diff --git a/changelog b/changelog index 631bee7..c9a6037 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20090608 tpd src/axiom-website/patches.html 20090608.03.tpd.patch +20090608 tpd src/input/Makefile tpieazas002 CATS test added +20090608 tpd src/input/tpieazas002.input added 20090608 tpd src/axiom-website/patches.html 20090608.02.tpd.patch 20090608 tpd books/bookvol10.4 MESH +-> conversion 20090608 tpd src/axiom-website/patches.html 20090608.01.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 8377d2d..7a6b416 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -1499,5 +1499,7 @@ tpieazas001.input add rm of output file
bookvol10.4 MATLIN +-> conversion
20090608.02.tpd.patch bookvol10.4 MESH +-> conversion
+20090608.03.tpd.patch +tpieazas002.input CATS test added
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 661fe9f..0b4f2bd 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -385,6 +385,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ tanhcoth.regress tanatan.regress tbagg.regress test.regress \ textfile.regress torus.regress \ tpiezas001.regress \ + tpiezas002.regress \ triglim.regress tsetcatvermeer.regress tutchap1.regress \ typetower.regress void.regress uniseg.regress \ unittest1.regress unittest2.regress unittest3.regress unittest4.regress @@ -694,6 +695,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/tanatan.input ${OUT}/tbagg.input ${OUT}/test.input \ ${OUT}/tetra.input ${OUT}/textfile.input ${OUT}/tknot.input \ ${OUT}/tpiezas001.input \ + ${OUT}/tpiezas002.input \ ${OUT}/tree.input \ ${OUT}/triglim.input ${OUT}/tschirn.input ${OUT}/tsetcatbutcher.input \ ${OUT}/tsetcatchemical.input ${OUT}/tsetcatvermeer.input \ @@ -1053,6 +1055,7 @@ DOCFILES= \ ${DOC}/textfile.input.dvi ${DOC}/tknot.input.dvi \ ${DOC}/torus.input.dvi \ ${DOC}/tpiezas001.input.dvi \ + ${DOC}/tpiezas002.input.dvi \ ${DOC}/tree.input.dvi \ ${DOC}/triglim.input.dvi ${DOC}/tschirn.input.dvi \ ${DOC}/tsetcatbutcher.input.dvi \ diff --git a/src/input/tpiezas002.input.pamphlet b/src/input/tpiezas002.input.pamphlet new file mode 100644 index 0000000..54fc7a3 --- /dev/null +++ b/src/input/tpiezas002.input.pamphlet @@ -0,0 +1,843 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input tpiezas002.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +This is from ``A Collection of Algebraic Identities'' +by Titus. Piezas III\cite{1} +\end{abstract} +\eject +\tableofcontents +\eject +\section{Part 2. Sums of Squares} +\subsection{Sums of two squares} +\begin{enumerate} +\item $x^2+y^2 = z^k$ +\item $x^2+ny^2 = z^k$ +\item $ad-bc = \pm 1$ +\item $x^2+y^2=z^2+1$ +\item $x^2+y^2=z^2-1$ +\item $x^2+y^2=z^2+nt^2$ +\item $x^2+y^2=z^2+nt^k$ +\item $x^2+y^2=mz^2+nt^2$ +\item $c_1(x^2+ny^2)=c_2(z^2+nt^2)$ +\item $mx^2+ny^2=mz^2+nt^2$ +\end{enumerate} +\begin{chunk}{*} +)set break resume +)sys rm -f tpiezas002.output +)spool tpiezas002.output +)set message test on +)set message auto off +)clear all + +\end{chunk} + +\subsection{Form: $x^2+y^2=z^k$} +The smallest primitive solutions (those with $a,b,c$ co-prime) are: +\{3,4,5\}, \{5,12,13\}, \{7,24,25\}, \{8,15,17\}, etc. + +{\bf Theorem}: For primitive triples $a^2+b^2=c^2$,\\ +Exactly one of a,b is odd, and c is always odd\\ +Exactly one of a,b is divisible by 3\\ +Exactly one of a,b is divisible by 4\\ +Exactly one of a,b,c is divisible by 5 + +Given one triple the others can be generated. + +{\bf Theorem}: If $a^2+b^2=c^2$ then $(a+2b+2c)^2+(2a+b+2c)^2=(2a+2b+3c)^2$ + +\begin{chunk}{*} +--S 1 of 71 +c:=sqrt(a^2+b^2) +--R +--R +--R +-------+ +--R | 2 2 +--R (1) \|b + a +--R Type: Expression Integer +--E 1 + +--S 2 of 71 +t1:=(a+2*b+2*c)^2 +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R (2) (8b + 4a)\|b + a + 8b + 4a b + 5a +--R Type: Expression Integer +--E 2 + +--S 3 of 71 +t2:=(2*a+b+2*c)^2 +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R (3) (4b + 8a)\|b + a + 5b + 4a b + 8a +--R Type: Expression Integer +--E 3 + +--S 4 of 71 +t3:=(2*a+2*b+3*c)^2 +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R (4) (12b + 12a)\|b + a + 13b + 8a b + 13a +--R Type: Expression Integer +--E 4 + +--S 5 of 71 +t1+t2-t3 +--R +--R +--R (5) 0 +--R Type: Expression Integer +--E 5 + +\end{chunk} +Starting with $\{a,b,c\}=\{\pm 3, \pm 4, 5\}$ it is possible to generate +all primitive Pythagorean triples (Barning, 1963, Roberts, 1977) + +Note1: There are 4 terms to generate so from $\{a,b,c\}=\{\pm 5,\pm 12,13\}$ +we see: +\begin{enumerate} +\item $\{a,b,c\} = \{55,48,73\}$ +\item $\{-a,b,c\} = \{45,28,53\}$ +\item $\{a,-b,c\} = \{7,24,25\}$ +\item $\{-a,-b,c\} = \{-3,4,5\}$ +\end{enumerate} +\begin{chunk}{*} +)clear all + +--S 6 of 71 +f(a,b,c)==[(a+2*b+2*c),(2*a+b+2*c),(2*a+2*b+3*c)] +--R +--R Type: Void +--E 6 + +--S 7 of 71 +f(5,12,13) +--R +--R Compiling function f with type (PositiveInteger,PositiveInteger, +--R PositiveInteger) -> List PositiveInteger +--R +--R (2) [55,48,73] +--R Type: List PositiveInteger +--E 7 + +--S 8 of 71 +f(-5,12,13) +--R +--R Compiling function f with type (Integer,PositiveInteger, +--R PositiveInteger) -> List Integer +--R +--R (3) [45,28,53] +--R Type: List Integer +--E 8 + +--S 9 of 71 +f(5,-12,13) +--R +--R Compiling function f with type (PositiveInteger,Integer, +--R PositiveInteger) -> List Integer +--R +--R (4) [7,24,25] +--R Type: List Integer +--E 9 + +--S 10 of 71 +f(-5,-12,13) +--R +--R Compiling function f with type (Integer,Integer,PositiveInteger) -> +--R List Integer +--R +--R (5) [- 3,4,5] +--R Type: List Integer +--E 10 + +\end{chunk} +In fact, we can prove the identity in general, +\begin{chunk}{*} +)clear all + +--S 11 of 71 +x:=(a^2-b^2) +--R +--R +--R 2 2 +--R (1) - b + a +--R Type: Polynomial Integer +--E 11 + +--S 12 of 71 +y:=(2*a*b) +--R +--R +--R (2) 2a b +--R Type: Polynomial Integer +--E 12 + +--S 13 of 71 +z:=(a^2+b^2) +--R +--R +--R 2 2 +--R (3) b + a +--R Type: Polynomial Integer +--E 13 + +--S 14 of 71 +x^2+y^2-z^2 +--R +--R +--R (4) 0 +--R Type: Polynomial Integer +--E 14 + +\end{chunk} +Note 2: It is possible the divisibility by 3,4,5 is contained in just one +term, as in the first term of +\begin{chunk}{*} +)clear all + +--S 15 of 71 +(60*v)^2+(900*v^2-1)^2 - (900*v^2+1)^2 +--R +--R +--R (1) 0 +--R Type: Polynomial Integer +--E 15 + +\end{chunk} +{\bf Theorem}: All odd numbers and multiples of 4 appear in a primitive +Pythagorean triple +\begin{chunk}{*} +)clear all + +--S 16 of 71 +n:=2*(m^2+m) +--R +--R +--R 2 +--R (1) 2m + 2m +--R Type: Polynomial Integer +--E 16 + +--S 17 of 71 +(2*m+1)^2 + n^2 - (n+1)^2 +--R +--R +--R (2) 0 +--R Type: Polynomial Integer +--E 17 + +)clear all + +--S 18 of 71 +n:=4*m^2-1 +--R +--R +--R 2 +--R (1) 4m - 1 +--R Type: Polynomial Integer +--E 18 + +--S 19 of 71 +(4*m)^2 + n^2 - (n+2)^2 +--R +--R +--R (2) 0 +--R Type: Polynomial Integer +--E 19 + +--S 20 of 71 +(a^2-b^2)^2+(2*a*b)^2-(a^2+b^2)^2 +--R +--R +--R (3) 0 +--R Type: Polynomial Integer +--E 20 + +--S 21 of 71 +(a^3-3*a*b^2)^2 + (3*a^2*b-b^3)^2 - (a^2+b^2)^3 +--R +--R +--R (4) 0 +--R Type: Polynomial Integer +--E 21 + +\end{chunk} +Working in the quaternions we note that if +\[(a + bi + cj + dk)^m = A + Bi + Cj + Dk\] +then +\[A^2 + B^2 + C^2 + D^2 = (a^2 + b^2 + c^2 + d^2)^m\] +so for $m=3$ we have: +\begin{chunk}{*} +)clear all + +--S 22 of 71 +t0:=quatern(a,b,c,d)^3 +--R +--R +--R (1) +--R 2 2 2 3 2 2 3 2 +--R - 3a d - 3a c - 3a b + a + (- b d - b c - b + 3a b)i +--R + +--R 2 3 2 2 3 2 2 2 +--R (- c d - c + (- b + 3a )c)j + (- d + (- c - b + 3a )d)k +--R Type: Quaternion Polynomial Integer +--E 22 + +--S 23 of 71 +A:=real t0 +--R +--R +--R 2 2 2 3 +--R (2) - 3a d - 3a c - 3a b + a +--R Type: Polynomial Integer +--E 23 + +--S 24 of 71 +B:=imagI t0 +--R +--R +--R 2 2 3 2 +--R (3) - b d - b c - b + 3a b +--R Type: Polynomial Integer +--E 24 + +--S 25 of 71 +C:=imagJ t0 +--R +--R +--R 2 3 2 2 +--R (4) - c d - c + (- b + 3a )c +--R Type: Polynomial Integer +--E 25 + +--S 26 of 71 +D:=imagK t0 +--R +--R +--R 3 2 2 2 +--R (5) - d + (- c - b + 3a )d +--R Type: Polynomial Integer +--E 26 + +--S 27 of 71 +A^2+B^2+C^2+D^2 - (a^2+b^2+c^2+d^2)^3 +--R +--R +--R (6) 0 +--R Type: Polynomial Integer +--E 27 + +\end{chunk} +\subsection{Euler} +\[(a^2+b^2)^2 + (b^2+d^2)^2 = (b^2+8c^2+d^2)^2\] +where +\[d=\frac{(b^2+3*c^2)}{(2*c)}\] +\[a^2-b^2=10c^2\] +\begin{chunk}{*} +)clear all + +--S 28 of 71 +d:=(b^2+3*c^2)/(2*c) +--R +--R +--R 2 2 +--R 3c + b +--R (1) -------- +--R 2c +--R Type: Fraction Polynomial Integer +--E 28 + +--S 29 of 71 +a:=sqrt(10*c^2+b^2) +--R +--R +--R +---------+ +--R | 2 2 +--R (2) \|10c + b +--R Type: Expression Integer +--E 29 + +--S 30 of 71 +(a^2+b^2)^2 + (b^2+d^2)^2 - (b^2+8*c^2+d^2)^2 +--R +--R +--R (3) 0 +--R Type: Expression Integer +--E 30 + +\end{chunk} +\subsection{Piezas} +A generalization of the Euler version: +\begin{chunk}{*} +)clear all + +--S 31 of 71 +p:=2*n^2 +--R +--R +--R 2 +--R (1) 2n +--R Type: Polynomial Integer +--E 31 + +--S 32 of 71 +q:=n^2-1 +--R +--R +--R 2 +--R (2) n - 1 +--R Type: Polynomial Integer +--E 32 + +--S 33 of 71 +d:=(b^2+q*c^2)/(2*c) +--R +--R +--R 2 2 2 2 +--R c n - c + b +--R (3) -------------- +--R 2c +--R Type: Fraction Polynomial Integer +--E 33 + +--S 34 of 71 +a:=sqrt(n*(n^2+1)*c^2 + (n-1)*b^2) +--R +--R +--R +----------------------+ +--R | 2 3 2 2 2 +--R (4) \|c n + (c + b )n - b +--R Type: Expression Integer +--E 34 + +--S 35 of 71 +(a^2+b^2)^2 + (b^2+d^2)^2-(b^2+p*c^2+d^2)^2 +--R +--R +--R (5) 0 +--R Type: Expression Integer +--E 35 + +\end{chunk} +Another approach is the parameterization: +\begin{chunk}{*} +)clear all + +--S 36 of 71 +a:=u^2-v^2-w^2 +--R +--R +--R 2 2 2 +--R (1) - w - v + u +--R Type: Polynomial Integer +--E 36 + +--S 37 of 71 +b:=2*u*v +--R +--R +--R (2) 2u v +--R Type: Polynomial Integer +--E 37 + +--S 38 of 71 +p:=u^2+v^2+w^2 +--R +--R +--R 2 2 2 +--R (3) w + v + u +--R Type: Polynomial Integer +--E 38 + +--S 39 of 71 +q:=2*u*w +--R +--R +--R (4) 2u w +--R Type: Polynomial Integer +--E 39 + +--S 40 of 71 +c:=sqrt(4*u*w*(u^2+v^2+w^2)-d^2) +--R +--R +--R +---------------------------+ +--R | 3 2 3 2 +--R (5) \|4u w + (4u v + 4u )w - d +--R Type: Expression Integer +--E 40 + +--S 41 of 71 +(a^2+b^2)^2 + (c^2+d^2)^2 - (p^2+q^2)^2 +--R +--R +--R (6) 0 +--R Type: Expression Integer +--E 41 + +\end{chunk} +\subsection{Fermat} +{\bf Theorem}: If $p^2+(p+1)^2=r^2$ then $q^2+(q+1)^2=(p+q+r+1)^2$, +where $q=3*p+2*r+1$ +\begin{chunk}{*} +)clear all + +--S 42 of 71 +q:=3*p+2*r+1 +--R +--R +--R (1) 2r + 3p + 1 +--R Type: Polynomial Integer +--E 42 + +--S 43 of 71 +r:=sqrt(p^2+(p+1)^2) +--R +--R +--R +------------+ +--R | 2 +--R (2) \|2p + 2p + 1 +--R Type: Expression Integer +--E 43 + +--S 44 of 71 +q^2 + (q+1)^2 - (p+q+r+1)^2 +--R +--R +--R +------------+ +--R | 2 2 +--R (3) (- 4r - 8p - 4)\|2p + 2p + 1 + 4r + (8p + 4)r +--R Type: Expression Integer +--E 44 + +\end{chunk} +Since the last result is not zero, something is wrong. +\subsection{Fibonacci} +\begin{chunk}{*} +)clear all + +--S 45 of 71 +e:=sqrt(c^2+d^2) +--R +--R +--R +-------+ +--R | 2 2 +--R (1) \|d + c +--R Type: Expression Integer +--E 45 + +--S 46 of 71 +(a*c+b*d)^2 + (a*d-b*c)^2 - (a*e)^2 - (b*e)^2 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 46 + +\end{chunk} +\subsection{Volpicelli} +\[(ac+bd)^2 + (ad-bc)^2 = (a^2+b^2)^2\] +where $a^2+b^2 = c^2+d^2$ +\begin{chunk}{*} +)clear all + +--S 47 of 71 +a:=sqrt(c^2+d^2-b^2) +--R +--R +--R +------------+ +--R | 2 2 2 +--R (1) \|d + c - b +--R Type: Expression Integer +--E 47 + +--S 48 of 71 +(a*c+b*d)^2 + (a*d-b*c)^2 - (a^2 + b^2)^2 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 48 + +\end{chunk} +\subsection{Fleck} +\[(a^2c-b^2c+2abd)^2 + (a^2d-b^2d-2abc)^2 = (a^2+b^2)^3\] +if $a^2+b^2 = c^2+d^2$ +\begin{chunk}{*} +)clear all + +--S 49 of 71 +a:=sqrt(c^2+d^2-b^2) +--R +--R +--R +------------+ +--R | 2 2 2 +--R (1) \|d + c - b +--R Type: Expression Integer +--E 49 + +--S 50 of 71 +(a^2*c-b^2*c+2*a*b*d)^2 + (a^2*d-b^2*d-2*a*b*c)^2 - (a^2+b^2)^3 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 50 + +\end{chunk} +A generalization of the above: +\begin{chunk}{*} +)clear all + +--S 51 of 71 +a:=sqrt(c^2+d^2-b^2) +--R +--R +--R +------------+ +--R | 2 2 2 +--R (1) \|d + c - b +--R Type: Expression Integer +--E 51 + +--S 52 of 71 +t1:=(a*c^3 - 3*b*c^2*d - 3*a*c*d^2 + b*d^3)^2 +--R +--R +--R (2) +--R +------------+ +--R 5 3 3 5 | 2 2 2 2 2 6 +--R (- 6b c d + 20b c d - 6b c d)\|d + c - b + (9c + b )d +--R + +--R 4 2 2 4 6 2 4 2 8 2 6 +--R (3c - 15b c )d + (- 5c + 15b c )d + c - b c +--R Type: Expression Integer +--E 52 + +--S 53 of 71 +t2:=(b*c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3)^2 +--R +--R +--R (3) +--R +------------+ +--R 5 3 3 5 | 2 2 2 8 2 2 6 +--R (6b c d - 20b c d + 6b c d)\|d + c - b + d + (- 5c - b )d +--R + +--R 4 2 2 4 6 2 4 2 2 6 +--R (3c + 15b c )d + (9c - 15b c )d + b c +--R Type: Expression Integer +--E 53 + +--S 54 of 71 +t1+t2-(a^2+b^2)^4 +--R +--R +--R (4) 0 +--R Type: Expression Integer +--E 54 + +\end{chunk} +\subsection{Piezas} +{\bf Theorem}: Let $F:=(p^2+q^2)(r^2+s^2)$. Then $F^k$ (for $k>0$) is +identically the sum of two squares in $k+1$ ways. +\begin{chunk}{*} +)clear all + +--S 55 of 71 +a:=(p^2+q^2)*(r^2+s^2) +--R +--R +--R 2 2 2 2 2 2 +--R (1) (q + p )s + (q + p )r +--R Type: Polynomial Integer +--E 55 + +--S 56 of 71 +b:=(p*r+q*s)^2 + (p*s-q*r)^2 +--R +--R +--R 2 2 2 2 2 2 +--R (2) (q + p )s + (q + p )r +--R Type: Polynomial Integer +--E 56 + +--S 57 of 71 +a-b +--R +--R +--R (3) 0 +--R Type: Polynomial Integer +--E 57 + +--S 58 of 71 +c:=(p*r-q*s)^2 + (p*s+q*r)^2 +--R +--R +--R 2 2 2 2 2 2 +--R (4) (q + p )s + (q + p )r +--R Type: Polynomial Integer +--E 58 + +--S 59 of 71 +a-c +--R +--R +--R (5) 0 +--R Type: Polynomial Integer +--E 59 + +\end{chunk} +\subsection{H. Mathieu} +\[(q^2(p^2-2))^2 + (2q^2)^3 = (pq)^4\quad {\rm\ if\ }q^2-2q^2=1\] +\[((p^4-p^2)/2)^2 + p^6 = (pq)^4\quad {\rm\ if\ }p^2-2q^2=-1\] +\begin{chunk}{*} +)clear all + +--S 60 of 71 +p:=sqrt(1+2*q^2) +--R +--R +--R +-------+ +--R | 2 +--R (1) \|2q + 1 +--R Type: Expression Integer +--E 60 + +--S 61 of 71 +(q^2*(p^2-2))^2 + (2*q^2)^3 - (p*q)^4 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 61 + +)clear all + +--S 62 of 71 +p:=sqrt(-1+2*q^2) +--R +--R +--R +-------+ +--R | 2 +--R (1) \|2q - 1 +--R Type: Expression Integer +--E 62 + +--S 63 of 71 +((p^4-p^2)/2)^2 + p^6 - (p*q)^4 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 63 + +\end{chunk} +\subsection{Piezas} +\[(4q^2d^4(p^2-2))^2+(4*q^2d^3)^3=(2pqd^2)^4\quad {\rm\ if\ }p^2-dq^2=1\] +\[(4*p^2*d^3(p^2-1))^2+(2*p*d)^6=(2*p*q*d^2)^4\quad {\rm\ if\ }p^2-dq^2=-1\] +\begin{chunk}{*} +)clear all + +--S 64 of 71 +p:=sqrt(1+d*q^2) +--R +--R +--R +--------+ +--R | 2 +--R (1) \|d q + 1 +--R Type: Expression Integer +--E 64 + +--S 65 of 71 +(4*q^2*d^4*(p^2-2))^2 + (4*q^2*d^3)^3 - (2*p*q*d^2)^4 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 65 + +)clear all + +--S 66 of 71 +p:=sqrt(-1+d*q^2) +--R +--R +--R +--------+ +--R | 2 +--R (1) \|d q - 1 +--R Type: Expression Integer +--E 66 + +--S 67 of 71 +(4*p^2*d^3*(p^2-1))^2 + (2*p*d)^6 - (2*p*q*d^2)^4 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 67 + +\end{chunk} +\subsection{K. Brown} +\[p^4 + (q^2-1)^3 = (q^3+3q)^2\quad {\rm\ if\ }p^2-3q^2=1\] +\begin{chunk}{*} +)clear all + +--S 68 of 71 +p:=sqrt(1+3*q^2) +--R +--R +--R +-------+ +--R | 2 +--R (1) \|3q + 1 +--R Type: Expression Integer +--E 68 + +--S 69 of 71 +p^4 + (q^2-1)^3 - (q^3+3*q)^2 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 69 + +)clear all + +--S 70 of 71 +p:=sqrt(1+3*d*q^2) +--R +--R +--R +---------+ +--R | 2 +--R (1) \|3d q + 1 +--R Type: Expression Integer +--E 70 + +--S 71 of 71 +p^4 + (d*q^2-1)^3 - d*(d*q^3+3*q)^2 +--R +--R +--R (2) 0 +--R Type: Expression Integer +--E 71 + +)spool +)lisp (bye) + +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} Titus Piezas, ``A Collection of Algebraic Identities''\\ +{\bf http://sites.google.com/site/tpiezas/002} +\end{thebibliography} +\end{document} + + +