integer quadruples with all pairwise distances being squares

[Max Alekseyev]

On Thu, Apr 17, 2008 at 4:32 PM, Max Alekseyev <maxale at gmail.com> wrote:
> SeqFaq,

Oh, that's funny ;)
Of course, I meant SeqFans.

>  There are quite interesting recent findings of quadruples of distinct
>  integers with all six pairwise distances being squares:
>  http://www.mathlinks.ro/viewtopic.php?t=33650
>
>  It is clear that the smallest element of such a quadruple can be taken
>  equal 0 (by shifting all 4 elements). Then the other elements are
>  perfect squares themselves and so are their pairwise distances.
>  In other words, each such quadruple corresponds to an unique triple of
>  distinct squares (x^2,y^2,z^2) such that x^2<y^2<z^2 and each of
>  y^2-x^2, z^2-x^2, z^2-y^2 is also a square.
>
>  Would anybody like to compute such triples (say, ordered by the value
>  of z) and add them to OEIS?

On the second thought, it's better to represent such triples as
(z,y,x) (i.e., from the largest to the smallest element) so that they
will naturally admit a lexicographical order.
It also makes sense to add separately the sequences of z values, y
values, and x values.

Regards,
Max




ma> From seqfan-owner at ext.jussieu.fr  Fri Apr 18 01:32:38 2008
ma> Date: Thu, 17 Apr 2008 16:32:22 -0700
ma> From: "Max Alekseyev" <maxale at gmail.com>
ma> To: SeqFan <seqfan at ext.jussieu.fr>
ma> Subject: integer quadruples with all pairwise distances being squares
ma> 
ma> SeqFaq,
ma> 
ma> There are quite interesting recent findings of quadruples of distinct
ma> integers with all six pairwise distances being squares:
ma> http://www.mathlinks.ro/viewtopic.php?t=33650
ma> 
ma> It is clear that the smallest element of such a quadruple can be taken
ma> equal 0 (by shifting all 4 elements). Then the other elements are
ma> perfect squares themselves and so are their pairwise distances.
ma> In other words, each such quadruple corresponds to an unique triple of
ma> distinct squares (x^2,y^2,z^2) such that x^2<y^2<z^2 and each of
ma> y^2-x^2, z^2-x^2, z^2-y^2 is also a square.
ma> 
ma> Would anybody like to compute such triples (say, ordered by the value
ma> of z) and add them to OEIS?
ma> 
ma> Thanks,
ma> Max

With columns z,y,x, sqrt(z^2-y^2), sqrt(z^2-x^2), sqrt(y^2-x^2), by definition
all integer, this starts
(for UNIXes pipe through
to get a b-file of the z which includes duplicates, or through
to suppress duplicates. Please check!

697 185 153 672 680 104
697 680 672 153 185 104
925 533 520 756 765 117
925 765 756 520 533 117
1073 952 448 495 975 840
1073 975 495 448 952 840
1105 520 264 975 1073 448
1105 561 264 952 1073 495
1105 1073 952 264 561 495
1105 1073 975 264 520 448
1394 370 306 1344 1360 208
1394 1360 1344 306 370 208
1850 1066 1040 1512 1530 234
1850 1530 1512 1040 1066 234
2091 555 459 2016 2040 312
2091 2040 2016 459 555 312
2146 1904 896 990 1950 1680
2146 1950 990 896 1904 1680
2165 725 644 2040 2067 333
2165 2067 2040 644 725 333
2210 1040 528 1950 2146 896
2210 1122 528 1904 2146 990
2210 2146 1904 528 1122 990
2210 2146 1950 528 1040 896
2665 2175 1092 1540 2431 1881
2665 2431 1540 1092 2175 1881
2775 1599 1560 2268 2295 351
2775 2295 2268 1560 1599 351
2788 740 612 2688 2720 416
2788 2720 2688 612 740 416
3219 2856 1344 1485 2925 2520
3219 2925 1485 1344 2856 2520
3277 2555 1925 2052 2652 1680
3277 2652 2052 1925 2555 1680
3315 1560 792 2925 3219 1344
3315 1683 792 2856 3219 1485
3315 3219 2856 792 1683 1485
3315 3219 2925 792 1560 1344
3485 925 533 3360 3444 756
3485 925 765 3360 3400 520
3485 2275 1428 2640 3179 1771
3485 2275 2261 2640 2652 252
3485 2652 2640 2261 2275 252
3485 3179 2640 1428 2275 1771
3485 3400 3360 765 925 520
3485 3444 3360 533 925 756
3700 2132 2080 3024 3060 468
3700 3060 3024 2080 2132 468
3965 3723 840 1364 3875 3627
3965 3875 1364 840 3723 3627
4181 1869 819 3740 4100 1680
4181 4100 3740 819 1869 1680
4182 1110 918 4032 4080 624
4182 4080 4032 918 1110 624
4225 615 468 4180 4199 399
4225 4199 4180 468 615 399
4292 3808 1792 1980 3900 3360
4292 3900 1980 1792 3808 3360
4330 1450 1288 4080 4134 666
4330 4134 4080 1288 1450 666
4420 2080 1056 3900 4292 1792
4420 2244 1056 3808 4292 1980
4420 4292 3808 1056 2244 1980
4420 4292 3900 1056 2080 1792
4453 3485 2275 2772 3828 2640
4453 3828 2772 2275 3485 2640
4625 2665 2600 3780 3825 585
4625 3825 3780 2600 2665 585
4879 1295 1071 4704 4760 728
4879 4760 4704 1071 1295 728
5330 4350 2184 3080 4862 3762
5330 4862 3080 2184 4350 3762
5365 4760 2240 2475 4875 4200
5365 4875 2475 2240 4760 4200
5525 2600 1320 4875 5365 2240
5525 2805 1320 4760 5365 2475
5525 3468 2880 4301 4715 1932
5525 4715 4301 2880 3468 1932
5525 5365 4760 1320 2805 2475
5525 5365 4875 1320 2600 2240
5550 3198 3120 4536 4590 702
5550 4590 4536 3120 3198 702
5576 1480 1224 5376 5440 832
5576 5440 5376 1224 1480 832
6005 3796 3404 4653 4947 1680
6005 4947 4653 3404 3796 1680


Table proudly produced by the following Maple prog. (Note that this
does not make any attempt to use Pythagorean squares as a basis):

A := proc(z)
end:
for z from 3 do
od:





Following recent discussion of the number of possible plays in chess,
consider the number of possible plays on the nth move in Mirror Chess in
which Black's play is always the mirror image of White (White must either
mate or play such that Black can mirror the move).

I find 20, 433, ...

But this would require more terms by computer analysis to be of interest to
the OEIS, I think.

Perhaps sequences derived from other variants of chess such as those played
on a 10x10 board might be interesting, although of course game rules are an
arbitrary invention.

Jeremy Gardiner