Text 74, 111 rader
Skriven 2004-08-08 18:52:14 av Jasen Betts (3:640/1042)
     Kommentar till text 61 av DAVID WILLIAMS (1:250/514)
Ärende: blunt triangles
=======================
Hello DAVID.

06 Aug 04 21:29, you wrote to me:

 DW> -> here's an impirical arithmetic approach, I have no idea why it
 DW> works.


 DW> A recipe for "square triangles"! Neat. Did you find it someplace, or
 DW> did you work it out yourself?

It's all my own work.

I attacked it by looking for a pattern in the differences between succesive C
terms after I found a formula for C I attacked B , (and A is always B-1)

 DW> Of course, n can't go above 12 because at 13 the numbers overflow
 DW> "long integers". Maybe I'll try it with double-precision floating
 DW> point. That should give a few more digits, if there are no errors.

yes, that'll get you a few more digits.... hmm it could be done using strings
if you want really big numbers, addition subtraction and multiplication by a
constant are fairly easy to do


In looking at the indivisible triangles with B less than 10000 I've noticed
that b-a seems to be restriccted in its factors, in that 2,3 and 5 are
forbidden 7 is allowed and 11 and 13 forbidden...

I don't know why this is. it may have some similarity to the rule for the
factors of C

Given that for bluntnesss the next best b-a (after 1) is 7 but the next greater
triangle (with b-a=1) is only a factor of 6 away from the last.
he triangles with b-a=1 will always be the bluntest  (closest to 45 degrees)


I also notice that my formula for b=a+1 triangles seems to work for b=a+n if
the correct starting point is used. but for b-a>1 there seem to be two
interleaved sequences


   table for A<B<10000, A^2+B^2=C^2,  A,B,C in _N_

      B      A       C       C-B     C-A     B-A

       4       3       5       1       2       1
      21      20      29       8       9       1
     120     119     169      49      50       1
     697     696     985     288     289       1
    4060    4059    5741    1681    1682       1
      12       5      13       1       8       7
      15       8      17       2       9       7
      55      48      73      18      25       7
      72      65      97      25      32       7
     304     297     425     121     128       7
     403     396     565     162     169       7
    1755    1748    2477     722     729       7
    2332    2325    3293     961     968       7
      24       7      25       1      18      17
      45      28      53       8      25      17
     105      88     137      32      49      17
     224     207     305      81      98      17
     572     555     797     225     242      17
    1265    1248    1777     512     529      17
    3293    3276    4645    1352    1369      17
    7332    7315   10357    3025    3042      17
      35      12      37       2      25      23
      56      33      65       9      32      23
     156     133     205      49      72      23
     275     252     373      98     121      23
     855     832    1193     338     361      23
    1548    1525    2173     625     648      23
    4928    4905    6953    2025    2048      23
    8967    8944   12665    3698    3721      23
      40       9      41       1      32      31
      91      60     109      18      49      31
     171     140     221      50      81      31
     460     429     629     169     200      31
     924     893    1285     361     392      31
    2607    2576    3665    1058    1089      31
    5311    5280    7489    2178    2209      31
      77      36      85       8      49      41
      80      39      89       9      50      41
     360     319     481     121     162      41
     377     336     505     128     169      41
    2001    1960    2801     800     841      41
    2100    2059    2941     841     882      41
      63      16      65       2      49      47
     132      85     157      25      72      47
     272     225     353      81     128      47
     663     616     905     242     289      47
    1475    1428    2053     578     625      47
    3752    3705    5273    1521    1568      47
    8484    8437   11965    3481    3528      47
      60      11      61       1      50      49
     153     104     185      32      81      49
     253     204     325      72     121      49
     780     731    1069     289     338      49
    1360    1311    1889     529     578      49
    4429    4380    6229    1800    1849      49
    7809    7760   11009    3200    3249      49
 etc....

=== Cut ===


Jasen

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