Pythagorean triangle-perimeter + square of any side is a square

 
 

Find a Pythagorean triangle the sum whose perimeter and square of any side is a square

 
 

Let the sides be   a \,x, \; b \,x, \; c \,x,   where   a^2 + b^2 = c^2

Then

x^2 \; + \; p \,x
x^2 \; + \; q \,x
x^2 \; + \; r \,x

are made squares

where    p = s/a^2,    q = s/b^2,    r = s/c^2,
                 s = a + b + c

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

2 Responses to Pythagorean triangle-perimeter + square of any side is a square

  1. Paul says:

    Here are a few, all of them are with a^2.

    {16,63,65} = 144 + 16^2 = 20^2
    {64,510,514} = 1088 + 64^2 = 72^2
    {144,1725,1731} = 3600 + 144^2 = 156^2
    {256,4092,4100} = 8448 + 256^2 = 272^2
    {400,7995,8005} = 16400 + 400^2 = 420^2
    {576,13818,13830} = 28224 + 576^2 = 600^2
    {784,21945,21959} = 44688 + 784^2 = 812^2
    {1024,32760,32776} = 66560 + 1024^2 = 1056^2
    {1296,46647,46665} = 94608 + 1296^2 = 1332^2
    {1936,85173,85195} = 172304 + 1936^2 = 1980^2
    {2304,110580,110604} = 223488 + 2304^2 = 2352^2
    {2704,140595,140621} = 283920 + 2704^2 = 2756^2
    {3136,175602,175630} = 354368 + 3136^2 = 3192^2
    {3600,215985,216015} = 435600 + 3600^2 = 3660^2
    {4096,262128,262160} = 528384 + 4096^2 = 4160^2
    {4624,314415,314449} = 633488 + 4624^2 = 4692^2
    {5184,373230,373266} = 751680 + 5184^2 = 5256^2
    {5776,438957,438995} = 883728 + 5776^2 = 5852^2
    {6400,511980,512020} = 1030400 + 6400^2 = 6480^2
    {7056,592683,592725} = 1192464 + 7056^2 = 7140^2
    {7744,681450,681494} = 1370688 + 7744^2 = 7832^2
    {8464,778665,778711} = 1565840 + 8464^2 = 8556^2
    {9216,884712,884760} = 1778688 + 9216^2 = 9312^2
    {10000,999975,1000025} = 2010000 + 10000^2 = 10100^2

    Paul.

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