some generalizations and closely related topics. I plan to discuss this at greater length in a future MO post when time permits. $ \; v.u = v_1 u_1 + v_2 u_2 - v_3 u_3$. Recall that here one defines the $\quad$ reflection of $v$ in $u$is given by $\quad\quad v \mapsto v - 2 \frac{v.u}{u.u} u$.Notice that this maps dfrac{v.u}{u.u} u \quad\quad$ Reflectivity is clear: $\; u \mapsto -u$, and $\; v \mapsto v$ if $v$ is perpendicular to $u$, \; v\perp u, \;$ i.e. $v.u = 0$, so that it does indeed reflect along $u$.. With $\; v = (x,y,z)$ and $\; u = (1,1,1)$ of norm 1, we obtain $\quad \quad\quad (x,y,z)\; \mapsto (x,y,z) - 2 \frac{(x,y,z).(1,1,1)}{(1,1,1).(1,1,1)} dfrac{(x,y,z).(1,1,1)}{(1,1,1).(1,1,1)} (1,1,1)$ $\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad = (x,y,z) - 2 \; (x+y-z) \; (1,1,1)$ $\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad = (-x-2y+2z,-2x-y+2z,-2x-2y+3z)$ which -x-2y+2z, \; -2x-y+2z, \; -2x-2y+3z)$ This is the nontrivial reflection that effects the descent in the triples tree. Translating this into more elementary geometric language, Said simpler: if $x^2 + y^2 = z^2$ then $(x/z, y/z)$ is a rational point P $P$ on the unit circle C.Draw $C$. A simple calculation shows that the line through P $P$ and (1,1). It will intersect C $(1,1)$ intersects $C$ in another "smaller" a smaller rational point, given projectively via the above reflection:, e.g. $\quad \quad\quad (x,y,z) 5,12,13) \mapsto (x,y,z) 5,12,13) - 2 \; (x+y-z) 5+12-13) \; (1,1,1)$ which descends 1,1,1) = (-3,4,5)$ We ascend the tree of Pythagorean triples; e.g. the triple $(5,12,13)$ descends toby inverting this reflection, combined with trivial sign-changing reflections:$\quad \quad\quad (5,12,13) -3,+4,5) \mapsto (-3,+4,5) - 2 \; (5+12-13) -3+4-5) \; (1,1,1) -3,+4,5) = ( -3,4,5)$ which is, projectively, simply the intersection of C with the line through $(5/13,12/13)$ and 5,12,13)$ $(1,1)$,
i.e. said line intersects C at the smaller rational point \quad\quad (-3,-4,5) \mapsto (-3,-4,5) - 2 \; (-3-4-5) \; (-3,-4,5) = (21,20,29)$ Inverting \quad\quad (+3,-4,5) \mapsto (+3,-4,5) - 2 \; (+3-4-5) \; (+3,-4,5) = (15,8,17)$Continuing in this reflection ascends manner one may reflectively generate the entire tree . Combined with the trivial negation reflections it generates all the of primitive Pythagorean triples, e.g. the top topmost edge of the triples tree corresponds to the ascending C-inscribed $C$-inscribed zigzag line $(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13) 5/12,12/13), (-5/12,12/13), (7/25,24/25), (-7/25,24/25) \ldots$Hence the reflective generation of primitive Pythagorean triples. In fact this can This technique easily be generalized generalizes to the form $ x_1^2 + x_2^2 + \cdots + x_{n-1}^2 = x_n^2$ distinct orbits under the automorphism group of the form (e.g. - see the paper by Cass and & Arpaia , PAMS, 109, 1990, 1-7).(1990) [1] THEOREM Let Suppose that the $n$-ary quadratic form $F(x)$ have has integral coefficients and has no nontrivial zero in ${\mathbb Z}^n$, and suppose further $|F(x-y)| \; |F(x-y)| < 1$. Then $F$ represents $m$ over $\mathbb Q$ iff $\iff$ $F$ represents The condition $|F(x-y)| < 1$ is closely connected to the Euclidean algorithm, in . These results are, in fact, merely very special cases of general results of Vinberg, Scharlau et.al et al. on arithmetic groups of isometries generated by reflections in hyperplanes - so-called "reflective lattices"lattices. However, the connections between these various strands of research appears to be little-known - if at all. 1 Daniel Cass; Pasquale J. Arpaia
Matrix Generation of Pythagorean n-Tuples.
Proc. Amer. Math. Soc. 109, 1, 1990, 1-7.
http://www.jstor.org/stable/2048355
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edited Jul 29 2010 at 1:19 |
One hidden gem of elementary number theory is that the tree of Pythagorean triples
has a beautiful geometric genesis in terms of reflections. This viewpoint should
clarify the points that you raise above. Below is a brief sketch extracted from
some emails where I explained this to John Conway and R. K. Guy. I plan
to discuss this at greater length in a future MO post when time permits
Consider the quadratic space $Z$ of the form $Q(x,y,z) = x^2 + y^2 - z^2$.
It has Lorentzian inner product
$(Q(x+y)-Q(x)-Q(y))/2$ given by $v.u = v_1 u_1 + v_2 u_2 - v_3 u_3$. Recall that the reflection of $v$ in $u$ is given by
$v \mapsto v - 2 \frac{v.u}{u.u} u$.
Notice that this maps $u \mapsto -u$, and $ v \mapsto v$ if $v$ is perpendicular to $u$,
i.e. $v.u = 0$, so that it does indeed reflect along $u$.
With $v = (x,y,z)$ and $u = (1,1,1)$ of norm 1, we obtain
$\quad (x,y,z)\; \mapsto (x,y,z) - 2 \frac{(x,y,z).(1,1,1)}{(1,1,1).(1,1,1)} (1,1,1)$
$\quad\quad\quad\quad\quad = (x,y,z) - 2 (x+y-z) (1,1,1)$
$\quad\quad\quad\quad\quad = (-x-2y+2z,-2x-y+2z,-2x-2y+3z)$
which is the nontrivial reflection that effects descent in the triples tree.
Translating this into more elementary geometric language,
if $x^2 + y^2 = z^2$ then $(x/z, y/z)$ is a rational point P on the unit circle C.
Draw the line through P and (1,1). It will intersect C in another "smaller" rational
point, given via the above reflection:
$\quad (x,y,z) \mapsto (x,y,z) - 2 (x+y-z) (1,1,1)$
which descends the tree of Pythagorean triples; e.g. the triple $(5,12,13)$ descends to
$\quad (5,12,13) - 2 (5+12-13) (1,1,1) = (-3,4,5)$
which is, projectively, simply the intersection of C with the line through $(5/13,12/13)$ and $(1,1)$,
i.e. said line intersects C at the smaller rational point $(-3/5,4/5)$.
Inverting this reflection ascends the tree. Combined with the trivial negation reflections it generates all the primitive triples,
e.g. the top edge of the tree corresponds to the ascending C-inscribed zigzag line
$(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13) (-5/12,12/13), (7/25,24/25), (-7/25,24/25) \ldots$
Hence the reflective generation of primitive Pythagorean triples.
In fact this can easily be generalized to the form $ x_1^2 + x_2^2 + \cdots + x_{n-1}^2 = x_n^2$
for $4 \le n \le 9$, but for
$n \ge 10 $ the Pythagorean n-tuples fall into at least $[(n+6)/8]$
distinct orbits under the automorphism group of the form (e.g. see the paper by Cass and Arpaia, PAMS, 109, 1990, 1-7).
There are also generalizations to different shape forms that were first used by Aubry in 1912
to give elementary proofs of the 3 & 4 square theorem, etc. These results show that if an
integer is represented by a form rationally, then it must also be so integrally. In
particular, the following class of forms is included $x^2+y^2, x^2 \pm 2y^2, x^2 \pm 3y^2,
x^2+y^2+2z^2, x^2+y^2+z^2+t^2,\ldots$ More precisely, essentially the same proof as for
Pythagorean triples shows
THEOREM Let the $n$-ary quadratic form $F(x)$ have integral
coefficients and no nontrivial zero in ${\mathbb Z}^n$ and suppose
that for any $x \in {\mathbb Q}^n$ there exists $y \in {\mathbb Z}^n$ such that
$|F(x-y)| < 1$. Then $F$ represents $m$ over $\mathbb Q$ iff $F$ represents
$m$ over $\mathbb Z$, for all nonzero integers $m$.
The condition $|F(x-y)| < 1$ is closely connected to the Euclidean algorithm,
in fact there is a function-field generalization by Cassels & Pfister that uses the Euclidean algorithm in place of this.
These results are very special cases of general results of Vinberg, Scharlau et.al
on arithmetic groups of isometries generated by reflections in hyperplanes - so-called "reflective lattices". However, the connections between these various strands of research appears to be little-known - if at all.
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edited Jul 28 2010 at 22:41 |
One hidden gem of elementary number theory is that the tree of Pythagorean triples
has a beautiful geometric genesis in terms of reflections. This viewpoint should
clarify the points that you raise above. Below is a brief sketch extracted from
some emails where I explained this to John Conway and R. K. Guy. I plan
to discuss this at greater length in a future MO post when time permits
Consider the quadratic space $Z$ of the form $Q(x,y,z) = x^2 + y^2 - z^2$.
It has Lorentzian inner product $(Q(x+y)-Q(x)-Q(y))/2$ given by $v.u = v_1 u_1 + v_2 u_2 - v_3 u_3$. Recall that the reflection of $v$ in $u$ is given by
$v \mapsto v - 2 \frac{v.u}{u.u} u$.
Notice that this maps $u \mapsto -u$, and $ v \mapsto v$ if $v$ is perpendicular to $u$,
i.e. $v.u = 0$, so that it does indeed reflect along $u$.
With $v = (x,y,z)$ and $u = (1,1,1)$ of norm 1, we obtain
$\quad (x,y,z)\; \mapsto (x,y,z) - 2 \frac{(x,y,z).(1,1,1)}{(1,1,1).(1,1,1)} (1,1,1)$
$\quad\quad\quad\quad\quad = (x,y,z) - 2 (x+y-z) (1,1,1)$
$\quad\quad\quad\quad\quad = (-x-2y+2z,-2x-y+2z,-2x-2y+3z)$
which is the nontrivial reflection that effects descent in the triples tree.
Translating this into more elementary geometric language,
if $x^2 + y^2 = z^2$ then $(x/z, y/z)$ is a rational point P on the unit circle C.
Draw the line through P and (1,1). It will intersect C in another "smaller" rational
point, given via the above reflection:
$\quad (x,y,z) \mapsto (x,y,z) - 2 (x+y-z) (1,1,1)$
which descends the tree of Pythagorean triples; e.g. the triple $(5,12,13)$ descends to
$\quad (5,12,13) - 2 (5+12-13) (1,1,1) = (-3,4,5)$
which is, projectively, simply the intersection of C with the line through $(5/13,12/13)$ and $(1,1)$,
i.e. said line intersects C at the smaller rational point $(-3/5,4/5)$.
Inverting this reflection ascends the tree. Combined with the trivial negation reflections it generates all the primitive triples,
e.g. the top edge of the tree corresponds to the ascending C-inscribed zigzag line
$(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13) (-5/12,12/13), (7/25,24/25), (-7/25,24/25) \ldots$
Hence the reflective generation of primitive Pythagorean triples.
In fact this can easily be generalized to the form $ x_1^2 + x_2^2 + \cdots + x_{n-1}^2 = x_n^2$
for $4 \le n \le 9$, but for $n \ge 10 $ the Pythagorean n-tuples fall into at least $[(n+6)/8]$
distinct orbits under the automorphism group of the form (e.g. see the paper by Cass and Arpaia, PAMS, 109, 1990, 1-7).
There are also generalizations to different shape forms that were first used by Aubry in 1912
to give elementary proofs of the 3 & 4 square theorem, etc. These results show that if an
integer is represented by a form rationally, then it must also be so integrally. In
particular, the following class of forms is included $x^2+y^2, x^2 \pm 2y^2, x^2 \pm 3y^2,
x^2+y^2+2z^2, x^2+y^2+z^2+t^2,\ldots$ More precisely, essentially the same proof as for
Pythagorean triples shows
THEOREM Let the $n$-ary quadratic form $F(x)$ have integral
coefficients and no nontrivial zero in ${\mathbb Z}^n$ and suppose
that for any $x \in {\mathbb Q}^n$ there exists $y \in {\mathbb Z}^n$ such that
$|F(x-y)| < 1$. Then $F$ represents $m$ over $\mathbb Q$ iff $F$ represents
$m$ over $\mathbb Z$, for all nonzero integers $m$.
The condition $|F(x-y)| < 1$ is closely connected to the Euclidean algorithm,
in fact there is a function-field generalization by Cassels & Pfister that uses the Euclidean algorithm in place of this.
These results are very special cases of general results of Vinberg, Scharlau et.al
on arithmetic groups of isometries generated by reflections in hyperplanes - so-called "reflective lattices". However, no one appears to have noticed the connections between these various worksstrands of research appears to be little-known - if at all.
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answered Jul 28 2010 at 22:35 |
One hidden gem of elementary number theory is that the tree of Pythagorean triples
has a beautiful geometric genesis in terms of reflections. This viewpoint should
clarify the points that you raise above. Below is a brief sketch extracted from
some emails where I explained this to John Conway and R. K. Guy. I plan
to discuss this at greater length in a future MO post when time permits
Consider the quadratic space $Z$ of the form $Q(x,y,z) = x^2 + y^2 - z^2$.
It has Lorentzian inner product $(Q(x+y)-Q(x)-Q(y))/2$ given by $v.u = v_1 u_1 + v_2 u_2 - v_3 u_3$. Recall that the reflection of $v$ in $u$ is given by
$v \mapsto v - 2 \frac{v.u}{u.u} u$.
Notice that this maps $u \mapsto -u$, and $ v \mapsto v$ if $v$ is perpendicular to $u$,
i.e. $v.u = 0$, so that it does indeed reflect along $u$.
With $v = (x,y,z)$ and $u = (1,1,1)$ of norm 1, we obtain
$\quad (x,y,z)\; \mapsto (x,y,z) - 2 \frac{(x,y,z).(1,1,1)}{(1,1,1).(1,1,1)} (1,1,1)$
$\quad\quad\quad\quad\quad = (x,y,z) - 2 (x+y-z) (1,1,1)$
$\quad\quad\quad\quad\quad = (-x-2y+2z,-2x-y+2z,-2x-2y+3z)$
which is the nontrivial reflection that effects descent in the triples tree.
Translating this into more elementary geometric language,
if $x^2 + y^2 = z^2$ then $(x/z, y/z)$ is a rational point P on the unit circle C.
Draw the line through P and (1,1). It will intersect C in another "smaller" rational
point, given via the above reflection:
$\quad (x,y,z) \mapsto (x,y,z) - 2 (x+y-z) (1,1,1)$
which descends the tree of Pythagorean triples; e.g. the triple $(5,12,13)$ descends to
$\quad (5,12,13) - 2 (5+12-13) (1,1,1) = (-3,4,5)$
which is, projectively, simply the intersection of C with the line through $(5/13,12/13)$ and $(1,1)$,
i.e. said line intersects C at the smaller rational point $(-3/5,4/5)$.
Inverting this reflection ascends the tree. Combined with the trivial negation reflections it generates all the primitive triples,
e.g. the top edge of the tree corresponds to the ascending C-inscribed zigzag line
$(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13) (-5/12,12/13), (7/25,24/25), (-7/25,24/25) \ldots$
Hence the reflective generation of primitive Pythagorean triples.
In fact this can easily be generalized to the form $ x_1^2 + x_2^2 + \cdots + x_{n-1}^2 = x_n^2$
for $4 \le n \le 9$, but for $n \ge 10 $ the Pythagorean n-tuples fall into at least $[(n+6)/8]$
distinct orbits under the automorphism group of the form (e.g. see the paper by Cass and Arpaia, PAMS, 109, 1990, 1-7).
There are also generalizations to different shape forms that were first used by Aubry in 1912
to give elementary proofs of the 3 & 4 square theorem, etc. These results show that if an
integer is represented by a form rationally, then it must also be so integrally. In
particular, the following class of forms is included $x^2+y^2, x^2 \pm 2y^2, x^2 \pm 3y^2,
x^2+y^2+2z^2, x^2+y^2+z^2+t^2,\ldots$ More precisely, essentially the same proof as for
Pythagorean triples shows
THEOREM Let the $n$-ary quadratic form $F(x)$ have integral
coefficients and no nontrivial zero in ${\mathbb Z}^n$ and suppose
that for any $x \in {\mathbb Q}^n$ there exists $y \in {\mathbb Z}^n$ such that
$|F(x-y)| < 1$. Then $F$ represents $m$ over $\mathbb Q$ iff $F$ represents
$m$ over $\mathbb Z$, for all nonzero integers $m$.
The condition $|F(x-y)| < 1$ is closely connected to the Euclidean algorithm,
in fact there is a function-field generalization by Cassels & Pfister that uses the Euclidean algorithm in place of this.
These results are very special cases of general results of Vinberg, Scharlau et.al
on arithmetic groups of isometries generated by reflections in hyperplanes - so-called "reflective lattices". However, no one appears to have noticed the connections between these various works.
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