A list of all primitive quads like this is hard to find on the web.

Primitive quads starting with the smallest side of (1) up too (88)
(*) = primitive
(1,2,2,3)*
(1,4,8,9)*
(1,6,18,19)*
(1,8,32,33)*
(1,10,50,51)*
(1,12,12,17)*
(1,12,72,73)*
(1,14,98,99)*
(1,18,30,35)*
(1,22,46,51)*
(1,28,76,81)*
(1,32,100,105)*
(1,34,38,51)*
(1,44,68,81)*
(1,68,80,105)*
(1,70,70,99)*
(2,3,6,7)*
(2,5,14,15)*
(2,6,9,11)*
(2,7,26,27)*
(2,9,42,43)*
(2,10,11,15)*
(2,10,25,27)*
(2,11,62,63)*
(2,13,86,87)*
(2,14,23,27)*
(2,14,49,51)*
(2,18,39,43)*
(2,18,81,83)*
(2,19,34,39)*
(2,21,42,47)*
(2,22,59,63)*
(2,26,29,39)*
(2,26,83,87)*
(2,29,82,87)*
(2,31,94,99)*
(2,34,53,63)*
(2,43,46,63)*
(2,44,92,102)
(2,49,86,99)*
(2,54,63,83)*
(2,59,94,111)*
(2,61,62,87)*
(2,66,99,119)*
(3,4,12,13)*
(3,6,22,23)*
(3,8,36,37)*
(3,10,54,55)*
(3,12,76,77)*
(3,14,18,23)*
(3,16,24,29)*
(3,24,28,37)*
(3,24,56,61)*
(3,26,66,71)*
(3,30,46,55)*
(3,36,68,77)*
(3,42,94,103)*
(3,46,54,71)*
(3,50,90,103)*
(3,56,84,101)*
(3,80,96,125)*
(3,92,96,133)*
(4,4,7,9)*
(4,5,20,21)*
(4,7,32,33)*
(4,8,19,21)*
(4,9,48,49)*
(4,11,68,69)*
(4,12,39,41)*
(4,13,16,21)*
(4,13,92,93)*
(4,16,67,69)*
(4,17,28,33)*
(4,23,52,57)*
(4,24,33,41)*
(4,27,72,77)*
(4,28,35,45)*
(4,32,47,57)*
(4,32,61,69)*
(4,33,36,49)*
(4,40,97,105)*
(4,44,53,69)*
(4,45,72,85)*
(4,52,77,93)*
(4,72,87,113)*
(4,77,88,117)*
(4,84,87,121)*
(5,6,30,31)*
(5,8,44,45)*
(5,10,62,63)*
(5,12,84,85)*
(5,38,50,63)*
(5,40,56,69)*
(5,48,60,77)*
(5,54,78,95)*
(5,70,86,111)*
(6,6,7,11)*
(6,6,17,19)*
(6,7,42,43)*
(6,9,18,21)*
(6,9,58,59)*
(6,10,15,19)*
(6,10,33,35)*
(6,11,78,79)*
(6,13,18,23)*
(6,14,27,31)*
(6,14,57,59)*
(6,15,42,45)*
(6,17,30,35)*
(6,18,27,33)*
(6,18,43,47)*
(6,18,89,91)*
(6,21,22,31)*
(6,21,78,81)*
(6,22,63,67)*
(6,23,54,59)*
(6,26,87,91)*
(6,27,38,47)*
(6,27,74,79)*
(6,33,58,67)*
(6,35,42,55)*
(6,38,69,79)*
(6,39,82,91)*
(6,41,42,59)*
(6,42,85,95)*
(6,43,66,79)*
(6,54,73,91)*
(6,58,75,95)*
(6,62,87,107)*
(6,67,78,103)*
(6,73,78,107)*
(6,74,93,119)*
(6,90,95,131)*
(7,8,56,57)*
(7,10,74,75)*
(7,12,96,97)*
(7,14,22,27)*
(7,16,28,33)*
(7,24,60,65)*
(7,26,70,75)*
(7,30,30,43)*
(7,40,40,57)*
(7,48,84,97)*
(7,74,98,123)*
(8,8,31,33)*
(8,9,12,17)*
(8,9,72,73)*
(8,11,16,21)*
(8,11,92,93)*
(8,12,51,53)*
(8,16,79,81)*
(8,21,48,53)*
(8,24,27,37)*
(8,28,49,57)*
(8,29,88,93)*
(8,31,100,105)*
(8,36,63,73)*
(8,36,81,89)*
(8,44,95,105)*
(8,49,64,81)*
(8,51,96,109)*
(8,53,76,93)*
(8,64,67,93)*
(8,69,84,109)*
(8,80,85,117)*
(9,10,90,91)*
(9,12,20,25)*
(9,18,38,43)*
(9,22,54,59)*
(9,24,32,41)*
(9,28,84,89)*
(9,30,50,59)*
(9,32,36,49)*
(9,42,98,107)*
(9,46,78,91)*
(9,62,66,91)*
(10,10,23,27)*
(10,10,49,51)*
(10,14,35,39)*
(10,14,73,75)*
(10,18,51,55)*
(10,22,71,75)*
(10,26,95,99)*
(10,37,50,63)*
(10,41,62,75)*
(10,45,54,71)*
(10,57,90,107)*
(10,65,74,99)*
(11,12,24,29)*
(11,18,42,47)*
(11,22,22,33)*
(11,22,58,63)*
(11,28,88,93)*
(11,36,48,61)*
(11,42,66,79)*
(11,44,52,69)*
(11,58,94,111)*
(11,74,82,111)*
(12,12,71,73)*
(12,14,84,86)*
(12,15,16,25)*
(12,16,21,29)*
(12,16,99,101)*
(12,19,48,53)*
(12,21,28,37)*
(12,21,56,61)*
(12,24,31,41)*
(12,24,41,49)*
(12,27,44,53)*
(12,29,96,101)*
(12,31,36,49)*
(12,32,69,77)*
(12,33,64,73)*
(12,36,37,53)*
(12,36,67,77)*
(12,39,88,97)*
(12,44,57,73)*
(12,44,99,109)*
(12,47,84,97)*
(12,48,59,77)*
(12,52,81,97)*
(12,59,60,85)*
(12,60,95,113)*
(12,88,99,133)*
(12,96,97,137)*
(13,14,34,39)*
(13,16,40,45)*
(13,24,72,77)*
(13,26,82,87)*
(13,50,70,87)*
(13,52,76,93)*
(13,66,78,103)*
(13,76,88,117)*
(14,14,47,51)*
(14,14,97,99)*
(14,17,46,51)*
(14,18,21,31)*
(14,18,63,67)*
(14,22,29,39)*
(14,22,83,87)*
(14,23,70,75)*
(14,27,90,95)*
(14,31,38,51)*
(14,35,98,105)*
(14,38,77,87)*
(14,39,42,59)*
(14,42,63,77)*
(14,42,93,103)*
(14,47,86,99)*
(14,58,79,99)*
(14,62,91,111)*
(14,70,85,111)*
(14,73,98,123)*
(15,18,26,35)*
(15,24,40,49)*
(15,30,58,67)*
(15,36,52,65)*
(15,36,80,89)*
(15,42,50,67)*
(15,42,70,83)*
(15,60,64,89)*
(15,90,94,131)*
(16,17,52,57)*
(16,20,37,45)*
(16,23,76,81)*
(16,27,96,101)*
(16,28,47,57)*
(16,28,61,69)*
(16,32,59,69)*
(16,36,93,101)*
(16,37,56,69)*
(16,41,68,81)*
(16,47,64,81)*
(16,55,88,105)*
(16,63,72,97)*
(16,69,72,101)*
(16,89,92,129)*
(17,20,20,33)*
(17,24,84,89)*
(17,26,94,99)*
(17,32,44,57)*
(17,46,86,99)*
(17,56,56,81)*
(18,18,79,83)*
(18,19,66,71)*
(18,21,38,47)*
(18,21,74,79)*
(18,22,99,103)*
(18,25,30,43)*
(18,26,45,55)*
(18,27,34,47)*
(18,33,74,83)*
(18,34,69,79)*
(18,39,98,107)*
(18,42,49,67)*
(18,46,51,71)*
(18,47,66,83)*
(18,51,90,105)
(18,54,71,91)*
(18,55,90,107)*
(18,66,77,103)*
(18,71,78,107)*
(19,22,26,39)*
(19,22,82,87)*
(19,30,42,55)*
(19,42,54,71)*
(19,48,96,109)*
(19,58,62,87)*
(20,24,57,65)*
(20,28,29,45)*
(20,32,85,93)*
(20,35,56,69)*
(20,36,45,61)*
(20,39,48,65)*
(20,43,80,93)*
(20,44,65,81)*
(20,52,89,105)*
(20,55,56,81)*
(20,79,100,129)*
(20,80,83,117)*
(21,22,90,95)*
(21,24,52,61)*
(21,28,84,91)*
(21,30,70,79)*
(21,36,92,101)*
(21,38,66,79)*
(21,40,72,85)*
(21,50,78,95)*
(21,52,84,101)*
(21,72,100,125)*
(21,78,98,127)*
(22,26,53,63)*
(22,31,34,51)*
(22,33,54,67)*
(22,34,77,87)*
(22,37,46,63)*
(22,45,90,103)*
(22,46,55,75)*
(22,54,75,95)*
(22,58,61,87)*
(22,71,98,123)*
(22,82,89,123)*
(23,24,24,41)*
(23,24,36,49)*
(23,28,44,57)*
(23,44,64,81)*
(23,64,80,105)*
(23,72,84,113)*
(23,84,84,121)*
(24,27,68,77)*
(24,28,63,73)*
(24,28,81,89)*
(24,29,48,61)*
(24,32,75,85)*
(24,33,88,97)*
(24,36,43,61)*
(24,45,68,85)*
(24,48,71,89)*
(24,57,64,89)*
(24,68,87,113)*
(24,80,93,125)*
(24,92,93,133)*
(25,32,40,57)*
(25,34,62,75)*
(25,36,48,65)*
(25,44,92,105)*
(25,60,72,97)*
(25,68,76,105)*
(25,70,98,123)*
(26,38,43,63)*
(26,39,78,91)*
(26,42,51,71)*
(26,49,82,99)*
(26,53,94,111)*
(26,57,66,91)*
(26,58,91,111)*
(26,65,70,99)*
(27,28,36,53)*
(27,30,86,95)*
(27,34,66,79)*
(27,40,60,77)*
(27,44,96,109)*
(27,64,84,109)*
(27,72,96,123)*
(27,88,96,133)*
(28,28,41,57)*
(28,36,57,73)*
(28,36,99,109)*
(28,41,64,81)*
(28,44,77,93)*
(28,68,91,117)*
(28,75,96,125)*
(29,48,84,101)*
(29,58,58,87)*
(30,30,41,59)*
(30,33,50,67)*
(30,33,70,83)*
(30,35,54,71)*
(30,42,65,83)*
(30,45,46,71)*
(30,54,97,115)*
(30,55,66,91)*
(30,58,69,95)*
(30,75,98,127)*
(30,78,79,115)*
(31,38,86,99)*
(31,40,92,105)*
(31,42,42,67)*
(31,46,82,99)*
(31,58,74,99)*
(31,90,90,131)*
(32,35,80,93)*
(32,43,76,93)*
(32,48,51,77)*
(32,49,56,81)*
(32,51,60,85)*
(32,56,67,93)*
(32,56,94,114)
(32,64,64,96)
(32,65,76,105)*
(32,72,81,113)*
(32,76,83,117)*
(32,80,100,132)
(32,81,84,121)*
(33,36,72,87)
(33,38,66,83)*
(33,44,48,73)*
(33,56,72,97)*
(33,66,66,99)
(33,68,84,113)*
(33,92,96,137)*
(34,37,38,63)*
(34,38,55,75)*
(34,40,40,66)
(34,42,63,83)*
(34,63,90,115)*
(34,64,88,114)
(34,66,93,119)*
(34,68,68,102)
(35,38,70,87)*
(35,40,44,69)*
(35,50,62,87)*
(35,60,84,109)*
(35,70,70,105)
(35,72,96,125)*
(35,86,98,135)*
(36,36,73,89)*
(36,37,96,109)*
(36,43,84,101)*
(36,44,93,109)*
(36,48,91,109)*
(36,55,60,89)*
(36,61,72,101)*
(36,76,87,121)*
(36,88,93,133)*
(37,46,94,111)*
(37,74,74,111)*
(38,41,50,75)*
(38,46,79,99)*
(38,54,99,119)*
(38,59,86,111)*
(39,48,64,89)*
(39,52,72,97)*
(39,54,62,91)*
(39,62,78,107)*
(40,41,88,105)*
(40,44,55,81)*
(40,45,48,77)*
(40,60,87,113)*
(40,64,73,105)*
(40,71,100,129)*
(40,91,100,141)*
(41,62,98,123)*
(42,42,89,107)*
(42,46,87,107)*
(42,47,54,83)*
(42,50,69,95)*
(42,54,77,103)*
(42,66,67,103)*
(42,66,73,107)*
(42,69,98,127)*
(42,70,81,115)*
(42,78,91,127)*
(42,81,94,131)*
(43,52,64,93)*
(43,64,88,117)*
(44,69,72,109)*
(44,79,92,129)*
(45,50,78,103)*
(46,47,74,99)*
(46,53,86,111)*
(46,54,57,91)*
(46,57,78,107)*
(46,59,82,111)*
(47,54,90,115)*
(47,60,60,97)*
(48,56,63,97)*
(48,56,69,101)*
(48,76,81,121)*
(48,84,97,137)*
(49,50,70,99)*
(49,72,72,113)*
(49,72,84,121)*
(49,76,92,129)*
(50,55,98,123)*
(50,81,90,131)*
(50,83,94,135)*
(51,64,72,109)*
(51,74,78,119)*
(52,53,56,93)*
(52,64,83,117)*
(52,72,99,133)*
(53,56,88,117)*
(53,60,96,125)*
(54,65,78,115)*
(54,66,83,119)*
(54,75,90,129)*
(54,87,94,139)*
(56,63,84,119)*
(56,64,97,129)*
(56,68,77,117)*
(56,71,92,129)*
(56,91,92,141)*
(57,62,66,107)*
(57,84,92,137)*
(58,59,74,111)*
(58,62,89,123)*
(58,71,82,123)*
(58,94,97,147)*
(59,80,100,141)*
(60,75,92,133)*
(61,66,78,119)*
(61,70,98,135)*
(63,66,70,115)*
(63,66,94,131)*
(63,84,88,137)*
(63,84,100,145)*
(64,68,89,129)*
(66,86,87,139)*
(67,70,94,135)*
(68,76,79,129)*
(69,72,88,133)*
(69,78,98,143)*
(70,70,73,123)*
(70,77,86,135)*
(70,90,99,151)*
(74,82,97,147)*
(75,82,90,143)*
(77,84,96,149)*
(80,84,87,145)*
(86,91,98,159)*
(88,88,89,153)*

Dan joyce <danj4084@gmail.com> wrote:
> A list of all primitive quads like this is hard to find on the web.
>
> Primitive quads starting with the smallest side of (1) up to (88)
> (*) = primitive
> (1,2,2,3)*
> (1,4,8,9)*
> (1,6,18,19)*
> (1,8,32,33)*
> (1,10,50,51)*
> (1,12,12,17)*
> (1,12,72,73)*
> (1,14,98,99)*
> (1,18,30,35)*
> (1,22,46,51)*
> (1,28,76,81)*
> (1,32,100,105)*
> (1,34,38,51)*
> (1,44,68,81)*
> (1,68,80,105)*
> (1,70,70,99)*
> (2,3,6,7)*
> (2,5,14,15)*

Do you have some criteria or restrictions that weren't mentioned?
There are infinitely many primitive Pythagorean quads with various
small sides. For example, your list doesn't show any of
(1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
(1,228,684,721), or (3,22347444,52144036,56731021).

> (2,6,9,11)*
> (2,7,26,27)*
> (2,9,42,43)*
> (2,10,11,15)*
(snip ~420 lines)
> (75,82,90,143)*
> (77,84,96,149)*
> (80,84,87,145)*
> (86,91,98,159)*
> (88,88,89,153)*
>
> Any errors or omissions?

On Mon, 31 Oct 2022 06:34:48 -0000 (UTC), James Waldby
<j-waldby@no.no> wrote:

>Dan joyce <danj4084@gmail.com> wrote:
>> A list of all primitive quads like this is hard to find on the web.
>>
>> Primitive quads starting with the smallest side of (1) up to (88)
>> (*) = primitive
>> (1,2,2,3)*
>> (1,4,8,9)*
>> (1,6,18,19)*
>> (1,8,32,33)*
>> (1,10,50,51)*
>> (1,12,12,17)*
>> (1,12,72,73)*
>> (1,14,98,99)*
>> (1,18,30,35)*
>> (1,22,46,51)*
>> (1,28,76,81)*
>> (1,32,100,105)*
>> (1,34,38,51)*
>> (1,44,68,81)*
>> (1,68,80,105)*
>> (1,70,70,99)*
>> (2,3,6,7)*
>> (2,5,14,15)*
>
>Do you have some criteria or restrictions that weren't mentioned?
>There are infinitely many primitive Pythagorean quads with various
>small sides. For example, your list doesn't show any of
>(1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
>(1,228,684,721), or (3,22347444,52144036,56731021).

There is some kind of error in the second example. The sum of the
squares of the first three terms is odd and the square of the fourth
is even.

--
Remove del for email

On Monday, October 31, 2022 at 2:34:57 AM UTC-4, James Waldby wrote:
> Dan joyce <danj...@gmail.com> wrote:
> > A list of all primitive quads like this is hard to find on the web.
> >
> > Primitive quads starting with the smallest side of (1) up to (88)
> > (*) = primitive
> > (1,2,2,3)*
> > (1,4,8,9)*
> > (1,6,18,19)*
> > (1,8,32,33)*
> > (1,10,50,51)*
> > (1,12,12,17)*
> > (1,12,72,73)*
> > (1,14,98,99)*
> > (1,18,30,35)*
> > (1,22,46,51)*
> > (1,28,76,81)*
> > (1,32,100,105)*
> > (1,34,38,51)*
> > (1,44,68,81)*
> > (1,68,80,105)*
> > (1,70,70,99)*
> > (2,3,6,7)*
> > (2,5,14,15)*
> Do you have some criteria or restrictions that weren't mentioned?
> There are infinitely many primitive Pythagorean quads with various
> small sides. For example, your list doesn't show any of
> (1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
> (1,228,684,721), or (3,22347444,52144036,56731021).
>
> > (2,6,9,11)*
> > (2,7,26,27)*
> > (2,9,42,43)*
> > (2,10,11,15)*
> (snip ~420 lines)
> > (75,82,90,143)*
> > (77,84,96,149)*
> > (80,84,87,145)*
> > (86,91,98,159)*
> > (88,88,89,153)*
> >
> > Any errors or omissions?
Sorry about that.
A limit was placed on the smallest of three sides as <100
Instead I wrote this and not explaining it better ---
"Primitive quads starting with the smallest side of (1) up too (88)"

On Monday, October 31, 2022 at 10:37:56 AM UTC-4, Dan joyce wrote:
> On Monday, October 31, 2022 at 2:34:57 AM UTC-4, James Waldby wrote:
> > Dan joyce <danj...@gmail.com> wrote:
> > > A list of all primitive quads like this is hard to find on the web.
> > >
> > > Primitive quads starting with the smallest side of (1) up to (88)
> > > (*) = primitive
> > > (1,2,2,3)*
> > > (1,4,8,9)*
> > > (1,6,18,19)*
> > > (1,8,32,33)*
> > > (1,10,50,51)*
> > > (1,12,12,17)*
> > > (1,12,72,73)*
> > > (1,14,98,99)*
> > > (1,18,30,35)*
> > > (1,22,46,51)*
> > > (1,28,76,81)*
> > > (1,32,100,105)*
> > > (1,34,38,51)*
> > > (1,44,68,81)*
> > > (1,68,80,105)*
> > > (1,70,70,99)*
> > > (2,3,6,7)*
> > > (2,5,14,15)*
> > Do you have some criteria or restrictions that weren't mentioned?
> > There are infinitely many primitive Pythagorean quads with various
> > small sides. For example, your list doesn't show any of
> > (1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
> > (1,228,684,721), or (3,22347444,52144036,56731021).
> >
> > > (2,6,9,11)*
> > > (2,7,26,27)*
> > > (2,9,42,43)*
> > > (2,10,11,15)*
> > (snip ~420 lines)
> > > (75,82,90,143)*
> > > (77,84,96,149)*
> > > (80,84,87,145)*
> > > (86,91,98,159)*
> > > (88,88,89,153)*
> > >
> > > Any errors or omissions?
> Sorry about that.
> A limit was placed on the smallest of three sides as <100
> Instead I wrote this and not explaining it better ---
> "Primitive quads starting with the smallest side of (1) up too (88)"

James,
Not a good answer to your question, It was really about the 1st and 2nd smallest term taken in
order <100

On Monday, October 31, 2022 at 11:02:12 AM UTC-4, Dan joyce wrote:
> On Monday, October 31, 2022 at 10:37:56 AM UTC-4, Dan joyce wrote:
> > On Monday, October 31, 2022 at 2:34:57 AM UTC-4, James Waldby wrote:
> > > Dan joyce <danj...@gmail.com> wrote:
> > > > A list of all primitive quads like this is hard to find on the web.
> > > >
> > > > Primitive quads starting with the smallest side of (1) up to (88)
> > > > (*) = primitive
> > > > (1,2,2,3)*
> > > > (1,4,8,9)*
> > > > (1,6,18,19)*
> > > > (1,8,32,33)*
> > > > (1,10,50,51)*
> > > > (1,12,12,17)*
> > > > (1,12,72,73)*
> > > > (1,14,98,99)*
> > > > (1,18,30,35)*
> > > > (1,22,46,51)*
> > > > (1,28,76,81)*
> > > > (1,32,100,105)*
> > > > (1,34,38,51)*
> > > > (1,44,68,81)*
> > > > (1,68,80,105)*
> > > > (1,70,70,99)*
> > > > (2,3,6,7)*
> > > > (2,5,14,15)*
> > > Do you have some criteria or restrictions that weren't mentioned?
> > > There are infinitely many primitive Pythagorean quads with various
> > > small sides. For example, your list doesn't show any of
> > > (1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
> > > (1,228,684,721), or (3,22347444,52144036,56731021).
> > >
> > > > (2,6,9,11)*
> > > > (2,7,26,27)*
> > > > (2,9,42,43)*
> > > > (2,10,11,15)*
> > > (snip ~420 lines)
> > > > (75,82,90,143)*
> > > > (77,84,96,149)*
> > > > (80,84,87,145)*
> > > > (86,91,98,159)*
> > > > (88,88,89,153)*
> > > >
> > > > Any errors or omissions?
> > Sorry about that.
> > A limit was placed on the smallest of three sides as <100
> > Instead I wrote this and not explaining it better ---
> > "Primitive quads starting with the smallest side of (1) up too (88)"
> James,
> Not a good answer to your question, It was really about the 1st and 2nd smallest term taken in
> order <100

Add in the third term.

On Monday, October 31, 2022 at 11:04:46 AM UTC-4, Dan joyce wrote:
> On Monday, October 31, 2022 at 11:02:12 AM UTC-4, Dan joyce wrote:
> > On Monday, October 31, 2022 at 10:37:56 AM UTC-4, Dan joyce wrote:
> > > On Monday, October 31, 2022 at 2:34:57 AM UTC-4, James Waldby wrote:
> > > > Dan joyce <danj...@gmail.com> wrote:
> > > > > A list of all primitive quads like this is hard to find on the web.
> > > > >
> > > > > Primitive quads starting with the smallest side of (1) up to (88)
> > > > > (*) = primitive
> > > > > (1,2,2,3)*
> > > > > (1,4,8,9)*
> > > > > (1,6,18,19)*
> > > > > (1,8,32,33)*
> > > > > (1,10,50,51)*
> > > > > (1,12,12,17)*
> > > > > (1,12,72,73)*
> > > > > (1,14,98,99)*
> > > > > (1,18,30,35)*
> > > > > (1,22,46,51)*
> > > > > (1,28,76,81)*
> > > > > (1,32,100,105)*
> > > > > (1,34,38,51)*
> > > > > (1,44,68,81)*
> > > > > (1,68,80,105)*
> > > > > (1,70,70,99)*
> > > > > (2,3,6,7)*
> > > > > (2,5,14,15)*
> > > > Do you have some criteria or restrictions that weren't mentioned?
> > > > There are infinitely many primitive Pythagorean quads with various
> > > > small sides. For example, your list doesn't show any of
> > > > (1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
> > > > (1,228,684,721), or (3,22347444,52144036,56731021).
> > > >
> > > > > (2,6,9,11)*
> > > > > (2,7,26,27)*
> > > > > (2,9,42,43)*
> > > > > (2,10,11,15)*
> > > > (snip ~420 lines)
> > > > > (75,82,90,143)*
> > > > > (77,84,96,149)*
> > > > > (80,84,87,145)*
> > > > > (86,91,98,159)*
> > > > > (88,88,89,153)*
> > > > >
> > > > > Any errors or omissions?
> > > Sorry about that.
> > > A limit was placed on the smallest of three sides as <100
> > > Instead I wrote this and not explaining it better ---
> > > "Primitive quads starting with the smallest side of (1) up too (88)"
> > James,
> > Not a good answer to your question, It was really about the 1st and 2nd smallest term taken in
> > order <100
> Add in the third term.

This would makes a great algorithm where the next group would stop @ <=200 in the third term.
Capturing all quads in the ordering of the 1st 2nd 3rd term. Yes, you need a GCD check for each Quad.
Then go on from there.

Barry Schwarz <schwarzb@delq.com> wrote:
> On Mon, 31 Oct 2022 06:34:48 -0000 (UTC), James Waldby
> <j-waldby@no.no> wrote:
>
>>Dan joyce <danj4084@gmail.com> wrote:
>>> A list of all primitive quads like this is hard to find on the web.
>>>
>>> Primitive quads starting with the smallest side of (1) up to (88)
>>> (*) = primitive
>>> (1,2,2,3)*
>>> (1,4,8,9)*
>>> (1,6,18,19)*
....
>>> (1,68,80,105)*
>>> (1,70,70,99)*
>>> (2,3,6,7)*
>>> (2,5,14,15)*
>>
>>Do you have some criteria or restrictions that weren't mentioned?
>>There are infinitely many primitive Pythagorean quads with various
>>small sides. For example, your list doesn't show any of
>>(1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
>>(1,228,684,721), or (3,22347444,52144036,56731021).
>
> There is some kind of error in the second example. The sum of the
> squares of the first three terms is odd and the square of the fourth
> is even.

Instead of (1,408,408,5770) it should be (1,408,408,577). Thanks!

On Monday, October 31, 2022 at 1:12:16 PM UTC-4, Dan joyce wrote:
> On Monday, October 31, 2022 at 11:04:46 AM UTC-4, Dan joyce wrote:
> > On Monday, October 31, 2022 at 11:02:12 AM UTC-4, Dan joyce wrote:
> > > On Monday, October 31, 2022 at 10:37:56 AM UTC-4, Dan joyce wrote:
> > > > On Monday, October 31, 2022 at 2:34:57 AM UTC-4, James Waldby wrote:
> > > > > Dan joyce <danj...@gmail.com> wrote:
> > > > > > A list of all primitive quads like this is hard to find on the web.
> > > > > >
> > > > > > Primitive quads starting with the smallest side of (1) up to (88)
> > > > > > (*) = primitive
> > > > > > (1,2,2,3)*
> > > > > > (1,4,8,9)*
> > > > > > (1,6,18,19)*
> > > > > > (1,8,32,33)*
> > > > > > (1,10,50,51)*
> > > > > > (1,12,12,17)*
> > > > > > (1,12,72,73)*
> > > > > > (1,14,98,99)*
> > > > > > (1,18,30,35)*
> > > > > > (1,22,46,51)*
> > > > > > (1,28,76,81)*
> > > > > > (1,32,100,105)*
> > > > > > (1,34,38,51)*
> > > > > > (1,44,68,81)*
> > > > > > (1,68,80,105)*
> > > > > > (1,70,70,99)*
> > > > > > (2,3,6,7)*
> > > > > > (2,5,14,15)*
> > > > > Do you have some criteria or restrictions that weren't mentioned?
> > > > > There are infinitely many primitive Pythagorean quads with various
> > > > > small sides. For example, your list doesn't show any of
> > > > > (1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
> > > > > (1,228,684,721), or (3,22347444,52144036,56731021).
> > > > >
> > > > > > (2,6,9,11)*
> > > > > > (2,7,26,27)*
> > > > > > (2,9,42,43)*
> > > > > > (2,10,11,15)*
> > > > > (snip ~420 lines)
> > > > > > (75,82,90,143)*
> > > > > > (77,84,96,149)*
> > > > > > (80,84,87,145)*
> > > > > > (86,91,98,159)*
> > > > > > (88,88,89,153)*
> > > > > >
> > > > > > Any errors or omissions?
> > > > Sorry about that.
> > > > A limit was placed on the smallest of three sides as <100
> > > > Instead I wrote this and not explaining it better ---
> > > > "Primitive quads starting with the smallest side of (1) up too (88)"
> > > James,
> > > Not a good answer to your question, It was really about the 1st and 2nd smallest term taken in
> > > order <100
> > Add in the third term.
> This would makes a great algorithm where the next group would stop @ <=200 in the third term.
> Capturing all quads in the ordering of the 1st 2nd 3rd term. Yes, you need a GCD check for each Quad.
> Then go on from there.

I checked further on my list and there is a flaw in my GCD algorithm.
Around 15-20 entries are not primitive.
See if you can find them in my list.

On Monday, October 31, 2022 at 10:25:25 PM UTC-4, Dan joyce wrote:
> On Monday, October 31, 2022 at 1:12:16 PM UTC-4, Dan joyce wrote:
> > On Monday, October 31, 2022 at 11:04:46 AM UTC-4, Dan joyce wrote:
> > > On Monday, October 31, 2022 at 11:02:12 AM UTC-4, Dan joyce wrote:
> > > > On Monday, October 31, 2022 at 10:37:56 AM UTC-4, Dan joyce wrote:
> > > > > On Monday, October 31, 2022 at 2:34:57 AM UTC-4, James Waldby wrote:
> > > > > > Dan joyce <danj...@gmail.com> wrote:
> > > > > > > A list of all primitive quads like this is hard to find on the web.
> > > > > > >
> > > > > > > Primitive quads starting with the smallest side of (1) up to (88)
> > > > > > > (*) = primitive
> > > > > > > (1,2,2,3)*
> > > > > > > (1,4,8,9)*
> > > > > > > (1,6,18,19)*
> > > > > > > (1,8,32,33)*
> > > > > > > (1,10,50,51)*
> > > > > > > (1,12,12,17)*
> > > > > > > (1,12,72,73)*
> > > > > > > (1,14,98,99)*
> > > > > > > (1,18,30,35)*
> > > > > > > (1,22,46,51)*
> > > > > > > (1,28,76,81)*
> > > > > > > (1,32,100,105)*
> > > > > > > (1,34,38,51)*
> > > > > > > (1,44,68,81)*
> > > > > > > (1,68,80,105)*
> > > > > > > (1,70,70,99)*
> > > > > > > (2,3,6,7)*
> > > > > > > (2,5,14,15)*
> > > > > > Do you have some criteria or restrictions that weren't mentioned?
> > > > > > There are infinitely many primitive Pythagorean quads with various
> > > > > > small sides. For example, your list doesn't show any of
> > > > > > (1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
> > > > > > (1,228,684,721), or (3,22347444,52144036,56731021).
> > > > > >
> > > > > > > (2,6,9,11)*
> > > > > > > (2,7,26,27)*
> > > > > > > (2,9,42,43)*
> > > > > > > (2,10,11,15)*
> > > > > > (snip ~420 lines)
> > > > > > > (75,82,90,143)*
> > > > > > > (77,84,96,149)*
> > > > > > > (80,84,87,145)*
> > > > > > > (86,91,98,159)*
> > > > > > > (88,88,89,153)*
> > > > > > >
> > > > > > > Any errors or omissions?
> > > > > Sorry about that.
> > > > > A limit was placed on the smallest of three sides as <100
> > > > > Instead I wrote this and not explaining it better ---
> > > > > "Primitive quads starting with the smallest side of (1) up too (88)"
> > > > James,
> > > > Not a good answer to your question, It was really about the 1st and 2nd smallest term taken in
> > > > order <100
> > > Add in the third term.
> > This would makes a great algorithm where the next group would stop @ <=200 in the third term.
> > Capturing all quads in the ordering of the 1st 2nd 3rd term. Yes, you need a GCD check for each Quad.
> > Then go on from there.
> I checked further on my list and there is a flaw in my GCD algorithm.
> Around 15-20 entries are not primitive.
> See if you can find them in my list.

A new simpler sort of primitive quads using the one odd side and then the two
even sides in the sort. Placing a limit on sides c,b,a when c>99 then when b>99
then when a>99 ending the sort. The new list below.
a is always odd and b and c are always even in the sort.
d, the space diagonal, is always odd but not in the sort.
Much higher limits of >99 depends on your software limitations.

a,b,c,d

1,2,2,3
1,4,8,9
1,6,18,19
1,8,32,33
1,10,50,51
1,12,12,17
1,12,72,73
1,14,98,99
1,18,30,35
1,22,46,51
1,28,76,81
1,34,38,51
1,44,68,81
1,34,38,51
1,44,68,81
1,68,80,105
1,70,70,99
3,2,6,7
3,4,12,13
3,6,22,23
3,8,36,37
3,10,54,55
3,12,76,77
3,14,18,23
3,16,24,29
3,24,28,37
3,24,56,61
3,26,66,71
3,30,46,55
3,36,68,77
3,42,94,103
3,46,54,71
3,50,90,103
3,56,84,101
3,80,96,125
3,92,96,133
5,2,14,15
5,4,20,21
5,6,30,31
5,8,44,45
5,10,62,63
5,12,84,85
5,38,50,63
5,40,56,69
5,48,60,77
5,54,78,95
5,70,86,111
7,2,26,27,
7,4,4,9
7,4,32,33
7,6,6,11
7,6,42,43
7,8,56,57
7,10,74,75
7,12,96,97
7,14,22,27
7,16,28,33
7,24,60,65
7,26,70,75
7,30,30,43
7,40,40,57
7,48,84,97
7,74,98.123
9,2,6,11
9,2,42,43
9,4,48,49
9,6,18,21
9,6,58,59
9,8,12,17
9,8,72,73
9,10,90,91
9,12,20,25
9,18,38,43
9,22,54,59
9,24,32,41
9,28,84,89
9,30,50,59
9,32,36,49
9,42,98,107
9,46,78,91
9,62,66,91
...

99,2,66,119
99,12,16,101
99,12,44,109
99,12,88,133
99,18,22,103
99,28,36,109
99,38,54,119
99,52,72,133
99,70,90,151

End of run for quads starting @ 1,3,5,7,..99
Dan

On Thursday, November 3, 2022 at 9:49:17 PM UTC-4, Dan joyce wrote:
> On Monday, October 31, 2022 at 10:25:25 PM UTC-4, Dan joyce wrote:
> > On Monday, October 31, 2022 at 1:12:16 PM UTC-4, Dan joyce wrote:
> > > On Monday, October 31, 2022 at 11:04:46 AM UTC-4, Dan joyce wrote:
> > > > On Monday, October 31, 2022 at 11:02:12 AM UTC-4, Dan joyce wrote:
> > > > > On Monday, October 31, 2022 at 10:37:56 AM UTC-4, Dan joyce wrote:
> > > > > > On Monday, October 31, 2022 at 2:34:57 AM UTC-4, James Waldby wrote:
> > > > > > > Dan joyce <danj...@gmail.com> wrote:
> > > > > > > > A list of all primitive quads like this is hard to find on the web.
> > > > > > > >
> > > > > > > > Primitive quads starting with the smallest side of (1) up to (88)
> > > > > > > > (*) = primitive
> > > > > > > > (1,2,2,3)*
> > > > > > > > (1,4,8,9)*
> > > > > > > > (1,6,18,19)*
> > > > > > > > (1,8,32,33)*
> > > > > > > > (1,10,50,51)*
> > > > > > > > (1,12,12,17)*
> > > > > > > > (1,12,72,73)*
> > > > > > > > (1,14,98,99)*
> > > > > > > > (1,18,30,35)*
> > > > > > > > (1,22,46,51)*
> > > > > > > > (1,28,76,81)*
> > > > > > > > (1,32,100,105)*
> > > > > > > > (1,34,38,51)*
> > > > > > > > (1,44,68,81)*
> > > > > > > > (1,68,80,105)*
> > > > > > > > (1,70,70,99)*
> > > > > > > > (2,3,6,7)*
> > > > > > > > (2,5,14,15)*
> > > > > > > Do you have some criteria or restrictions that weren't mentioned?
> > > > > > > There are infinitely many primitive Pythagorean quads with various
> > > > > > > small sides. For example, your list doesn't show any of
> > > > > > > (1,72,144,161), (1,408,408,5770), (1,543339720,543339720,768398401),
> > > > > > > (1,228,684,721), or (3,22347444,52144036,56731021).
> > > > > > >
> > > > > > > > (2,6,9,11)*
> > > > > > > > (2,7,26,27)*
> > > > > > > > (2,9,42,43)*
> > > > > > > > (2,10,11,15)*
> > > > > > > (snip ~420 lines)
> > > > > > > > (75,82,90,143)*
> > > > > > > > (77,84,96,149)*
> > > > > > > > (80,84,87,145)*
> > > > > > > > (86,91,98,159)*
> > > > > > > > (88,88,89,153)*
> > > > > > > >
> > > > > > > > Any errors or omissions?
> > > > > > Sorry about that.
> > > > > > A limit was placed on the smallest of three sides as <100
> > > > > > Instead I wrote this and not explaining it better ---
> > > > > > "Primitive quads starting with the smallest side of (1) up too (88)"
> > > > > James,
> > > > > Not a good answer to your question, It was really about the 1st and 2nd smallest term taken in
> > > > > order <100
> > > > Add in the third term.
> > > This would makes a great algorithm where the next group would stop @ <=200 in the third term.
> > > Capturing all quads in the ordering of the 1st 2nd 3rd term. Yes, you need a GCD check for each Quad.
> > > Then go on from there.
> > I checked further on my list and there is a flaw in my GCD algorithm.
> > Around 15-20 entries are not primitive.
> > See if you can find them in my list.
> A new simpler sort of primitive quads using the one odd side and then the two
> even sides in the sort. Placing a limit on sides c,b,a when c>99 then when b>99
> then when a>99 ending the sort. The new list below.
> a is always odd and b and c are always even in the sort.
> d, the space diagonal, is always odd but not in the sort.
> Much higher limits of >99 depends on your software limitations.
>
> a,b,c,d
>
> 1,2,2,3
> 1,4,8,9
> 1,6,18,19
> 1,8,32,33
> 1,10,50,51
> 1,12,12,17
> 1,12,72,73
> 1,14,98,99
> 1,18,30,35
> 1,22,46,51
> 1,28,76,81
> 1,34,38,51
> 1,44,68,81
> 1,34,38,51
> 1,44,68,81
> 1,68,80,105
> 1,70,70,99
> 3,2,6,7
> 3,4,12,13
> 3,6,22,23
> 3,8,36,37
> 3,10,54,55
> 3,12,76,77
> 3,14,18,23
> 3,16,24,29
> 3,24,28,37
> 3,24,56,61
> 3,26,66,71
> 3,30,46,55
> 3,36,68,77
> 3,42,94,103
> 3,46,54,71
> 3,50,90,103
> 3,56,84,101
> 3,80,96,125
> 3,92,96,133
> 5,2,14,15
> 5,4,20,21
> 5,6,30,31
> 5,8,44,45
> 5,10,62,63
> 5,12,84,85
> 5,38,50,63
> 5,40,56,69
> 5,48,60,77
> 5,54,78,95
> 5,70,86,111
> 7,2,26,27,
> 7,4,4,9
> 7,4,32,33
> 7,6,6,11
> 7,6,42,43
> 7,8,56,57
> 7,10,74,75
> 7,12,96,97
> 7,14,22,27
> 7,16,28,33
> 7,24,60,65
> 7,26,70,75
> 7,30,30,43
> 7,40,40,57
> 7,48,84,97
> 7,74,98.123
> 9,2,6,11
> 9,2,42,43
> 9,4,48,49
> 9,6,18,21
> 9,6,58,59
> 9,8,12,17
> 9,8,72,73
> 9,10,90,91
> 9,12,20,25
> 9,18,38,43
> 9,22,54,59
> 9,24,32,41
> 9,28,84,89
> 9,30,50,59
> 9,32,36,49
> 9,42,98,107
> 9,46,78,91
> 9,62,66,91
> ..
>
> 99,2,66,119
> 99,12,16,101
> 99,12,44,109
> 99,12,88,133
> 99,18,22,103
> 99,28,36,109
> 99,38,54,119
> 99,52,72,133
> 99,70,90,151
>
> End of run for quads starting @ 1,3,5,7,..99
> Dan

The trick is getting the right algorithm for GCD has to =1 to be a primitive quad.
I believe all above are primitive

On Thu, 3 Nov 2022 18:49:13 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

<snip>

>A new simpler sort of primitive quads using the one odd side and then the two
>even sides in the sort. Placing a limit on sides c,b,a when c>99 then when b>99
>then when a>99 ending the sort. The new list below.
>a is always odd and b and c are always even in the sort.
>d, the space diagonal, is always odd but not in the sort.
>Much higher limits of >99 depends on your software limitations.
>
>a,b,c,d
>
>1,2,2,3
>1,4,8,9
>1,6,18,19

<snip>

>9,2,6,11
>9,2,42,43
>9,4,48,49
>9,6,18,21

Obviously not primitive since the GCD is 3.

>9,6,58,59
>9,8,12,17
>9,8,72,73

<snip>

--
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On Friday, November 4, 2022 at 3:08:16 AM UTC-4, Barry Schwarz wrote:
> On Thu, 3 Nov 2022 18:49:13 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
>
> <snip>
> >A new simpler sort of primitive quads using the one odd side and then the two
> >even sides in the sort. Placing a limit on sides c,b,a when c>99 then when b>99
> >then when a>99 ending the sort. The new list below.
> >a is always odd and b and c are always even in the sort.
> >d, the space diagonal, is always odd but not in the sort.
> >Much higher limits of >99 depends on your software limitations.
> >
> >a,b,c,d
> >
> >1,2,2,3
> >1,4,8,9
> >1,6,18,19
> <snip>
>
> >9,2,6,11
> >9,2,42,43
> >9,4,48,49
> >9,6,18,21
>
> Obviously not primitive since the GCD is 3.
>
> >9,6,58,59
> >9,8,12,17
> >9,8,72,73
>
> <snip>
> --
> Remove del for email

You are right, 9,6,18,21 is not primitive. The only one?