## Pythagorean triangle – perimeter and square of any side

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

math grad - Interest: Number theory
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### 5 Responses to Pythagorean triangle – perimeter and square of any side

1. paul says:

Here is a list of PT, who’s small leg is <=10000.

It would seem the solutions only happen using the small leg as the square.

{16,63,65}… P = 144 + 16^2 = 20^2 = 400
{64,510,514}… P = 1088 + 64^2 = 72^2 = 5184
{144,1725,1731}… P = 3600 + 144^2 = 156^2 = 24336
{256,4092,4100}… P = 8448 + 256^2 = 272^2 = 73984
{400,7995,8005}… P = 16400 + 400^2 = 420^2 = 176400
{576,13818,13830}… P = 28224 + 576^2 = 600^2 = 360000
{784,21945,21959}… P = 44688 + 784^2 = 812^2 = 659344
{1024,32760,32776}… P = 66560 + 1024^2 = 1056^2 = 1115136
{1296,46647,46665}… P = 94608 + 1296^2 = 1332^2 = 1774224
{1600,63990,64010}… P = 129600 + 1600^2 = 1640^2 = 2689600
{1936,85173,85195}… P = 172304 + 1936^2 = 1980^2 = 3920400
{2304,110580,110604}… P = 223488 + 2304^2 = 2352^2 = 5531904
{2704,140595,140621}… P = 283920 + 2704^2 = 2756^2 = 7595536
{3136,175602,175630}… P = 354368 + 3136^2 = 3192^2 = 10188864
{3600,215985,216015}… P = 435600 + 3600^2 = 3660^2 = 13395600
{4096,262128,262160}… P = 528384 + 4096^2 = 4160^2 = 17305600
{4624,314415,314449}… P = 633488 + 4624^2 = 4692^2 = 22014864
{5184,373230,373266}… P = 751680 + 5184^2 = 5256^2 = 27625536
{5776,438957,438995}… P = 883728 + 5776^2 = 5852^2 = 34245904
{6400,511980,512020}… P = 1030400 + 6400^2 = 6480^2 = 41990400
{7056,592683,592725}… P = 1192464 + 7056^2 = 7140^2 = 50979600
{7744,681450,681494}… P = 1370688 + 7744^2 = 7832^2 = 61340224
{8464,778665,778711}… P = 1565840 + 8464^2 = 8556^2 = 73205136
{9216,884712,884760}… P = 1778688 + 9216^2 = 9312^2 = 86713344
{10000,999975,1000025}… P = 2010000 + 10000^2 = 10100^2 = 102010000

Paul.

• benvitalis says:

At first, I thought I could find solutions using the longest leg, too.
I made some errors in my equations. I still wonder whether it would be possible?

• paul says:

I checked all PT’s with a short leg <=2,000,000 with no solutions other leg than the short leg.
P.

2. paul says:

All this can be generated by the following:-

Format is{a, b, c, Perimeter, root of sum of P + square side, sum of p + square side} for all n>0

{(4n)^2, 64n^3 – n, 64n^3 + n, 16 n^2 (1 + 8 n), 4 n (1 + 4 n), 16 n^2 (1 + 4 n)^2}

this prints out the following

{16,63,65,144,20,400}
{64,510,514,1088,72,5184}
{144,1725,1731,3600,156,24336}
{256,4092,4100,8448,272,73984}
{400,7995,8005,16400,420,176400}
{576,13818,13830,28224,600,360000}
{784,21945,21959,44688,812,659344}
{1024,32760,32776,66560,1056,1115136}
{1296,46647,46665,94608,1332,1774224}
{1600,63990,64010,129600,1640,2689600}

Paul.

3. paul says:

Using these generators for all n>0 produces PT’s whos perimeter is a square + the square of the small side also a square.

{4 n^2 (1+n)^2, 1/2 n (1+2 n)^2 (-1+3 n+8 n^2+4 n^3), 1/2 n (1+n+16 n^2+48 n^3+48 n^4+16 n^5), 4 n^2 (1+3 n+2 n^2)^2, 4 n^2 (1+3 n+4 n^2+2 n^3)^2}

The first 10 are

{16,63,65}… P = 12^2 = 144 + 16^2 = 20^2 = 400
{144,1725,1731}… P = 60^2 = 3600 + 144^2 = 156^2 = 24336
{576,13818,13830}… P = 168^2 = 28224 + 576^2 = 600^2 = 360000
{1600,63990,64010}… P = 360^2 = 129600 + 1600^2 = 1640^2 = 2689600
{3600,215985,216015}… P = 660^2 = 435600 + 3600^2 = 3660^2 = 13395600
{7056,592683,592725}… P = 1092^2 = 1192464 + 7056^2 = 7140^2 = 50979600
{12544,1404900,1404956}… P = 1680^2 = 2822400 + 12544^2 = 12656^2 = 160174336
{20736,2985948,2986020}… P = 2448^2 = 5992704 + 20736^2 = 20880^2 = 435974400
{32400,5831955,5832045}… P = 3420^2 = 11696400 + 32400^2 = 32580^2 = 1061456400
{48400,10647945,10648055}… P = 4620^2 = 21344400 + 48400^2 = 48620^2 = 2363904400

Paul.